The World Tree : Slavic

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Slavic mythology

Slavic mythology

Unlike Greek or Egyptian mythology, there are no first-hand records for the study of Slavic mythology. Despite some controversial theories (for instance, the Book of Veles), it cannot be proven that the Slavs had any sort of writing system prior to Christianisation; therefore, all their original religious beliefs and traditions were likely passed down orally over generations, and potentially forgotten over the centuries following the arrival of Christianity. Prior to that, sparse records of Slavic religion were mostly written by non-Slavic Christian missionaries who were uninterested in accurately portraying pagan beliefs. Archaeological remains of old Slavic Cult images and shrines have been found, though little can be yielded from them without proper knowledge of their contexts, other than confirming existing historical records. Fragments of old mythological beliefs and pagan festivals survive up to this day in folk customs, songs, and stories of all the Slavic nations.

A modern artistic representation of Saxo Grammaticus.

Written sources

There are no known written accounts of Slavic mythology predating the fragmentation of the Proto-Slavic people into WesternEastern, and Southern Slavs, with the possible exception of a short note in Herodotus’ Histories, mentioning a tribe of Neuri in the far north, whose men, Herodotus claims, transform themselves into wolves for several days each year. Some researchers have interpreted this through the Slavic folk belief in werewolves, whilst others believe that Herodotus actually referred to ancient Slavic carnival festivals, when groups of young men roamed the villages in masks, sometimes referred to as vucari (wolf-humans)a. The identification of “Neuri” with Proto-Slavs remains controversial, however.

The first definitive reference to the Slavs and their mythology in written history was made by the 6th century Byzantine historian Procopius, whose Bellum Gothicum described the beliefs of a South Slavic tribe that crossed the Danube heading south in just two days. According to Procopius, these Slavs worshipped a single deity, who crafted lightning and thunder. Though not named explicitly, it can be deduced this is a reference to the deity known as Perun in later historic sources, as in many Slavic languages today (Polish ‘piorun’ for example) Perunsimply means “thunder” or “lightning bolt”. He also mentions the belief in various demons and nymphs (i.e. vilas), but does not mention any other names.

The Slavic Primary Chronicle is a major work with many valuable references to the pagan beliefs of Eastern Slavs. The chronicle treats the history of the early Eastern Slavic state. Even though the manuscript was compiled at the beginning of the 12th century, it contains references to, and copies of, older documents, and describes events predating the Baptism of Kiev. Two deities, Perun and Veles/Volos, are mentioned in the text of the early 10th century peace treaties between pagan rulers of East Slavs and Byzantine Emperors. Later, Nestor the Chronicler describes a state pantheon introduced by Prince Vladimir in Kiev in 980 CE. Vladimir’s pantheon included PerunHorsDažbogStribogSimargl, and Mokosh. The Hypatian Codex of the Primary Chronicle also mentions Svarog, compared to Greek Hephaestus. Also very interesting are the passages in the East Slavic epic The Tale of Igor’s Campaignreferring to Veles, Dažbog, and Hors. The original epic has been dated to the end of the 12th century, although there are marginal disputes over the authenticity of this work.

The most numerous and richest written records are of West Slavic paganism, particularly of Wendish and Polabian tribes, who were forcibly Christianised only at the end of the 12th century. The German missionaries and priests who assailed pagan religion left extensive records of old mythological systems they sought to overcome. However, they hardly restrained themselves from “pious lies”, claiming pagan Slavs were idolatrous, blood-thirsty barbarians. As none of those missionaries learned any Slavic language, their records are confused and exaggerated.

Major works include a chronicle of Thietmar of Merseburg from the beginning of the 11th century, who described a temple in the city of Riedegost (Radagast) where the great deityZuarasic (Svarožič) was worshipped. According to Thietmar, this was the most sacred place in the land of pagan Slavs, and Svarožič was their most important deity.

See Radegast (god).

Another very valuable document is the Chronica Slavorum written in the late 12th century by Helmold, a German priest. He mentions ‘the devil’ Zerneboh (Chernobog), goddess Živa, godPorenut, some unnamed gods whose statues had multiple heads and, finally, the great god Svantevit, worshipped on the island of Rügen who, according to Helmod, was the most important of all (Western) Slavic deities.

The third, and arguably the most important record, comes from the Danish chronicler Saxo Grammaticus, who in his Gesta Danorumdescribed the war fought in 1168 by the Danish king Valdemar I against the Wends of Rügen, the conquest of their city at cape Arkona and the destruction of the grand temple of Svantevit that stood there. Saxo meticulously described the worship of Svantevit, the customs associated with it and, the tall four-headed statue of the god. He also mentioned multi-headed deities of other Slavic tribes; Rugievit,Porewit and Porentius.

The fourth major source are three biographies of the German warrior-bishop St Otto, who in the early 12th century led several military-pastoral expeditions into the regions of Slavic tribes living near the Baltic Sea. According to the manuscript, the most important Slavic deity was Triglav, whose temples in the city of Szczecin were respected oracles. In the cities of Wolgast and Havelberg, the war god Gerovit was worshiped, a likely corruption of Jarovit, a Slavic deity possibly identical to Jarilo of the East Slavic folklore.

Another source authenticity of which is being disputed particularly by the Russian Orthodox Church is the Book of Veles.

Archaeological remains

Statues of several Slavic deities were discovered in 1848, on the banks of theZbruch river, a tall stone statue was found, with four faces under a single stone hat. Because of its likehood with Saxo’s description of the great image in the temple of Rügen, the statue was immediately proclaimed a representation of Svantevit, although it was clear it could not be the original Svantevit of Rügen. Several other multi-headed statues were discovered elsewhere. A tiny four-headed statue from the 10th century, carved out of bone, was unearthed amongst the ruins of Preslav, a capital of medieval Bulgarian tsars. A two-headed, human-sized wooden statue was discovered on an island in the Tollensesee lake near Neubrandenburg: in the Middle Ages, this was the land of Slavic Dolenain tribe, whose name survives in the name of the lake. Furthermore, a three-headed statue was discovered inDalmatia (Croatia) on the hill bearing the name of Suvid, not far from the peak of Mt. Dinara called Troglav.

The remains of several Slavic shrines have also been discovered. Some archeological excavations on the cape of Arkona on Rügen island have uncovered vestiges of a great temple and a city, identified with those described by Saxo. In Novgorod, at the ancient Peryn skete, archeologists discovered the remains of a pagan shrine likely dedicated to Perun. The shrine consisted of a wide circular platform centred around a statue. The platform was encircled by a trench with eight apses, which contain remains of sacrificial altars. Remains of a citadel with a more or less identical layout were discovered on a location with the suggestive name Pohansko (Paganic), near Břeclav in the Czech Republic.

All these archeological remains have the multiplicity of aspects in common. Statues of gods with multiple faces and remains of shrines with multiple sacrificial altars confirm written reports of Christian missionaries about the Slavs worshipping polycephalic gods, and also indicate that ancient Slavic mythology apparently put great emphasis on worship of deities with more aspects than one.

Also quite important are remains of several pieces of pottery from 4th century Chernyakhov culture. Russian archeologist Boris Rybakov identified and interpreted symbols inscribed onto them as records of the ancient Slavic calendar.

It is claimed usually that worshiping in woods was more common to Slavic people than praying in shrines. Those woods were called in PSlav. *gaje (conf. Polish Nom. sg. m. gaj ‘small wood, thicket, bush, grove’; see: sacred grove), they were sometimes encircled by a fence which created a sacred area, both a natural and social sphere. Sometimes they would be cemeteries as well (conf. Kleczanów Wood).

[edit]Folklore traces

As various Slavic populations were Christianised between the 7th and 12th centuries, Christianity was introduced as a religion of the elite, flourishing mostly in cities and amongst the nobility. Amongst the rural majority of the medieval Slavic population, old myths remained strong. Christian priests and monks in Slavic countries, particularly in Russia, for centuries fought against the phenomenon called dvoeverie (double faith). On the one hand, peasants and farmers eagerly accepted baptism, masses and the new Christian holidays. On the other hand, they still persisted performing ancient rites and worshiping old pagan cults, even when the ancient deities and myths on which those were based were completely forgotten.

This was because, from a perspective of the Slavic peasant, Christianity was not a replacement of old Slavic mythology, but rather an addition to it. Christianity may have offered a hope of salvation, and of blissful afterlife in the next world, but for survival in this world, for yearly harvest and protection of cattle, the old religious system with its fertility rites, its protective deities, and its household spirits was taken to be necessary. This was a problem the Christian church never really solved; at best, it could offer a Christian saint or martyr to replace the pagan deity of a certain cult, but the cult itself thrived, as did the mythological view of the world through which natural phenomena were explained.

While folk beliefs and traditions of all Slavic peoples indeed are the richest resource for reconstructing the ancient pagan beliefs, these may very likely have lost their original mythology and sanctity. People entertained a vague idea that some festivals must be celebrated in a certain way, some stories must be told or some songs must be sung, merely in accordance with tradition. Cults of old deities were mixed with worship of new Christian saints, and old rituals blended among new Christian holidays.

This led scholars to analyse the structure of folklore itself, and to devise methodologies through which they could reconstruct the lost mythology from this structure. We can roughly divide the folklore accounts into two groups:

  • Fairy tales about various fantastical characters and creatures such as AlkonostBaba YagaKoschei the Deathless, FirebirdZmey songs and tales of legendary heroes such as Russian bogatyrs, and superstitions about various demons and spirits such as domovoilikhovilas,vampiresvodyanoyrusalkas etc. Many of these tales and beliefs may be quite ancient, and probably contain at least some elements of old mythical structure, but they are not myths themselves. They lack a deeper, sacral meaning and religious significance, and furthermore they tend to vary greatly among various Slavic populations.
  • Folk celebrations of various Christian festivals and popular beliefs in various saints. It is, for instance, quite clear that a popular saint in many Slavic countries, St Elijah the Thunderer, is a replacement of old thunder-god Perun. Likewise, traces of ancient deities can also be found in cults of many other saints, such as St MarySt VitusSt GeorgeSt BlaiseSt Nicholas, and it is also obvious that various folk celebrations, such as the spring feast of Jare or Jurjevo and the summer feast of Ivanje or Ivan Kupala, both very loosely associated with Christian holidays, are abundant with pre-Christian elements. These beliefs have considerable religious and sacral significance to the people still performing them. The problem is, of course, that the elements of pre-Christian religion are hopelessly mixed into popular Christianity.

Reconstruction of original Slavic myths is thus a true detective work, requiring a considerable knowledge of various scientific disciplines such assemioticslinguisticsphilologycomparative mythology and ethnology. Folklore accounts must be analysed on level of structure, not merely as songs or stories, but as groups of signs and symbols which contain some internal structural logic. Each of these signs is composed of some key words, which are more than simply names of characters, places or artifacts. One important aspect of symbols is that they are almost impossible to change; while their names may be altered, their structure may not. Changing or losing of key words would result in a change of symbol, which would then invalidate the internal structural logic of a text and render the entire tale meaningless. It would then soon be forgotten, because the pattern, or logic, through which it was transmitted over generations would be lost.

For example: as stated already, the Slavic god of thunder, Perun, was mostly equated with St Elijah the Thunderer in Christian folklore. But he was also sometimes equated with St Michael, and sometimes even with the Christian God, whilst in some of Russian or Belarusian folk stories, he was downgraded to various fairy characters such as Tsar Ogin (Tsar Flame) or Grom (Thunder). Notwithstanding changes in the name itself, there are always some key words present which were used to describe Perun as a symbol in ancient mythical texts, and have survived through folklore. Perun is always gore (up, above, high, on the top of the mountain or in heaven; Perun is a heavenly god, and he is also the ‘highest’ deity of old Slavic pantheon), he is suh (dry, as opposite of wet; he is god of thunder and lightning, which causes fire), he treska/razbija/goni/ubija(strikes/sunders/pursues/kills; he is a god of thunder and storms, destructive and furious) with strela/kamen/molnija (arrow/stone/lightning; Perun’s weapons, are of course, his bolts of lightning. He fires them as arrows which are so powerful they explode and blow up stones when they hit). These key words are always preserved in folklore traces, even if the true name of Perun has been long ago forgotten. Consequently, the structure of this symbol allowed the identification of Perun with similar characters either from Christian religion or from later folklore, which share these similarities in structure of their own symbols.

Following similar methodology, and drawing parallels with structure of other, related Indo-European mythologies (particularly Baltic mythology), and occasionally using some hints found in historical records of Slavic paganism, some of the ancient myths could be reconstructed. Significant progress in the study of Slavic mythology was made during last 30 years, mostly through the work of the Russian philologists Vladimir Toporov and Vyacheslav Vsevolodovich Ivanov, as well as that of the Croatian scientists Radoslav Katičić and Vitomir Belaj. Also very valuable are the studies of Russian scholar Boris Uspensky and of Serbian philologist and ethnologist Veselin Čajkanović.

However, uncritical interpretation of folklore or unskilled reconstruction of myths can lead to disastrous effects, as we shall see.

[edit]Inauthentic sources

When dealing with Slavic mythology, one cannot be too careful or too critical about the validity and authenticity of sources. Scholarly interest in beliefs of ancient Slavs has been continually waxing since the times of Renaissance, and with it the overall number of confusions, errors, misinterpretations, and unsupported reconstructions (not to mention inventions) has also increased.

No valid scientific methodology by which folklore accounts could be interpreted was known before the mid-20th century, and with sparse historical and archeological sources, the doors were thus opened to wild and unwarranted speculation. One of the best examples of overall confusion and complete misinterpretation is a fake deity of love, Lada or Lado, constructed from meaningless exclamations in Slavic wedding songs. Gods such as Koleda and Kupala were constructed from misinterpreted names of popular Slavic folk festivals; Koledo was the Slavic name for Christmas processions of carol singers, whilst the source of the name Kupala is unknown. Christian sources claim that it comes from Ivan Kupala (literally: John the Baptist) however this claim is as baseless as the claim of those who choose to interpret it as a pagan holiday. This festival day is celebrated at the summer solstice in many Slavic, and also western European countries, such as France and Italy. These customs indeed do have more than a few elements of pre-Christian beliefs, but simply inventing gods based on names of customs is not considered a valid method for reconstruction of lost beliefs.

Misinterpretation of Thiethmar’s historic description of Wendish paganism led to confusion between a god, Svarožič, and a city in which his temple stood, Radegast. Since the name Radegast can be easily etymologised as meaning “Dear guest”, this led to the construction of Radegast as the supposed Slavic god of hospitality. Likewise, to pair up with a deity with the sinister sounding name of Chernobog (Black god) mentioned by Helmod, the White God, or Belobog, was invented. That name is not found in any reliable historic or ethnographic record; rather, it was simply assumed that, since there already was a Black God, there simply had to be a White God as well. Again, this is clearly not a scientific approach to the study of Slavic mythology, but pages and pages have been written about the supposed Belobog-Chernobog dualism so far, and many books and scholarly references even today take for granted that such gods were truly worshipped by ancient Slavs.

Even more questionable than confusions or misinterpretations are deliberate forgeries. In the nineteenth and twentieth century, the general population became increasingly interested in Slavic mythology, fuelled by various romanticnationalistic, and, in modern times, neopaganmovements. Forging evidence of ancient mythology, for a time, became almost a sort of hobby among various social groups, often with the aim to promote their own topical agendas. For instance, statues of ancient Slavic gods were “discovered”, inscribed with Germanic runes, or folk songs and stories were “recorded” in which half of the Slavic pantheon is described as picking flowers or merrily dancing around a bonfire.

The 19th century Veda Slovena is a heavy mystification of Bulgarian folk songs, with many alleged references to Slavic mythology, which most scholars consider a forgery. A more recent example is a controversial Book of Veles, which claims to be an authentic written record of old Slavic religion from the 9th or 10th century CE, written in the Cyrillic alphabet, whereas it cannot be proven that the Slavs had any sort of writing system prior to Christianisation, let alone that they used Cyrillic alphabet (named, of course, after St Cyril, who coined the first known writing system for Slavs when he was sent together with his brother Methodius to baptise them in 9th century). Some of the Slavic neopagansuse the Book of Veles as their sacred text, and consequently, insist that the document is authentic. However, the original book, supposedly written on birch barks, was lost (if indeed it ever existed), and thus its authenticity cannot be established at present.

[edit]Calendar and festivals

Slavic myths were cyclical, repeating every year over a series of festivities that followed changes of nature and seasons. Thus, to understand their mythology, it is important to understand their concept of calendar. On the basis of archeological and folklore remains, it is possible to reconstruct some elements of pre-Christian calendar, particularly major feastivals.

  • The year was apparently lunar, and began in early March, similar to other Indo-European cultures whose old calendar systems are better known to us. The names for the last night of old year and the first day of new year are reconstructed as Velja Noc(*Velja Notj)/Velik Dan(Velikŭ dĭnĭ) (Great Night/Great Day). After Christianization, these names were probably passed onto Easter. In Slavic countries belonging to Orthodox Churches, Easter is known as Velik Dan/Great Day, whilst amongst Catholic Slavs, it is known as Velika Noc/Great Night. The names blend nicely with the translation of the Greek Megale Evthomada, Great Week, the Christian term for the week in which Easter falls. In pagan times, however, this was a holiday probably quite like Halloween. Certain people (shamans[citation needed]) donned grotesque masks and coats of sheep wool, roaming around the villages, as during the Great Night, it was believed, spirits of dead ancestors travelled across the land, entering villages and houses to celebrate the new year with their living relatives. Consequently, the deity of the last day of the year was probably Veles, god of Underworld.
  • There was a large spring festival dedicated to Jarilo, god of vegetation and fertility. Processions of young men or girls used to go round villages on this day, carrying green branches or flowers as symbols of new life. They would travel from home to home, reciting certain songs and bless each household with traditional fertility rites. The leader of procession, usually riding on horse, would be identified with Jarilo. The custom of creation of pisanki or decorated eggs, also symbols of new life, was another tradition associated with this feast, which was later passed on Christian Easter.
  • The summer solstice festival is known today variously as Pust, IvanjeKupala or Kries. It was celebrated pretty much as a huge wedding, and, according to some indications from historical sources, in pagan times likely followed by a general orgy. There was a lot of eating and drinking on the night before, large bonfires (in Slavic — Kres) were lit, and youngsters were coupling and dancing in circles, or jumped across fires. Young girls made wreaths from flowers and fern (which apparently was a sacred plant for this celebration), tossed them into rivers, and on the basis of how and where they floated, foretold each other how they would get married. Ritual bathing on this night was also very important; hence the name of Kupala (from kupati= to bathe), which probably fit nicely with folk translation of the future patron saint the Church installed for this festival, John the Baptist (Ivan Kupala Day). Overall, the whole festivity probably celebrated a divine wedding of a fertility god, associated with growth of plants for harvesting.
  • In the middle of summer, there was a festival associated with thunder-god Perun, in post-Christian times transformed into a very important festival of Saint Elijah. It was considered the holiest time of the year, and there are some indications from historic sources that it involved human sacrifices. The harvest probably began afterwards.
  • It is unclear when exactly the end of harvest was celebrated, but historic records mention interesting tradition associated with it that was celebrated at Svantevit temple on the island of Ruyana (present-day Rugen), a survived through later folklore. People would gather in front of the temple, where priests would place a huge wheat cake, almost the size of a human. The high priest would stand behind the cake and ask the masses if they saw him. Whatever their answer was, the priest would then plead that the next year, people could not see him behind the ritual cake, i.e. that the next year’s harvest would be even more bountiful.
  • There probably also was an important festival around winter solstice, which later became associated with Christmas. Consequently, in many Slavic countries, Christmas is calledBozhich, which simply means little god. While this name fits very nicely with the Christian idea of Christmas, the name is likely of pagan origin; it indicated the birth of a young and new god of the sun to the old and weakened solar deity during the longest night of the year. The old sun god was identified as Svarog, and his son, the young and new sun, asDažbog[citation needed]. An alternative (or perhaps the original) name for this festival was Korochun.

[edit]Cosmology

A fairly typical cosmological concept among speakers of Indo-European languages, that of the World Tree, is also present in Slavic mythology. It is either an oak tree, or some sort ofpine tree. The mythological symbol of the World Tree was a very strong one, and survived throughout the Slavic folklore for many centuries after Christianisation. Three levels of the universe were located on the tree. Its crown represented the sky, the realm of heavenly deities and celestial bodies, whilst the trunk was the realm of mortals. They were sometimes combined together in opposition to the roots of the tree, which represented the underworld, the realm of the dead.

The pattern of three realms situated vertically on the axis mundi of the World Tree parallels the horizontal, geographical organisation of the world. The world of gods and mortals was situated in the centre of the earth (considered to be flat, of course), encircled by a sea, across which lay the land of the dead, where birds would fly to every winter and return from in spring. In many folklore accounts, the concepts of going across the sea (idit) versus coming from across the sea (dolazit) are equated with dying versus returning to life. This echoes an ancient mythological concept that the afterlife is reached by crossing over a body of water. Additionally, on the horizontal axis, the world was also split; in this case by four cardinal points, representing the four wind directions (north, east, south, west). These two divisions of the world, into three realms on the vertical axis and into four points on the horizontal, were quite important in mythology; they can be interpreted in statues of Slavic gods, particularly those of the three-headed Triglav and the four-headed Svantevit.

[edit]Pantheon

As noted in the description of historical sources, a very wide range of deities was worshipped by Slavs, on a huge geographical area from the shores of the Baltic to the shores of theWhite Sea, in a time span of over 600 years. Historic sources also show that each Slavic tribe worshipped its own gods, and thus probably had its own pantheon. Overall, ancient Slavic religion seems to be fairly local and cultic in nature, with gods and beliefs varying from tribe to tribe. However, just as in the case of the various Slavic languages — it can be shown that they originate from a single, Proto-Slavic language — it is also possible to establish some sort of Proto-Slavic Olympus and, through careful study of folklore, reconstruct some elements of this original pantheon, from which the various gods of the various Slavic tribes originated.

[edit]Supreme god

There are various modern theories about a supreme Slavic deity being Rod or Svarog, and historic sources show that deities such as SvarožičSvantevit or Triglav were worshipped as supreme by certain tribes. But overall by far the best candidate for the position of supreme deity is Perun. His name is the most common in all historic records of Slavic religion; in fact, he is the first Slavic god mentioned in written history (Procopius in his short note mentions that the god of thunder and lightning is the only god of Slavs, lord of all). The Primary Chronicleidentifies him as chief god of Kievan Rus prior to Christianisation. A short note in Helmold‘s Chronica Slavorum states that West Slavs believe in a single deity in heaven who rules over all the other deities on earth; the name of this deity is not mentioned, but nevertheless it seems quite possible this was a reference to Perun. And even though we do not find the name of Perun in any of the extensive records of West Slavic religion, he was known by all branches of Slavs, as shown by a vast number of toponyms that still bear his name in all Slavic countries today. Finally, by analysing the folklore texts, one will notice that Perun is the only Slavic deity who was equated with the Christian God. These are very strong indications that Perun was indeed the supreme god of the original Proto-Slavic pantheon.

Perun, however, had a match. As Roman Jakobson pointed out, whenever Perun is mentioned in historic texts, he is always “accompanied” by another god, Veles. This relationship can be observed in toponyms as well. Wherever we find a hill or a mountain peak whose name can be associated with Perun, below it, in the lowlands, usually near a river, there will be a place with a name reminiscent of Veles. Consequently, as Perun was sometimes identified with the Сhristian God in folklore accounts, Veles was identified with the Devil.

Further information: List of Slavic deities

Perun and Veles

Ivanov and Toporov reconstructed the ancient myth involving the two major gods of the Proto-Slavic pantheon, Perun and Veles. The two of them stand in opposition in almost every way. Perun is a heavenly god of thunder and lightning, fiery and dry, who rules the living world from his citadel high above, located on the top of the highest branch of the World Tree. Veles is a chthonicgod associated with waters, earthly and wet, lord of the underworld, who rules the realm of the dead from down in the roots of the World Tree. Perun is a giver of rain to farmers, god of war and weapons, invoked by fighters. Veles is a god of cattle, protector of shepherds, associated with magic and commerce.

A cosmic battle fought between two of them echoes the ancient Indo-European myth of a fight between a storm god and adragon. Attacking with his lightning bolts from sky, Perun pursues his serpentine enemy Veles who slithers down over earth. Veles taunts Perun and flees, transforming himself into various animals, hiding behind trees, houses, or people. In the end, he is killed by Perun, or he flees into the water, into the underworld. This is basically the same thing; by killing Veles, Perun does not actually destroy him, but simply returns him to his place in the world of the dead. Thus the order of the world, disrupted by Veles’s mischief, is established once again by Perun. The idea that storms and thunder are actually a divine battle between the supreme god and his arch-enemy was extremely important to Slavs, and continued to thrive long after Perun and Veles were replaced by the Сhristian God and Devil. A lightning bolt striking down a tree or burning down a peasant’s house was always explained through the belief of a raging heavenly deity bashing down on his earthly, underworldly, enemy.

The enmity of the two gods was explained by Veles’ theft of Perun’s cattle, or by Perun’s theft of Veles’ cattle (since Veles was the god of cattle, the matter of ownership here is not clear). The motif of stealing divine cattle is also a common one in Indo-European mythology; the cattle in fact may be understood simply as a metaphor for heavenly water or rain. Thus, Veles steals rain water from Perun, or Perun steals water for rain from Veles (again, since Veles is associated with waters, and Perun with sky and clouds, it is unclear to whom rain should belong). An additional reason for this enmity may be wife-theft. From folklore accounts it seems that the Sun was sometimes considered to be Perun’s wife (an odd idea, as all Slavic sun-gods, like Hors and Dažbog, are male). However, since the Sun, in the mythic view of the world, dies every evening, as it descends beyond the horizon and into the underworld where it spends the night, this was understood by Slavs as Veles’ theft of Perun’s wife (but again, the rebirth of the Sun in the morning could also be understood as Perun’s theft of Veles’ wife).

Jarilo and Morana

Katicic and Belaj continued down the path laid by Ivanov and Toporov and reconstructed the myth revolving around the fertility and vegetation god, Jarilo, and his sister and wife, Morana, goddess of nature and death. Jarilo is associated with the Moon and Morana is considered a daughter of the Sun. Both of them are children of Perun, born on the night of the new year (Great Night). However, on the same night, Jarilo is snatched from the cradle and taken to the underworld, where Veles raises him as his own. At the Spring festival of Jare/Jurjevo, Jarilo returns from the world of the dead (from across the sea), bringing spring from the ever-green underworld into the realm of the living. He meets his sister Morana and courts her. At the beginning of summer, the festival later known as Ivanje/Ivan, Kupala celebrated their divine wedding. The sacred union between brother and sister, children of the supreme god, brings fertility and abundance to earth, ensuring a bountiful harvest. Also, since Jarilo is a (step)son of Veles, and his wife daughter of Perun, their marriage brings peace between two great gods; in other words, it ensures there will be no storms which could damage the harvest.

After the harvest, however, Jarilo is unfaitfhul to his wife, and she vengefully slays him (returns him into the underworld), renewing the enmity between Perun and Veles. Without her husband, god of fertility and vegetation, Morana — and all of nature with her — withers and freezes in the upcoming winter; she turns into a terrible, old, and dangerous goddess of darkness and frost, and eventually dies by the end of year. The whole myth would repeat itself anew each following year, and retelling of its key parts was accompanied by major yearly festivals of the Slavic calendar. The story also shows numerous parallels to similar myths of Baltic and Hittite mythology.

Svarog, Svarožič, Dažbog

The name of Svarog is found only in East Slavic manuscripts, where it is usually equated with the Greek smith god Hephaestus. However, the name is very ancient, indicating that Svarog was a deity of Proto-Slavic pantheon. The root svar means bright, clear, and the suffix -og denotes a place. Comparison with Vedic Svarga indicates that Svarog simply meant (daylight) sky. It is possible he was the original sky god of the pantheon, perhaps a Slavic version of Proto-Indo-European *Dyēus Ph2ter. Svarog can be also understood as meaning a shining, fiery place; a forge. This, and identification with Hephaestus from historic sources, indicates he was also a god of fire and blacksmithing. According to the interpretation by Ivanov and Toporov, Svarog had two sons: Svarožič, who represented fire on earth, and Dažbog, who represented fire in the sky and was associated with Sun. Svarog was believed to have forged the Sun and have given it to his son Dažbog to carry it across the sky.

In Russian manuscripts he is equated with Sun, and folklore remembers him as a benevolent deity of light and sky. Serbian folklore, however, presents a far darker picture of him; he is remembered as Dabog, a frightful and lame deity guarding the doors of the underworld, associated with mining and precious metals. Veselin Čajkanović pointed out that these two aspects fit quite nicely into a symbolism of Slavic solar deity; a benevolent side represents the Dažbog during day, when he carries the Sun across the sky. The malevolent and ugly Dabog carries the Sun through the underworld at night. This pattern can also be applied to Sun’s yearly cycle; a benevolent aspect is associated with young, summer Sun, and a malevolent one with old, winter Sun.

Svarožič was worshipped as a fire spirit by Russian peasants well after Christianisation. He was also known amongst Western Slavs, but there he was worshipped as a supreme deity in the holy city of Radegast. Svarožič is a simply diminutive of Svarog’s name, and thus it may simply be another aspect (a surname, so to speak) of Dažbog. There is also a point of view that Svarog was the ancestor of all other Slavic gods, and thus Svarožič could simply be an epithet of any other deity, so that Dažbog, Perun, Veles, and so on, were possibly all Svarožičs.

Svantevit and Triglav

It is somewhat ironic that for now we cannot clearly determine the position of these two gods in the Proto-Slavic pantheon, yet we have the most extensive historic accounts written about them. That they were important to all pagan Slavs is indicated by a significant number of toponyms whose names can be associated with them and by discoveries of multi-headed statues in various Slavic lands. Both of these gods were considered supreme in various locations; they were associated with divination and symbolized by the horse. A possibly significant difference is that Svantevit had a white horse whilst Triglav a black one, and Svantevit was represented with four heads whilst Triglav (whose name simply means three-headed) with three. Svantevit was also associated with victory in war, harvest, and commerce.

Various hypotheses about them were proposed: that they are in fact one and the same deity, being somewhat similar; that they are not gods at all but compounds of three or four gods, a kind of mini-pantheons. Slavic neopagans tend to think of Triglav in particular as a concept ofTrinity. Svantevit has also been proclaimed as a late West Slavic alternation of Perun or Jarilo, or compared with Svarožič and deemed a solar deity. None of these hypotheses is quite satisfactory, and mostly they are just wild speculation, another attempt to reconstruct Slavic mythology as it should be, rather than discovering what it was really like. Further research is necessary before more can be said of these deities.

It is claimed that Slovenia’s highest mountain, Triglav, is named after the god Triglav.

[edit]Zorica and Danica

These names mean simply Dawn and Daystar, but in folklore accounts of all Slavic nations, they are often described as persons, or associated with persons, in pretty much the same way as Sun and Moon. Danica is often called Sun’s younger sister or daughter, and was probably associated with Morana. Consequently, Zorica was either Sun’s mother or older sister. It is quite possible this was a Slavic relic of the Proto-Indo-European dawn god.

Further developments

Ivanov and Toporov also schematically periodised various stages of development of Slavic mythology, attempting to show how it evolved from the original pantheon:

  • The first subsequent development occurred after the Proto-Slavs had split into East, West, and South Slavs. Each branch of the Slavic family devised various feminine deities of household (e.g. Mokosh), and deities associated with crafts, agriculture, and fertility (e.g. Rod and Chur). Deities such as Hors and Simargl are sometimes interpreted as the East Slavic borrowings from their Iranian neighbours.
  • At the level of abstract personification of divine functions, we have such concepts as Pravda/Krivda (Right/Wrong), Dobra Kob/Zla Kob(Good Fortune/Evil Fortune). These concepts, found in many Slavic fairy tales, are presumed to have originated at a time when old myths were already being downgraded to the level of legends and stories. Loius Leger pointed out that various Slavic words describing success, destiny, or fortune are all connected with the ancient Slavic word for God — “bog”. Although used to denote the God of Christianity, the word is of pagan origin and quite ancient. It originates from the Proto-Indo-European root *bhag (meaning fortune), being cognate to Avestic baga and Sanskrit bhagah (epithets of deities).
  • The next level of development is a mythologisation of historical traditions. Beginning in pagan times, it continued well after the advent of Сhristianity. It is characterised by tales and songs of legendary heroes, ranging from purely legendary founders of certain tribes, such as the stories about Lech, Czech, and Rus, to quite historical persons such as the 15th century Croatian-Hungarian kingMatthias Corvinus or the Serbian Prince Marko, who were both immortalised in folk legend or poetry. Russian bylinas about bogatyrs, Polish legends of Krak the Dragonslayer, Czech legends about Libuše, and the foundation of Prague all fall into this category. Various elements of these tales will still reveal elements of old myths (such as a hero slaying a dragon, a faint echo of an ancient concept of a cosmic battle between Perun the Thunderer and the serpentine Veles).
  • On an even lower level, certain mythical archetypes evolved into fairy-tale characters. These include Baba YagaKoschei the Immortal,Nightingale the RobberVodyanoyZmey Gorynych, and so on. At this point of development, one can hardly speak of mythology anymore. Rather, these are legends and stories which contain some fragments of old myths, but their structure and meaning are not so clear.
  • The lowest level of development of Slavic mythology includes various groups of home or nature spirits and magical creatures, which vary greatly amongst different Slavic nations. Mythic structure on this level is practically incomprehensible, but some of the beliefs nevertheless have a great antiquity. As early as the 5th century, Procopius mentioned that Slavs worshipped river and nature spirits, and traces of such beliefs can still be recognised in the tales about vilasvampireswitches, and werewolves.
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The Világfa (Életfa) Tree : Hungarian

FUTHER READING: http://en.wikipedia.org/wiki/Hungarian_mythology

The Religion in Brief

The world is divided into three spheres: the first is the Upper World (Felső világ), the home of the gods; the second is the Middle World (Középső világ) where is the world we know, and finally the underworld (Alsó világ). In the center of the world, a tall tree is standing: the World Tree / Tree of Life / Life Tree (Világfa/Életfa). Its foliage is the Upper World. The Middle World is located at its trunk and the underworld is around its roots. In some stories, the tree has fruits: these are the golden apples.

[edit]Upper World

The gods and the good souls live in the Upper World. Gods have the same rank, although the most important figure of them is Isten (meaning ‘God’ in Hungarian). He controls the world, shapes the fate of humans, observes the Middle World from the sky, and sometimes gives warning bylightning (mennykő). Isten created the world with the help of Ördög (“the devil” Evil). Other gods include: Istenanya (‘Mother God’), also known as Boldogasszony (‘Blessed Lady’; later identified with the Virgin Mary), and Hadúr (War Lord or Army Lord).

The major celestial bodies, (the Sun and the Moon), are also located in the Upper World. The sky was thought to be a big tent held up by the Tree of Life. There are several holes on it: those are the stars.

[edit]Middle World

The Middle World is shared among humans and many mythological creatures, the latter are often supernatural. There are ghosts of the forests and waters, who are ordered to scare humans. They have different names in different places. There are females, for example, the sellő (mermaid), which lives in waters and has a human torso with the tail of a fish. The wind is controlled by an old lady called Szélanya (Wind Mother) or Szélkirály (Wind King). The Sárkány (dragon) is a frightening beast: he is the enemy of many heroes in fairy tales, symbolising the psychical inner struggle of the hero. The lidérc is a ghostly, mysterious creature with several different appearances, its works are always malicious. The manók (elves / goblins) and the törpék (dwarfs) are foxy beings living in woods or under the ground. Óriások (giants) live in the mountains. They have both good and bad qualities. The most favourite creatures are thetündérek (fairies), who are beautiful and young virgins or female creatures. They aid humans, who sometimes can ask three wishes from them. Their opposites are the bábák, who are equated with catty, old witches. (Bába means ‘midwife’ in Hungarian, and originally they were wise old women, later equated with witches as Christianity became widespread.)

[edit]Underworld

The Underworld is the place of bad souls (this includes evil spirits and the souls of dead people who were cruel and evil in their lives) and the home of Ördög. He is the creator of everything that is bad for humans: for example, the creator of the annoying animals (such as fleaslice, and flies).

One of the theory of the ancient Hungarian religion is that it was a form of Tengriism, a shamanistic religion common among the early Turkic, Uralic and Mongol people, that was influenced by Zoroastrianism from the Persians whom the Magyars had encountered during their westward migration.

TALTOS

The shaman role was filled by the táltos. Their souls were thought to be able to travel between the three spheres (révülés). Táltos’ were also doctors. They were selected by fate; their slight abnormalities at birth (neonatal teeth, caulbearer, additional fingers, etc.) were believed to be the sign of a divine order. The steps of their introduction:

  1. Climbing up on the “shaman ladder/shaman tree” symbolized the World Tree;
  2. Drenching the ghosts: drinking the blood of the sacrificed animal.

According to general consensus, the táltos were considered as part of pagan religion, and were persecuted in a witch-hunt during the reign of King Stephen I of Hungary. There is evidence, though, that the táltos were still existing until the Habsburg era, when this tradition was terminated. Maria Theresa made a law requiring that all babies born with teeth or with six fingers be reported and killed, a deliberate act against surviving táltos.[1] The painted ceiling of the church of Székelyderzsi had a figure with six fingers, it was renovated, “correcting” the picture to five fingers.[2]

According to folklore tradition, the égi táltos (or heavenly táltos) is Jesus Christ[3].

FUTHER READING:

The two dragon világfán; Irish or imperial representation. The wood (világtengelynek appropriate) strain visible szvasztikák the sky designated by movement. Both dragon twelve days or is upon, which include the twelve month. A century miniature VIII on the basis of leaves from Northumberlandból subject entitled, in which the Würzburgi University directory is located.

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The Assyrian Tree of Life

FUTHER READING: http://www.symbolonpress.com/htm/Tree_of_Life.html

TREE OF LIFE, MYTHICAL ARCHETYPE

The oldest scriptures at the heart of every major religion make reference to a mysterious tree at the center of the world. Its fruits, guarded by an evil serpent, confer immortality. A nearby stream of water divides into four rivers flowing into the four cardinal directions. The vicinity of this tree is said to be the birthplace of the first human ancestors. This legend is the oldest, most widely dispersed, and most mysterious religious idea known to mankind.


The Tree also appears with other symbols on artifacts found at the ancient city of Troy and on the oldest examples of Greek ceramic art. The decipherment of these Bronze Age symbols, described for the first time in this book, leads to the discovery of an archaic theme pervading much of world mythology. An understanding of this archetype, and of the natural phenomenon that inspired it, unlocks mythological enigmas that for centuries have eluded interpretation.

Read the Foreword by Michael Witzel, PhD

Tree of Life, Mythical Archetype contains 350 illustrations. These images trace the fascinating evolution of the myth through its many permutations in world art and religion. Click here to see the book’s complete Table of Contents.

Assyrian Tree of Life with winged god sprinkling it with sacred water

An eagle-headed Assyrian divinity sprinkling sacred water onto the Tree of Life. Variations of this motif have been found in Mesopotamia, Egypt, Greece, and pre-Columbian Mexico.

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The Acacia Tree : Saosis

The Egyptian creation Myth

Egyptian Culture: Acacia tree of ‘Saosis’ is considered by Egyptians as the Tree of Life. As per the Ennead system (nine deities) of ancient Egyptian culture, Isis and Osiris are believed to be the first couple. They emerged from the Tree of Life i.e. the acacia of Saosis. Read more on ‘Creation Theories‘.

see Ennead: http://en.wikipedia.org/wiki/Ennead

FURTHER READING: http://www.aldokkan.com/religion/creation.htm

http://vincentbridges.com/?tag=raising-the-djed

see also Djed: http://en.wikipedia.org/wiki/Djed

Ancient Egypt

  • In Egyptian mythology, in the Ennead system of Heliopolis, the first couple, apart from ShuTefnut (moisture & dryness) and GebNuit (earth & sky), are IsisOsiris. They were said to have emerged from the acacia tree of Saosis, which the Egyptians considered the “tree of life”, referring to it as the “tree in which life and death are enclosed”. A much later myth relates how Set killed Osiris, putting him in a coffin, and throwing it into the Nile, the coffin becoming embedded in the base of a tamarisk tree.
  • The Egyptians’ Holy Sycamore also stood on the threshold of life and death, connecting the two worlds.

Atum (alternatively spelled TemTemuTum, and Atem) is an important deity in Egyptian mythology, whose cult centred on the city of Heliopolis (EgyptianAnnu). His name is thought to be derived from the word ‘tem’ which means to complete or finish.

—————–

Anthes (1957) translates Atum as “he who is integral”Bonnet as “he who is not yet complete”Kees(1941) opts for “he who is not present yet” or “he who does not yet exist completely”, whereas Hornung(1986) chooses “he who is differentiated”, eliminating the important connotation of the alternation-point between a mere genetic potential (in precreation) and the beginning of its activity (or escape from latency). Atum completes everything in precreation, not in creation. Indeed, the whole creative process of Atum “hatching” out of his “egg” and initiating the “first occasion” (“zep tepi”) takes place in an interstitial “First Time”, the “Golden Age” of the gods and goddesses who, in the beginning of time and space itself, differentiate Atum in so many frequencies of natural differentials (natural laws and the accidents they describe).

—————-

Thus he has been interpreted as being the ‘complete one’ and also the finisher of the world, which he returns to watery chaos at the end of the creative cycle. As creator he was seen as the underlying substance of the world, the deities and all things being made of his flesh or alternatively being his ka.

Sometimes he also is shown as a serpent, the form which he returns to at the end of the creative cycle and also occasionally as a mongooselion,bulllizard, or ape.

In the Heliopolitan creation myth established in the sixth dynasty, he was considered to be the first god, having created himself, sitting on a mound (benben) (or identified with the mound itself), from the primordial waters (Nu). Early myths state that Atum created the god Shu and goddess Tefnutfrom spitting or from his semen by masturbation in Heliopolis.[1]

Atum was a self-created deity, the first being to emerge from the darkness and endless watery abyss that girdled the world before creation. A product of the energy and matter contained in this chaos, he created divine and human beings through loneliness: alone in the universe, he produced from his own semen Shu, the god of air, and Tefnut, the goddess of moisture. The brother and sister, curious about the primeval waters that surrounded them went to explore the- and disappeared into the darkness. Unable to bear his loss, Atum sent a fiery messenger to find his children. The tears of joy he shed on their return were the first human beings.

He is generally represented in human form and as the source of the Pharaoh’s power he wears the double crown of Egypt- red for Lower Egypt, White for Upper Egypt and he also carries a tall cross, the symbol of eternal life.

Mound/Pyramid of Atum.

FUTHER READING: http://www.braden.org/modules/news/article.php?storyid=17

The Acacia is a tree long intertwined with the metaphor and mystery of the human mind. In particular, the religious myths of the desert-dwelling peoples of Africa and the middle east.

In the old testament it is sometimes referred to as the shittum tree. It was used in the construction of the ark of the covenant and God’s tabernacle. When God first appeared to Moses in the desert, it was a burning acacia bush that carried the message. It is widely accepted that a crown of acacia thorns that was placed on Jesus’ head as he was crucified on a cross of acacia wood.

Frederick Dalcho explains that a sprig of Acacia was used by Hebrews to mark the graves of dear departed friends while explaining its use in Masonic funeral rites.

The bulk of the ancient Egyptian pantheon was said to have be born beneath the branches of the goddess Saosis’ Acacia tree north of Heliopolis. The thorns of the tree are said to represent the goddess Nieth and symbolize birth and death.

When Osiris was betrayed and murdered, his coffin was thrown into the river. It lodged in the rocks where an acacia tree grew around it, enveloping it completely. This tree was the key to Isis’ finding and restoring his body. Upon his resurrection, Osiris was given rulership of the underworld.

Some versions of the myth record the birth of the god Horus as “out of the branches of the acacia”, to represent the continued lineage of Osiris. In Egyptian politics, the sitting pharoh was considered to be a living representation of Horus, ruler of the land of the living, and the recently deceased was thought to be Osiris, ruler of the land of the dead.

In “The Hiram Key” Christopher Knight and Robert Lomas speculate on the nature of the Egyptian king making ceremony and submit that the rituals of freemasonry are a modern analogue of these ancient rites of passage.

In the Egyptian ritual, Knight and Lomas conjecture that the pharaoh-to-be would undergo an elaborate ritual in order to seal and ordain his right to rule. The ritual would involve members of an elect group of the high priesthood and would culminate in the administration of a powerful hallucinogenic drug. The drug would induce a catatonic state for a known period of time, set to wear off at the dawning of the bright star of the morning.  While in this trance, the new pharaoh would travel to Orion’s belt and commune with the gods, who would pass on the necessary knowledge and power to assume his own God-ship on earth.

This claim is somewhat substantiated in the translation of utterance 294 from the east wall of the sarcophagus antechamber of the pyramid of Unas.

“436: To say the words : ‘Unas is a Horus who came out of the acacia (SnD) [House of the Acacia, linked with funerary ritual / mummification], who came out of the acacia, to whom it was ordered : “Beware of the lion!”. He comes out to whom it was ordered : “Beware of the lion !”. 437: Unas has come out of this Dnj.t-jar after he has passed the night in his Dnj.t-jar. Unas appears in the morning. He has come out of his Dnj.t-jar after he has passed the night in his Dnj.t-jar. Unas appears in the morning.”

What I find even more compelling is the addition of the knowledge that the acacia tree would have been a viable and ready source of  the chemical Di-Methyl Tryptamine, or DMT.

DMT is a naturally occurring chemical in the human brain, in small quantities. There is much that is not known about where it comes from or why. Some suspect it is produced in the pineal gland and production stimulated through the attainment of certain meditative states. In 1988 Jace Callaway speculated that release of this chemical is responsible for the visual imagery of dreaming.  When taken in large doses it is a powerful hallucinogen.

Persons who have taken DMT for recreational psychedelic experiences report a short lived, intense visual experience.

“1 minute – 2 – 5 minutes – depending on dosage: DMT hyperspace. For all practical purposes, you will no longer be embodied. You will be part of the intergalactic information network. You may experience any of the following:

  • Sense of transcending time or space
  • Strange plants or plantlike forms
  • The universe of formless vibration
  • Strange machines
  • Alien music
  • Alien languages, understandable or      not
  • Intelligent entities in a variety of      forms (http://nepenthes.lycaeum.org)

Rick Straussman, author of “The Spirit Molecule”, a book exploring the biology of DMT, believes that pineal DMT release at 49 days after conception marks the entrance of the soul into the human fetus.

The pineal has mostly been implicated for its role in melatonin production, which is important in seasonal and age-related reproduction issues. Also, there are some mood, sleep, and body temperature effects. Some skin coloration ones in reptiles and amphibians. I have suggested it’s involved in DMT production, because the precursors and enzymes necessary for its formation are quite high in the pineal. However, there are no hard data to suggest this actually happens.

There are some interesting coincidental findings around the pineal and spiritual issues. Descartes believed that because it’s the only unpaired organ in the brain, and because we can only have one thought in our mind at one time, and since thought seems to be a function of the soul (our relationship to the divine), that the pineal was a valve for the transfer/conduction of divine and human communication.

The pineal is first seen in the embryo at 49 days, the same amount of time the Tibetan Buddhists believe the soul requires from death to its next rebirth. Also, the first sign of clearly differentiated male and female gonads in the human occurs are 49 days. Thus, there’s some interesting relationship among spirituality and gender/reproduction here.

Recent speculation has centered around the role of naturally occurring DMT being responsible for the mystic visions of saints and biblical prophets.

(from DMT, Moses, and the Quest for Transcendence by Cliff PickoverReality Carnival )The molecule DMT (N,N-Dimethyltryptamine) is a psychoactive chemical that causes intense visions and can induce its users to quickly enter a completely different “environment” that some have likened to an alien or parallel universe. The transition from our world to theirs occurs with no cessation of consciousness or quality of awareness. In this environment, beings often appear who interact with the person who is using DMT. The beings appear to inhabit this parallel realm. The DMT experience has the feel of reality in terms of detail and potential for exploration. The creatures encountered are often identified as being alienlike or elflike. Some of the creatures appear to be three-dimensional. Others appear to lack depth.

Author Terence McKenna has used DMT and feels that, “Right here and now, one quanta away, there is raging a universe of active intelligence that is transhuman, hyperdimensional, and extremely alien… What is driving religious feeling today is a wish for contact with this other universe.” The aliens seen while using DMT present themselves “with information that is not drawn from the personal history of the individual.”

Given what we know about the pharmacological prowess of the ancient Egyptian priesthood, I think it’s safe to assume they had knowledge of the effects of the chemicals derivable from Acacia. I also think it’s safe to say that they were familiar with ways to combine it with other hallucinogens in order to prolong and magnify the effects. Could a DMT trip be the means by which a new pharoh communicated with the Gods and received the secrets of kingship? Did the ancient Jews use the chemical in their own rites? Is naturally occurring DMT the key to mystical experience? At this time, we can only speculate.

From Wikipedia, the free encyclopedia

For other uses, see Tree of life (disambiguation).
Not to be confused with Tree of life (science).
Not to be confused with Tree of life journal.
This article does not cite any references or sources.
Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed(January 2009)

An 1847 depiction of the Norse Yggdrasilas described in the Icelandic Prose Edda by Oluf Olufsen Bagge.

The concept of a tree of life as a many-branched tree illustrating the idea that all life on earth is related has been used in sciencereligion,philosophymythology, and other areas. A tree of life is variously;

  1. motif in various world theologiesmythologies, and philosophies;
  2. a metaphor for the livelihood of the spirit.
  3. a mystical concept alluding to the interconnectedness of all life on our planet; and
  4. metaphor for common descent in the evolutionary sense.

According to the Encyclopædia Britannica, the tree of knowledge, connecting heaven and the underworld, and the tree of life, connecting all forms of creation, are both forms of the world tree or cosmic tree.[1] According to some scholars, the tree of life and the tree of the knowledge of good and evil, portrayed in various religions and philosophies, are the same tree.[2]

Contents

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The White Tree: Tolkien

TOLKIEN’S TREES

THE WHITE TREE OF GONDOR

In J. R. R. Tolkien‘s fantasy universe of Middle-earth, the White Tree of Gondor stood as a symbol of Gondor in the Court of the Fountain in Minas Tirith. The White Tree also appears as a motif upon Gondor’s flag and throughout its heraldry combined with the seven stars of the House of Elendil and the crown of the King.

(see Gondor)

THE GATES of MORIA

Taken from “Lord of the Rings: The Fellowship of the Ring”

“TREE AND LEAF”

Tree and Leaf (1964) A collection of stories by J R R Tolkien

A new edition of Tolkien’s Tree and Leaf, complete with his rare translation and commentary of The Homecoming of Beorhtnoth, the Battle of Maldon. Fairy-stories are not just for children, as anyone who has read Tolkien will know. In his essay On Fairy-Stories, Tolkien discusses the nature of fairy-tales and fantasy and rescues the genre from those who would relegate it to juvenilia. The haunting short story, Leaf by Niggle, recounts the story of the artist, Niggle, who has ‘a long journey to make’ and is seen as an allegory of Tolkien’s life. The poem Mythopoeia relates an argument between two unforgettable characters as they discuss the making of myths. Lastly, and published for the very first time, we are treated to the translation of Tolkien’s account of the Battle of Maldon, known as The Homecoming of Beorhtnoth. Tree and Leaf is an eclectic, amusing, provocative and entertaining collection of works which reveals the diversity of J.R.R. Tolkien’s imagination, the depth of his knowledge of English history, and the breadth of his talent as a creator of fantastic fiction.

“LEAVES FROM THE TREE”

Leaves from the Tree: J.R.R. Tolkien’s Shorter Fiction. The Proceedings of the 4th Tolkien Society Workshop.
Peter Roe Memorial Booklet No. 2.
Shippey, et al.
1st Edition 1991.
The Tolkien Society.
ISBN 0905520033.
Booklet with card covers.
Hammond p.387.

Includes a paper (“Dragons from Andrew Lang’s Retelling of Sigurd to Tolkien’s Chrysophylax” by Christina Scull) that reproduces an extract from Tolkien’s unpublished lecture on dragons.

Two impressions issued?  The 1st Impression had a yellow-green coloured cover, while the later unidentified reprint (shown opposite) was issued in a yellow card cover.

FUTHER READING: LINKS – [Home] [The Bibliography] [The Early Years 1910-1937] [The Hobbit 1937] [The Middle Years 1938-1953] [The Lord of the Rings 1954-1955] [The Later Years 1955-1973] [The Tale Goes On 1974-Onwards] [Uncertain Date] [Slipcased & Boxed Sets] [Other Credits] [Book Index] [Articles] [Site History] [Links] [Contact]


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The Fractal Tree : Julia Sets

The Fractal Tree : Julia Sets

A fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”[1] a property called self-similarity. Roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Karl WeierstrassGeorg Cantor and Felix Hausdorff in studying functions that were analyticbut not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractusmeaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedbackbased on recursion.[2]

This principle of self-similarity recursion can be seen in the Spiral of the Golden Mean. The Fibonacci Sequence is the iterating pattern that geanerates the Golden mean. SEE ALSO; Newton iteration ; Newton’s method ;

The Glynn Set is a special Julia Set, named after Earl Glynn. It kind of looks like a tree. See also my 3D Glynn Julia set. Here is some Mathematica code:
(* runtime: 48 seconds *)
Julia[z0_] := Module[{z = z0, i = 0}, While[i < 100 && Abs[z] < 2, z = z^1.5 + c; i++]; i];
c = -0.2;
DensityPlot[Julia[-x + I y], {y, -0.2, 0.2}, {x, 0.35, 0.75}, PlotPoints -> 275, Mesh -> False, Frame -> False]

Here is some code to plot this using the Modified Inverse Iteration Method (MIIM). Note that special care must be taken to verify each root’s validity:
(* runtime 7 seconds *)
Pow[z_, n_, k_] := Module[{theta = Arg[z]}, theta = n(theta + 2Pi (k – Floor[(theta/Pi +Abs[1/n])/2])); If[Abs[theta] > Pi, Null, Abs[z]^n Exp[I theta]]];
power = 1.5; nroot = 3; c = -0.2; zlist = {}; dzmax = 25.0; imax =1000; z = Table[-0.61, {imax}]; dz = Table[1, {imax}]; roots = Table[1, {imax}]; i = 2;
While[i > 1, z[[i]] = Pow[z[[i – 1]] – c, 1/power, roots[[i]] – 1]; If[z[[i]] === Null, prune = True, dz[[i]] = Abs[power]Abs[z[[i]]]^(power – 1)dz[[i – 1]]; zlist = Append[zlist, z[[i]]]; prune = (i == imax || dz[[i]] > dzmax)]; If[prune, While[i > 1 && roots[[i]] == nroot, roots[[i]] = 1; i–]; roots[[i]]++, i++; roots[[i]] = 1]];
ListPlot[{Re[#], Im[#]} & /@ zlist, PlotStyle -> PointSize[0.005], AspectRatio ->Automatic]

FUTHER READING : WIKIPEDIA

From Wikipedia, the free encyclopedia

A Julia set.

Julia set 3d slice animation.ogg

Three-dimensional slices through the (four-dimensional) Julia set of a function on thequaternions.

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly underrepeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is ‘regular’, while on the Julia set its behavior is ‘chaotic‘.

The Julia set of a function ƒ is commonly denoted J(ƒ), and the Fatou set is denoted F(ƒ).[1] These sets are named after the French mathematicians Gaston Julia,[2] and Pierre Fatou[3] whose work began the study of complex dynamics during the early 20th century.

Contents

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[edit]Formal definition

Let f(z) be a complex rational map from the plane into itself, that is, f(z) = p(z) / q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F_i, i = 1, \dots, r, that are left invariant by f(z) and are such that:

  1. the union of the Fi‘s is dense in the plane and
  2. f(z) behaves in a regular and equal way on each of the sets Fi.

The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.

These sets Fi are the Fatou domains of f(z), and their union is the Fatou set F(f) of f(z). Each of the Fatou domains contains at least one critical point of f(z), that is, a (finite) point zsatisfying f‘(z) = 0, or z = ∞, if the degree of the numerator p(z) is at least two larger than the degree of the denominator q(z), or if f(z) = 1 / g(z) + c for some c and a rational functiong(z) satisfying this condition.

The complement of F(f) is the Julia set J(f) of f(z). J(f) is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). LikeF(f), J(f) is left invariant by f(z), and on this set the iteration is repelling, meaning that | f(z) − f(w) | > | z − w | for all w in a neighbourhood of z (within J(f)). This means that f(z)behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

For f(z) = z2 the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.

For f(z) = z2 − 2 the Julia set is the line segment between -2 and 2, and the iteration corresponds to x \to 4(x - 0.5)^{2} in the unit interval – a very used method for generation of random numbers. There is one Fatou domain: the points not on the line segment iterate towards ∞.

These two functions are of the form z2c, where c is a complex number. For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes. See the pictures below.

Julia set (in white) for the rational function associated to Newton’s method for ƒ:zz3−1. Coloring of Fatou set according to attractor (the roots of ƒ)

For some functions f(z) we can say beforehand that the Julia set is a fractal and not a simple curve. This is because of the following main theorem on the iterations of a rational function:

Each of the Fatou domains has the same boundary, which consequently is the Julia set

This means that each point of the Julia set is a point of accumulation for each of the Fatou domains. Therefore, if there are more than two Fatou domains, each point of the Julia set must have points of more than two different open sets infinitely close, and this means that the Julia set cannot be a simple curve. This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equationzn = 1(n > 2):  f(z) = z − f(z) / f‘(z) = (1 + (n − 1)zn) / (nzn − 1). The image on the right shows the case n = 3.

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.[4] Each component of the Fatou set of a rational map can be classified into one of four different classes.[5]

[edit]Equivalent descriptions of the Julia set

  • J(f) is the smallest closed set containing at least three points which is completely invariant under f.
  • J(f) is the closure of the set of repelling periodic points.
  • For all but at most two points z\in X, the Julia set is the set of limit points of the full backwards orbit \bigcup_n f^{-n}(z). (This suggests a simple algorithm for plotting Julia sets, see below.)
  • If f is an entire function – in particular, when f is a polynomial, then J(f) is the boundary of the set of points which converge to infinity under iteration.
  • If f is a polynomial, then J(f) is the boundary of the filled Julia set; that is, those points whose orbits under iterations of f remain bounded.

[edit]Properties of the Julia set and Fatou set

The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f, i.e.

f − 1(J(f)) = f(J(f)) = J(f)

and

f − 1(F(f)) = f(F(f)) = F(f).[6]

[edit]Quadratic polynomials

A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as

f_c(z) = z^2 + c\,

where c is a complex parameter.

Filled Julia set for fc, c=1−φ where φ is thegolden ratio

Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i

Julia set for fc, c=0.285+0i

Julia set for fc, c=0.285+0.01i

Julia set for fc, c=0.45+0.1428i

Julia set for fc, c=-0.70176-0.3842i

Julia set for fc, c=-0.835-0.2321i

Julia set for fc, c=-0.8+0.156i

A Julia set plot showing julia sets for different values of c, the plot resembles theMandelbrot set

The parameter plane of quadratic polynomials – that is, the plane of possible c-values – gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all c such that J(fc) is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor set: in this case it is sometimes referred to as Fatou dust.

In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called‘Misiurewicz’ parameters, i.e. parameters c for which the critical point is pre-periodic. For instance:

  • At ci, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
  • At c = − 2, the tip of the long spiky tail, the Julia set is a straight line segment.

In other words the Julia sets J(fc) are locally similar around Misiurewicz points.[7]

[edit]Generalizations

The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Epstein’s ‘finite-type maps’.

Julia sets are also commonly defined in the study of dynamics in several complex variables.

[edit]The potential function and the real iteration number

The Julia set for f(z) = z2 is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = log | z | . The equipotential lines for this function are concentric circles. As | f(z) | = | z2 we have \phi(z) = \lim_{k\to\infty} \log|z_k|/2^{k}, where zk is the sequence of iteration generated by z. For the more general iteration f(z) = z2c, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that | ψ(f(z)) | = | ψ(z) | 2[8]. This means that the potential function on the outer Fatou domain defined by this correspondence is given by:

\phi(z) = \lim_{k\to\infty} \log|z_k|/2^k. \,

This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by:

\phi(z) = \lim_{k\to\infty} \log|z_k|/d^k, \,

where d = m – n is the degree of the rational function[9].

If N is a very large number (e.g. 10100), and if k is the first iteration number such that | zk | > N, we have that log | zk | / dk = log(N) / dν(z), for some real number ν(z), which should be regarded as the real iteration number, and we have that:

ν(z) = k − log(log | zk | / log(N)) / log(d),

where the last number is in the interval [0, 1).

For iteration towards a finite attracting cycle of order r, we have that if z* is a point of the cycle, then f(f(…f(z * ))) = z * (the r-fold composition), and the number \alpha = 1/|(d(f(f(\cdots f(z))))/dz)_{z=z*}| (> 1) is the attraction of the cycle. If w is a point very near z* and w’ is w iterated r times, we have that \alpha = \lim_{k\to\infty} |w - z*|/|w' - z*|. Therefore the number | zkr − z * | αk is almost independent of k. We define the potential function on the Fatou domain by:

\phi(z) = \lim_{k\to\infty} 1/(|z_{kr} - z*|\alpha^{k}).

If ε is a very small number and k is the first iteration number such that | zk − z * | < ε, we have that \phi(z) = 1/(\varepsilon \alpha^{\nu(z)}) for some real number ν(z), which should be regarded as the real iteration number, and we have that:

\nu(z) = k - \log(\varepsilon/|z_k - z*|)/\log(\alpha).

If the attraction is ∞, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a critical point, we must replace α by \alpha = \lim_{k\to\infty} \log|w' - z*|/\log|w - z*| (where w’ is w iterated r times) and the formula for φ(z) by:

\phi(z) = \lim_{k\to\infty} \log(1/|z_{kr} - z*|)/\alpha^k. \,

And now the real iteration number is given by:

\nu(z) = k - \log(\log|z_k - z*|/\log(\varepsilon))/\log(\alpha). \,

For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H-1 (H = 500, for instance). We multiply the real number ν(z) by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H.

The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the iteration.

[edit]Field lines

The equipotential lines for iteration towards infinity

Field lines for an iteration of the form(1 − z3 / 6) / (z − z2 / 2)2c

In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the equipotential lines (for the potential function or the real iteration number) and the field lines.

If we colour the Fatou domain according to the iteration number (and not the real iteration number), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration z2c), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the x-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently – at least not by the method we describe here. In this case a field line is also called an external ray.

Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence of iteration is a finite cycleC, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r be the order of the cycle C (its number of points) and let z* be a point in C. We have f(f(\dots f(z*))) = z* (the r-fold composition), and we define the complex number α by

\alpha = (d(f(f(\dots f(z))))/dz)_{z=z*}. \,

If the points of C are z_i, i = 1, \dots, r (z_1 = z*), α is the product of the r numbers f‘(zi). The real number 1/ | α | is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1/|α| < ∞. The point z* is a fixed point for f(f(\dots f(z))), and near this point the map f(f(\dots f(z))) has (in connection with field lines) character of a rotation with the argument β of α (that is, α = | α | eβi).

In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration z_k (k = 0, 1, 2, \dots, z_0 = z)to stop when | zk − z * | < ε, and we colour the point z according to the number k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z* given by an angle θ, the field line issuing from z* in this direction consists of the points z such that the argument ψ of the number zk − z * satisfies the condition that

\psi - k\beta = \theta \mod \pi. \,

For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ψ is increased by β, therefore the number \psi - k\beta \mod \pi is constant along the field line.

Pictures in the field lines for an iteration of the form z2c

A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from z*, and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the centre line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture).

A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number). Therefore we can put pictures into the field lines (third picture).

[edit]Distance estimation

Julia set drawn by distance estimation, the iteration is of the form1 − z2z5 / (2 + 4z) + c

Three-dimensional rendering of Julia set using distance estimation.

As a Julia set is infinitely thin we cannot draw it effectively by backwards iteration from the pixels. It will appear fragmented because of the impracticality of examining infinitely many startpoints. Since the iteration count changes vigorously near the Julia set, a partial solution is to imply the outline of the set from the nearest color contours, but the set will tend to look muddy.

A better way to draw the Julia set in black and white is to estimate the distance of pixels from the set and to color every pixel whose center is close to the set. The formula for the distance estimation is derived from the formula for the potential function φ(z). When the equipotential lines for φ(z) lie close, the number | φ(z) | is large, and conversely, therefore the equipotential lines for the function δ(z) = φ(z) / | φ(z) | should lie approximately regularly. It has been proven that the value found by this formula (up to a constant factor) converges towards the true distance for z converging towards the Julia set [10].

We assume that f(z) is rational, that is, f(z) = p(z) / q(z) where p(z) and q(z) are complex polynomials of degrees m and n, respectively, and we have to find the derivative of the above expressions for φ(z). And as it is only zk that varies, we must calculate the derivative zk of zk with respect to z. But as z_k = f(f(\cdots f(z))) (the k-fold composition), zk is the product of the numbers f‘(zk), and this sequence can be calculated recursively by zk + 1f‘(zk)zk, starting with z0 = 1 (before the calculation of the next iteration zk + 1f(zk)).

For iteration towards ∞ (more precisely when m ≥ n + 2, so that ∞ is a super-attracting fixed point), we have

|\phi^'(z)| = \lim_{k\to\infty} |z'_k|/|z_k|d^k, \,

(dm − n) and consequently:

\delta(z) = \phi(z)/|\phi^'(z)| = \lim_{k\to\infty} \log|z_k||z_k|/|z'_k|. \,

For iteration towards a finite attracting cycle (that is not super-attracting) containing the point z* and having order r, we have

|\phi^'(z)| = \lim_{k\to\infty} |z'_{kr}|/(|z_{kr} - z*|^{2}\alpha^{k}), \,

and consequently:

\delta(z) = \phi(z)/|\phi^'(z)| = \lim_{k\to\infty} |z_{kr} - z*|/|z'_{kr}|. \,

For a super-attracting cycle, the formula is:

\delta(z) = \lim_{k\to\infty} \log|z_{kr} - z*||z_{kr} - z*|/|z'_{kr}|. \,

We calculate this number when the iteration stops. Note that the distance estimation is independent of the attraction of the cycle. This means that it has meaning for transcendental functions of “degree infinity” (e.g. sin(z) and tan(z)).

Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal landscape.

[edit]Plotting the Julia set

[edit]Using backwards (inverse) iteration (IIM)

A Julia set plot, generated using random IIM

A Julia set plot, generated using MIIM

As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point z we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of z under some high iterate fn of f.

Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally. However, we can adjust this method, in a similar way as the “random game” method for iterated function systems. That is, in each step, we choose at random one of the inverse images of f\,.

For example, for the quadratic polynomial f_c \,, the backwards iteration is described by

z_{n-1} = \sqrt{z_n - c} .

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite difficult to access with the reverse Julia algorithm. For this reason, one must modify IIM/J ( it is called MIIM/J) or use other methods to produce better images.

[edit]Using DEM/J

Julia set : image with C source code using DEM/J

[edit]See also

Search Wikimedia Commons Wikimedia Commons has media related to: Julia set
Search Wikibooks Wikibooks has a book on the topic of

[edit]Notes

  1. ^ Note that for other areas of mathematics the notation J(f)\, can also represent the Jacobian matrix of a real valued mapping f\, between smooth manifolds.
  2. ^ Gaston Julia (1918) “Mémoire sur l’iteration des fonctions rationnelles,” Journal de Mathématiques Pures et Appliquées, vol. 8, pages 47–245.
  3. ^ Pierre Fatou (1917) “Sur les substitutions rationnelles,” Comptes Rendus de l’Académie des Sciences de Paris, vol. 164, pages 806-808 and vol. 165, pages 992–995.
  4. ^ Beardon, Iteration of Rational Functions, Theorem 5.6.2
  5. ^ Beardon, Theorem 7.1.1
  6. ^ Beardon, Iteration of Rational Functions, Theorem 3.2.4
  7. ^ Lei.pdf Tan Lei, “Similarity between the Mandelbrot set and Julia Sets”, Communications in Mathematical Physics 134 (1990), pp. 587–617.
  8. ^ Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d’Orsay 2/4 (1984 / 1985)
  9. ^ Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0.
  10. ^ Peitgen, Heinz-Otto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: Springer-Verlag. ISBN 0-387-15851-0.

[edit]References

[edit]External links

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CONT.

From Wikipedia, the free encyclopedia

The Mandelbrot set is a famous example of a fractal

Frost crystals formed naturally on cold glass illustrate fractal process development in a purely physical system

fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,”[1] a property called self-similarity. Roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Karl WeierstrassGeorg Cantor and Felix Hausdorff in studying functions that were analyticbut not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractusmeaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedbackbased on recursion.[2]

A fractal often has the following features:[3]

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

Images of fractals can be created using fractal-generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, such as when it is possible to zoom into a region of the fractal that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.

Contents

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[edit]History

To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral “bump.” One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite. For this reason, the Koch snowflake and similar constructions were sometimes called “monster curves.”

The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibnizconsidered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).

It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass’s abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. (The image at right is three Koch curves put together to form what is commonly called the Koch snowflake.) Waclaw Sierpinskiconstructed his triangle in 1915 and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, theLévy C curveGeorg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri PoincaréFelix KleinPierre Fatou and Gaston Julia. Without the aid of modern computer graphics, however, they lacked the means to visualize the beauty of many of the objects that they had discovered.

In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word “fractal” to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term “fractal”.

[edit]Examples

Julia set, a fractal related to the Mandelbrot set

A class of examples is given by the Cantor setsSierpinski triangle and carpetMenger spongedragon curvespace-filling curve, andKoch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can bedeterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.

Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (seeattractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that theboundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set.

[edit]Generating fractals

The whole Mandelbrot set
Mandelbrot zoomed 6x
Mandelbrot Zoomed 100x
Mandelbrot Zoomed 2000xEven 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

Four common techniques for generating fractals are:

[edit]Classification

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

  • Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
  • Quasi-self-similarity – This is a looser form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar. The Mandelbrot set is quasi-self-similar, as the satellites are approximations of the entire set, but not exact copies.
  • Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of “fractal” trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar. The coastline of Britain is another example; one cannot expect to find microscopic Britains (even distorted ones) by looking at a small section of the coast with a magnifying glass.

Possessing self-similarity is not the sole criterion for an object to be termed a fractal. Examples of self-similar objects that are not fractals include the logarithmic spiral and straight lines, which do contain copies of themselves at increasingly small scales. These do not qualify, since they have the same Hausdorff dimension as topological dimension.

[edit]In nature

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakescrystals,mountain rangeslightningriver networkscauliflower or broccoli, and systems of blood vessels and pulmonary vesselsCoastlines may be loosely considered fractal in nature.

Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or afrond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees.[5]

In 1999, certain self similar fractal shapes were shown to have a property of “frequency invariance”—the same electromagnetic properties no matter what the frequency—from Maxwell’s equations (see fractal antenna).[6]

A fractal that models the surface of a mountain (animation)

Photograph of a romanesco broccoli, showing a naturally occurring fractal

Fractal pentagram drawn with a vectoriteration program

[edit]In creative works

Further information: Fractal art

Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock’s paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.[7]

Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[8] It involves pressing paint between two surfaces and pulling them apart.

Fractals are also prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[9]

In a 1996 interview David Foster Wallace admitted that the structure of his novel Infinite Jest was inspired by fractals, specifically the Sierpinski triangle.[10]

The song “Hilarious Movie of the 90’s” from Pause (album) by the artist Four Tet employs the use of fractals.[11]

[edit]Gallery

A fractal is formed when pulling apart two glue-covered acrylic sheets.

High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure.

Fractal branching occurs in a fractured surface such as a microwave-irradiatedDVD.[12]

DLA cluster grown from a copper(II) sulfate solution in an electrodepositioncell

A “woodburn” fractal

A magnification of the phoenix set

fractal flame created with the programApophysis

Fractal made by the program Sterling

A fractal created using the programApophysis and a julian transform

A double fractal found in nature. Ice on a vine. Two fractals naturally occurring at once

[edit]Applications

Main article: Fractal analysis

As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:[13]

[edit]See also

[edit]References

  1. ^ Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN 0-7167-1186-9.
  2. ^ Briggs, John (1992). Fractals:The Patterns of Chaos. London : Thames and Hudson, 1992.. pp. 148. ISBN 0500276935, 0500276935.
  3. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv. ISBN 0-470-84862-6.
  4. ^ The Hilbert curve map is not a homeomorhpism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.
  5. ^ “Hunting the Hidden Dimension.” Nova. PBS. WPMB-Maryland. 28 October 2008.
  6. ^ Hohlfeld R, Cohen N (1999). “Self-similarity and the geometric requirements for frequency independence in Antennae”. Fractals 7 (1): 79–84. doi:10.1142/S0218348X99000098.
  7. ^ Richard Taylor, Adam P. Micolich and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?
  8. ^ A Panorama of Fractals and Their Uses by Michael Frame and Benoît B. Mandelbrot
  9. ^ Ron Eglash. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999.
  10. ^ http://www.kcrw.com/etc/programs/bw/bw960411david_foster_wallace
  11. ^ http://lala.com/zVPSY
  12. ^ Peng, Gongwen; Decheng Tian (21 July 1990). “The fractal nature of a fracture surface”Journal of Physics A 23 (14): 3257–3261. doi:10.1088/0305-4470/23/14/022. Retrieved 2007-06-02.
  13. ^ “Applications”. Retrieved 2007-10-21.

[edit]Further reading

[edit]External links

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From Wikipedia, the free encyclopedia

ray-traced image of the 3D Mandelbulb
for the iteration z ↦ z8c.

Daniel White and Paul Nylander constructed the Mandelbulb, a 3-dimensional analog of the Mandelbrot set, using a hypercomplex algebra based on spherical coordinates.[1]

White and Nylander’s formula for the nth power of the 3d hypercomplex number \langle x, y, z\rangle is:

\langle x, y, z\rangle^n = r^n\langle\cos(n\theta)\cos(n\phi),\sin(n\theta)\cos(n\phi),\sin(n\phi)\rangle

where

\begin{align}r&=\sqrt{x^2+y^2+z^2} \\  \theta&=\arctan(y/x) \\  {\rm and\ } \phi&=\arctan(z/\sqrt{x^2+y^2})=\arcsin(z/r).\end{align}

They use the iteration z\mapsto z^n+c where z and c are 3-dimensional hypercomplex numbers with the power map z\mapsto z^n defined as above.[2] For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of “lobes” controlled by the parameter n. Many of their graphic renderings use n = 8.

[edit]References

[edit]External links

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The World Ash : Yggdrasil

Yggdrasil: The Nine Worlds of Norse Mythology

In Norse mythology, Yggdrasil (pronounced /ˈɪɡdrəsɪl/; from Old Norse Yggdrasill, pronounced [ˈyɡːˌdrasilː]) is an immense tree that is central in Norse cosmology; the world tree, and around the tree exist nine worlds. It is generally considered to mean “Ygg‘s (Odin‘s) horse”.

Yggdrasil is attested in the Poetic Edda, compiled in the 13th century from earlier traditional sources, and the Prose Edda, written in the 13th century by Snorri Sturluson. In both sources, Yggdrasil is an immense ash tree that is central and considered very holy. The gods go to Yggdrasil daily to hold their courts. The branches of Yggdrasil extend far into the heavens, and the tree is supported by three roots that extend far away into other locations; one to the well Urðarbrunnr in the heavens, one to the spring Hvergelmir, and another to the well Mímisbrunnr. Creatures live within Yggdrasil, including the wyrm (dragon) Níðhöggr, an unnamed eagle, and the stags Dáinn, Dvalinn, Duneyrr and Duraþrór.

From Wikipedia, the free encyclopedia

For other uses, see Yggdrasil (disambiguation).

“The Ash Yggdrasil” (1886) by Friedrich Wilhelm Heine.

Conflicting scholarly theories have been proposed about the etymology of the name Yggdrasill, the possibility that the tree is of another species than ash, the relation to tree lore and to Eurasian shamanic lore, the possible relation to the trees Mímameiðr and LæraðrHoddmímis holt, the sacred tree at Uppsala, and the fate of Yggdrasil during the events of Ragnarök.

Contents

[hide]

[edit]Terminology

Yggdrasil (1895) by Lorenz Frølich.

Yggdrasil comes from Old Norse Yggdrasill.[1] In English, the spellings YggdrasilYggdrasill, and Ygdrasil are used, as shown by entries in English dictionaries and encyclopedias.[2][3][4] It is usually pronounced /ˈɪɡdrəsɪl/ in English, and only rarely [ˈyɡdrəsiːl][5] because the sound[y] does not exist in English and most English speakers are therefore unaccustomed to producing it.

The generally accepted meaning of Old Norse Yggdrasill is “Odin’s horse”, based on the etymology that drasill means “horse” and Ygg(r) is one of Odin’s many names. The Poetic Edda poem Hávamál describes how Odin sacrificed himself to himself by hanging in a tree, making this tree Odin’s gallows. This tree was apparently Yggdrasil, and gallows can be called “the horse of the hanged”, so Odin’s gallows developed into the expression “Odin’s horse”, which then became the name of the tree.[1]

Nevertheless, scholarly opinions regarding the precise meaning of the name Yggdrasill vary, particularly on the issue of whether Yggdrasill is the name of the tree itself or if only the full term askr Yggdrasils refers specifically to the tree, where Old Norse askr means “ash tree”. According to this interpretation, askr Yggdrasils means “the world tree upon which ‘the horse [Odin’s horse] of the highest god [Odin] is bound'”. Both of these etymologies rely on a presumed but unattested *Yggsdrasill.[1]

A third interpretation, presented by F. Detter, is that the name Yggdrasill refers to the word yggr (“terror”), yet not in reference to the Odinic name, and so Yggdrasill would then mean “tree of terror, gallows”. F. R. Schröder has proposed a fourth etymology according to which yggdrasill means “yew pillar”, deriving yggia from*igwja (meaning “yew-tree“), and drasill from *dher- (meaning “support”).[1]

[edit]Attestations

[edit]Poetic Edda

In the Poetic Edda, the tree is mentioned in the three poems VöluspáHávamál, and Grímnismál.

[edit]Völuspá

“Norns” (1832) from Die Helden und Götter des Nordens, oder das Buch der Sagen.

In the second stanza of the Poetic Edda poem Völuspá, the völva (a shamanic seeress) reciting the poem to the god Odin says that she remembers far back to “early times”, being raised by jötnar (giants), recalls nine worlds and “nine wood-ogresses” (Old Norse nío ídiðiur), and when Yggdrasil was a seed (“glorious tree of good measure, under the ground”).[6] In stanza 19, the völva says:

An ash I know there stands,
Yggdrasill is its name,
a tall tree, showered
with shining loam.
From there come the dews
that drop in the valleys.
It stands forever green over
Urðr’s well.[7]

In stanza 20, the völva says that from the lake under the tree come three “maidens deep in knowledge” named UrðrVerðandi, and Skuld. The maidens “incised the slip of wood,” “laid down laws” and “chose lives” for the children of mankind and the destinies (ørlǫg) of men.[8] In stanza 27, the völva details that she is aware that “Heimdallr‘s hearing is couched beneath the bright-nurtured holy tree.”[9] In stanza 45, Yggdrasil receives a final mention in the poem. The völva describes, as a part of the onset of Ragnarök, that Heimdallr blows Gjallarhorn, that Odin speaks with Mímir‘s head, and then:

Yggdrasill shivers,
the ash, as it stands.
The old tree groans,
and the giant slips free.[10]

[edit]Hávamál

Odin sacrificing himself upon Yggdrasil (1895) by Lorenz Frølich.

In stanza 34 of the poem Hávamál, Odin describes how he once sacrificed himself to himself by hanging on a tree. The stanza reads:

I know that I hung on a windy tree
nine long nights,
wounded with a spear, dedicated to Odin,
myself to myself,
on that tree of which no man knows
from where its roots run.[11]

In the stanza that follows, Odin describes how he had no food nor drink there, that he peered downward, and that “I took up the runes, screaming I took them, then I fell back from there.”[11] While Yggdrasil is not mentioned by name in the poem and other trees exist in Norse mythology, the tree is near universally accepted as Yggdrasil, and if the tree is Yggdrasil, then the name Yggdrasil directly relates to this story.[12]

[edit]Grímnismál

In the poem Grímnismál, Odin (disguised as Grímnir) provides the young Agnar with cosmological lore. Yggdrasil is first mentioned in the poem in stanza 29, where Odin says that, because the “bridge of the Æsir burns” and the “sacred waters boil,” Thor must wade through the riversKörmt and Örmt and two rivers named Kerlaugar to go “sit as judge at the ash of Yggdrasill.” In the stanza that follows, a list of names of horses are given that the Æsir ride to “sit as judges” at Yggdrasil.[13]

In stanza 31, Odin says that the ash Yggdrasil has three roots that grow in three directions. He details that beneath the first lives Hel, under the second live frost jötnar, and beneath the third lives mankind. Stanza 32 details that a squirrel named Ratatoskr must run across Yggdrasil and bring “the eagle’s word” from above to Níðhöggr below. Stanza 33 describes that four harts named Dáinn, Dvalinn, Duneyrr and Duraþrór consume “the highest boughs” of Yggdrasil.[13]

In stanza 34, Odin says that more serpents lie beneath Yggdrasil “than any fool can imagine” and lists them as Góinn and Móinn (possibly meaning Old Norse “land animal”[14]), which he describes as sons of Grafvitnir (Old Norse, possibly “ditch wolf”[15]), Grábakr (Old Norse “Greyback”[14]), Grafvölluðr (Old Norse, possibly “the one digging under the plain” or possibly amended as “the one ruling in the ditch”[15]), Ófnir (Old Norse “the winding one, the twisting one”[16]), and Sváfnir (Old Norse, possibly “the one who puts to sleep = death”[17]), who Odin adds that he thinks will forever gnaw on the tree’s branches.[13]

In stanza 35, Odin says that Yggdrasil “suffers agony more than men know”, as a hart bites it from above, it decays on its sides, and Níðhöggr bites it from beneath.[18] In stanza 44, Odin provides a list of things that are what he refers to as the “noblest” of their kind. Within the list, Odin mentions Yggdrasil first, and states that it is the “noblest of trees”.[19]

[edit]Prose Edda

The title page of Olive Bray’s 1908 translation of the Poetic Edda by W. G. Collingwood.

The norns Urðr, Verðandi, and Skuld beneath the world tree Yggdrasil (1882) byLudwig Burger.

Yggdrasil is mentioned in two books in the Prose Edda, in Gylfaginning and Skáldskaparmál. In Gylfaginning, Yggdrasil is introduced in chapter 15. In chapter 15, Gangleri (described as king Gylfi in disguise) asks where is the chief or holiest place of the gods. High replies “It is the ash Yggdrasil. There the gods must hold their courts each day”. Gangleri asks what there is to tell about Yggdrasil. Just-As-High says that Yggdrasil is the biggest and best of all trees, that its branches extend out over all of the world and reach out over the sky. Three of the roots of the tree support it, and these three roots also extend extremely far: one “is among the Æsir, the second among the frost jötnar, and the third over Niflheim. The root over Niflheim is gnawed at by the wyrm Níðhöggr, and beneath this root is the spring Hvergelmir. Beneath the root that reaches the frost jötnar is the well Mímisbrunnr, “which has wisdom and intelligence contained in it, and the master of the well is called Mimir“. Just-As-High provides details regarding Mímisbrunnr and then describes that the third root of the well “extends to heaven” and that beneath the root is the “very holy” well Urðarbrunnr. At Urðarbrunnr the gods hold their court, and every day the Æsir ride to Urðarbrunnr up over the bridgeBifröst. Later in the chapter, a stanza from Grímnismál mentioning Yggdrasil is quoted in support.[20]

In chapter 16, Gangleri asks “what other particularly notable things are there to tell about the ash?” High says there is quite a lot to tell about. High continues that an eagle sits on the branches of Yggdrasil and that it has much knowledge. Between the eyes of the eagle sits a hawk called Veðrfölnir. A squirrel called Ratatoskr scurries up and down the ash Yggdrasil carrying “malicious messages” between the eagle and Níðhöggr. Four stags named Dáinn, Dvalinn, Duneyrr, and Duraþrór run between the branches of Yggdrasil and consume its foilage. In the spring Hvergelmir are so many snakes along with Níðhöggr “that no tongue can enumerate them”. Two stanzas from Grímnismál are then cited in support. High continues that the norns that live by the holy well Urðarbrunnr each day take water from the well and mud from around it and pour it over Yggdrasil so that the branches of the ash do not rot away or decay. High provides more information about Urðarbrunnr, cites a stanza from Völuspá in support, and adds that dew falls from Yggdrasil to the earth, explaining that “this is what people call honeydew, and from it bees feed”.[21]

In chapter 41, the stanza from Grímnismál is quoted that mentions that Yggdrasil is the foremost of trees.[22] In chapter 54, as part of the events of Ragnarök, High describes that Odin will ride to the well Mímisbrunnr and consult Mímir on behalf of himself and his people. After this, “the ash Yggdrasil will shake and nothing will be unafraid in heaven or on earth”, and then the Æsir and Einherjar will don their war gear and advance to the field of Vígríðr. Further into the chapter, the stanza in Völuspá that details this sequence is cited.[23]

In the Prose Edda book Skáldskaparmál, Yggdrasil receives a single mention, though not by name. In chapter 64, names for kings and dukesare given. “Illustrious one” is provided as an example, appearing in a Christianity-influenced work by the skald Hallvarðr Háreksblesi: “There is not under the pole of the earth [Yggdrasil] an illustrious one closer to the lord of monks [God] than you.”[24]

[edit]Theories

This large tree in the Viking AgeÖverhogdal tapestries may be Yggdrasil with Gullinkambi on top.[25]

[edit]Shamanic origins

Hilda Ellis Davidson comments that the existence of nine worlds around Yggdrasil is mentioned more than once in Old Norse sources, but the identity of the worlds is never stated outright, though it can be deduced from various sources. Davidson comments that “no doubt the identity of the nine varied from time to time as the emphasis changed or new imagery arrived”. Davidson says that it is unclear where the nine worlds are located in relation to the tree; they could either exist one above the other or perhaps be grouped around the tree, but there are references to worlds existing beneath the tree, while the gods are pictured as in the sky, a rainbow bridge (Bifröst) connecting the tree with other worlds. Davidson opines that “those who have tried to produce a convincing diagram of the Scandinavian cosmos from what we are told in the sources have only added to the confusion”. [26]

Davidson notes parallels between Yggdrasil and shamanic lore in northern Eurasia:

[…] the conception of the tree rising through a number of worlds is found in northern Eurasia and forms part of the shamanic lore shared by many peoples of this region. This seems to be a very ancient conception, perhaps based on the Pole Star, the centre of the heavens, an the image of the central tree in Scandinavia may have been influenced by it […]. Among Siberian shamans, a central tree may be used as a ladder to ascend the heavens […].[26]

Davidson says that the notion of an eagle atop a tree and the world serpent coiled around the roots of the tree has parallels in other cosmologies from Asia. She goes on to say that Norse cosmology may have been influenced by these Asiatic cosmologies from a northern location. Davidson adds, on the other hand, that it is attested that the Germanic peoples worshiped their deities in open forest clearings and that a sky god was particularly connected with the oak tree, and therefore “a central tree was a natural symbol for them also”.[26]

[edit]Mímameiðr, Hoddmímis holt and Ragnarök

Líf and Lífþrasir after emerging from Hoddmímis holt (1895) by Lorenz Frølich

Connections have been proposed between the wood Hoddmímis holt (Old Norse “Hoard-Mímir‘s”[27] holt) and the tree Mímameiðr (“Mímir’s tree”), generally thought to refer to the world tree Yggdrasil, and the spring Mímisbrunnr.[27] John Lindow concurs that Mímameiðr may be another name for Yggdrasil and that if the Hoard-Mímir of the name Hoddmímis holt is the same figure as Mímir (associated with the spring named after him, Mímisbrunnr), then the Mímir’s holt—Yggdrasil—and Mímir’s spring may be within the same proximity.[28]

Carolyne Larrington notes that it is nowhere expressly stated what will happen to Yggdrasil during the events of Ragnarök. Larrington points to a connection between the primordial figure of Mímir and Yggdrasil in the poem Völuspá, and theorizes that “it is possible that Hoddmimir is another name for Mimir, and that the two survivors hide in Yggdrasill.”[29]

Rudolf Simek theorizes that the survival of Líf and Lífþrasir through Ragnarök by hiding in Hoddmímis holt is “a case of reduplication of the anthropogeny, understandable from the cyclic nature of the Eddic escatology.” Simek says that Hoddmímis holt “should not be understood literally as a wood or even a forest in which the two keep themselves hidden, but rather as an alternative name for the world-tree Yggdrasill. Thus, the creation of mankind from tree trunks (Askr, Embla) is repeated after the Ragnarǫk as well.” Simek says that in Germanic regions, the concept of mankind originating from trees is ancient. Simek additionally points out legendary parallels in a Bavarian legend of a shepherdwho lives inside a tree, whose descendants repopulate the land after life there has been wiped out by plague (citing a retelling by F. R. Schröder). In addition, Simek points to an Old Norse parallel in the figure of Örvar-Oddr, “who is rejuvenated after living as a tree-man (Ǫrvar-Odds saga 24–27)”.[30]

[edit]Warden trees, Irminsul, and sacred trees

A tree grows atop Mysselhøj,
Nordic Bronze Age burial mound inRoskildeDenmark.

Continuing as late as the 19th century, warden trees were venerated in areas of Germany and Scandinavia, considered to be guardians and bringers of luck, and offerings were sometimes made to them. A massive birch tree standing atop a burial mound and located beside a farm in western Norway is recorded as having had ale poured over its roots during festivals. The tree was felled in 1874.[31]

Davidson comments that “the position of the tree in the centre as a source of luck and protection for gods and men is confirmed” by these rituals to Warden Trees. Davidson notes that the gods are described as meeting beneath Yggdrasil to hold their things, and that the pillars venerated by the Germanic peoples, such as the pillar Irminsul, were also symbolic of the center of the world. Davidson details that it would be difficult to ascertain whether a tree or pillar came first, and that this likely depends on if the holy location was in a thickly wooded area or not. Davidson notes that there is no mention of a sacred tree at Þingvellir in Iceland yet that Adam of Bremen describes a huge tree standing next to the Temple at Uppsala in Sweden, which Adam describes as remaining green throughout summer and winter, and that no one knew what type of tree it was. Davidson comments that while it is uncertain that Adam’s informant actually witnessed that the tree’s type is unknown, the existence of sacred trees in pre-Christian Germanic Europe is further evidenced by records of their destruction by early Christian missionaries, such as Thor’s Oak by Saint Boniface.[31]

Ken Dowden comments that behind Irminsul, Thor’s Oak in Geismar, and the sacred tree at Uppsala “looms a mythic prototype, an Yggdrasil, the world-ash of the Norsemen”.[32]

[edit]Modern influence

The world ash Ygdrasil (as Richard Wagner spelled it) appears in the ominous opening scene of Götterdämmerung, in which the three Norns tell how Wotan had long ago broken off a branch to fashion himself the spear that gave him mastery over men and gods, and in which Wotan soon comes to wake Erda, Mother Earth, from her sleep with urgent questioning.

Modern works of art depicting Yggdrasil include Die Nornen (painting, 1888) by K. Ehrenberg; Yggdrasil (fresco, 1933) by Axel Revold, located in the University of Oslo library auditorium in OsloNorwayHjortene beiter i løvet på Yggdrasil asken (wood relief carving, 1938) on the Oslo City Hall by Dagfin Werenskjold; and the bronze relief on the doors of the Swedish Museum of National Antiquities (around 1950) by B. Marklund in Stockholm, Sweden. Poems mentioning Yggdrasil include Vårdträdet by Viktor Rydberg and Yggdrasill by J. Linke.[33]

Many modern video games or otherwise fantasy themed books or role-playing games have included references or versions of Yggdrasil. One such reference is in the Dragon Quest series; a gigantic tree known as Yggrasil stands in the midst of a desert in the games. Resurecting items known as “Yggdrasil Leaves” return life to fallen characters, as well as “Yggdrasil Dew” which heals all characters. Another such game to feature Yggdrasil is the Warcraft series; the night elven capital is based in a giant tree called Teldrassil, often referred to as “world tree”.

[edit]See also

Futher Reading:

http://odinsvolk.ca/O.V.A.%20-%20COSMOLOGY.htm

Yggdrasil – The home cosmic ‘Tree of Life’ that binds the universe together

Asgard – The home to the Gods and Goddesses of the Aesir
Alfheim – The realm where the Light Elves dwell
Vanaheim – The home to the Gods and Goddesses of the Vanir
Midgard – The home of the mankind
Jotunheim – The realm where the Giants dwell
Muspellheim – The world of primal fire where the Muspilli Fire Giants dwell
Niflheim – The “world of mists” and primal ice
SvartAlfheim – The realm where the Black Elves / Dwarfs dwell
Hel – The land of the dead, ruled by the goddess Hella.

The  S A C R E D   C O S M O L O G Y  of

A S A T R U

The ancient Northern Europeans did not see a simple universe with a heaven above and a hell below. Instead they saw a complex system of multiple planes and enclosures interconnected with our own. According to the ancient Eddas, these planes or worlds were born when the realm of fire, Muspelheim, in the South moved north to meet the icy realm of Niflheim in the North. They met in what is known as the Ginnungapap “the yawning void.” From this union sprang forth two beings Ymir the primeval giant and Audhumla, the giant primeval cow. By licking the ice, Audhumla made a new being appear, Buri. From Buri sprang Borr who married Bestla, who gave birth to Odin, Villi and Ve. These three brother Gods slew Ymir and from him created the Nine Worlds and the World Tree that supports the worlds. Although the Nine Worlds are linked by the World Tree, they by no means lie near each other, for there are hills, valleys, mountains, and even rivers between them formed by the bark of the tree. Beyond the Nine Worlds are unknown worlds resting in the Útgard “that outside the enclosure”.

FUTHER READING : http://www.native-science.net/Creation.Myth.Norse.htm

THE NORSE CREATION MYTH – UP TO DATE

By comparing the old mythological explanation to modern Astronomy and Cosmology, it seems for me probable to show a common knowledge, which, for the first part, is experienced by physically and spiritually senses and for the other part is knowledge based on modern instrumental Astronomical and Cosmological measurements.

If such a comparison is successful, a great implication must be: The human spirit is able to gain Cosmological knowledge about the Earth, the Solar system and the Galaxy without the use of any instruments. And it is my claim that there is a big different between getting Cosmological knowledge spiritually and instrumentally. Instrumental measurements and knowledge tends to bond to the Linear World picture and the spiritual knowledge to the Circular World picture. And, although the modern scientists of course can be very ecstatic and emotional when new knowledge is gained, the spiritual way of experiencing the Cosmos creates a great respect for the creative forces, which gained intuitively, gives the basic understanding how to live concordantly with the creative forces on the Earth and outside the Earth.

The 3-fold common knowledge in 1

The (Accumulated) Norse mythological family.

LAYERS UPON LAYERS OF STORY TELLING

When you are dealing with the Norse mythology, you must have in mind that, except from the Creation Myth itself, there is many layers of telling added throughout the time. Gods and Goddesses have been described from one time to another, and maybe the Gods and Goddesses from other cultures have been added with the increasing movements of populations. In order to distinguish and categorize the Gods and Goddesses, one must concentrate on their specific and common attributes. In this matter, the Comparative Mythology and Religion is of great importance to study.

And it is astoundingly firstly how identical the Story of Creation is told all over the World and secondly how identical the Gods and Goddesses are described also all over the World. Globally the stories are very similar! And only local conditions use some different animals – and anthropomorphic beings – to describe the meaning of the different myths. Of course there is this common knowledge! We all live on the same planet under the same sky under the same cosmological conditions!

THE ATTEMPT OF A MODERN INTERPRETATION OF THE NORSE CREATION MYTH STARTS HERE:

(The Mythological Story itself in blueprint – comments and explanations in ordinary print)

GINNUNGAGAB:

The Story of Creation in the Norse Mythology begins in the great emptiness, called Ginnungagap.

This opening of the story can immediately be compared to the modern theory of Big Bang: “Before there was something, there was nothing”! Or: “Out of nothing emerged everything”.

Both explanations should of course not been taking literally – and, in my opinion, both telling should be understood as a “technique” to explain the basics of how creative forces merges the material and later on expands in cosmos in a rhythmic and cyclic movement.

THE BASIC ELEMENTS OF CREATION

1 is the number for Everything. 2 for Light and Matter, for Warmth and Cold and for expansion and contraction. 3 is for the combined worlds of the Earth, the Solar System and our Galaxy. 4 is for Air, Fire, Water and Soil. 8 is for the interaction of the 4 elements. By this description one can easily imagine even modern atomic principles which is used in many moderns scientific branches. And just think about how the weather changes and interacts throughout the seasons.

THE STORY OF CREATION

We set the number in (1 = Everything) and begin the Norse Story of Creation with the 2 basic qualities:

In the warm Muspelheim in the South, sparkles and glowing embers from Fire are flying out and spreads towards the northern part in Ginnungagap, Niflheim, where darkness and coldness has deep-frozen all matter.

Now: In order to describe the originally creative and characteristically powers, one can only describe the movement and progression of these powers in familiar terms. And the familiar terms comes from the seasonal changes of the Year. And when using the seasonal changing’s as a story telling technique, our ancestors also have described that “everything above is like below”. That is: The same forces and laws works both in Macro Cosmos and Micro Cosmos.

– The fire, light and warmth from Muspelheim meets the frozen matter from Niflheim in “the middle of Ginnungagap = “the centre where Creation in our Galaxy” began.

The frozen matter gets warmer and moist shrouds everything. Using the terms of modern Astrophysics, the “cold+moist” and the “hot+dry” directly can be interpreted as Hydrogen and Helium. When two hydrogen atoms collides, Helium is created releasing light (let there be light) and warmth and thereby accelerate the matter of the Creation.

In the course of this events, the Story concerns the very basics of Creation: When cold and warm matter meets and sets of a beginning of movement.

Out from the middle of Ginnungagap grows the cow Audhumbla, the first symbol of Creation. From Audhumbla´s udder floats rivers of milk.

Why “rivers of milk”? Because the color of the Milky Way is white. And that´s why a Cow is such a great symbol of Creation – of course together with many other symbols.

A Rock Carving from Norway and Turkey, both marking the centre of the Milky Way.

An Egyptian female, The Great Mother, radiating matter from the Womb compared to a Star Atlas with the southern contours of the Milky Way and the centre of the Milky Way marked with an inserted Spiral.

From the centre in “Audhumbla´s womb floats the rivers out and gives nourishment to the Giant Ymer, the second symbol of Creation.

On both sides of the Egyptian picture in the middle, the northern and the southern contours of our Milky Way. The Egyptian female picture are covered with Stars which clearly tells us that she have something to do with the Night Sky, and her name is Nut, queen of the Night and she is the Mother Goddess. If the left Star Atlas picture is placed under the right Atlas picture, we have the very same motif and meaning as on the Egyptian mythological picture.

Ymer drank of the 4 rivers of Milk and, while sleeping, 2 human beings, one woman and one man, grow out of his arm pits. And out from the giant Ymer, the whole Sky and World was created.

From the Danish Wessel of the Gundestrup Cauldron: The giant Ymer holding the first 2 humans, who are symbolized above in the Star Atlas pictures and on the Egyptian picture.

From Muspelheim in the south came more sparks of light which created the Stars, the Sun and the Moon. This indicates very strongly that our Solar System is created in the Centre of our Galaxy.

THE 3 WORLD DIMENSIONS

The number 3 symbolizes the 3 dimensions in the Norse Mythology, namely Midgaard where humans live, Asgaard above with Star Constellation fantasy pictures of both human-  and animal like beings. The third dimension, Udgaard, belongs to all giant beings directly connected to the fantasy pictures of the Milky way contours on the northern and the southern hemisphere. Every of these 3 dimensions or Worlds was mythological divides up in 3 subdivisions which gives the holy number of 9 Worlds in which for instants Odin and Balder traveled in order to gain knowledge from all dimensions.

The schematic drawing in the middle show the 3 dimensions or 3 Rings of worlds. Number 1 is your location on Earth, 2 is the Earth itself, 3 is the Star Heaven and 4 with its grey/white vaulting band on the night Sky. The Rock Carving pictures are from Ireland and Sweden.

This Rock Art Carving was found 2009.10.10 at the location of “Anebjerg” on the Northern part of the Baltic Island of Bornholm, Denmark, very close to my location.
In the Rings, dots are engraved as symbols of the Sun, Moon, Planets and Stars encircling the Day- and Night Sky on the 3 dimensions or Worlds in the Norse Mythology. Such Cup Marks carved in the rings, are not that common. The figures in the image are symbolizing the revolving contours of the Milky Way. (Location explanation: “Anebjerg = “Ane” = Danish for “Ancient/forefathers/foremothers” and “bjerg” = Danish for mountain, the mythological archetype of the Primeval Mound, “the place where the Human live and rise from”, the center of the Milky Way Galaxy.

THE WORLD TREE AND ITS CREATIONAL MOVEMENTS

In the middle of the World stands a tall ash tree, called Yggdrasil. Its crown reach up in Heaven and its roots stands in the Underworld. (Originally, the Tree of Life meant the the Galaxy Tree in the centre of our Galaxy, but the Tree of Life is also symbolized as the cooperative forces between the Sun and the Earth as told below:)

There have been many attempts to describe how our Nordic ancestors have imagined their perception of the World.

A Rock Art carving from Sweden shoving 2 Trees and the whole “Noah Arch” or the Norse Mythology great Ship “Skibladnir” sailing on the Heavenly Oceans. (http://www.native-science.net/Ship.Mythical.htm)

The Story of the World Tree is about how the creative forces works throughout the seasonal changes on the Earth. The giant tree Yggdrasil is standing in the middle of the Earth with its stem going TROUGH the Earth axis and its crown and roots spreading out in the Earth atmosphere! All the mythological animal figures mentioned in connection with the Ash Yggdrasil, are Star Constellations or Milky Way figures. Except from one special animal, namely the squirrel “Ratatosk”

The story of the World Tree is specifically dealing with the powers that works inside and outside the Earth throughout the seasons. It’s first and foremost about how the geomagnetic forces inside and outside the Earth is working and how this force is creating all vegetable life on Earth. And it’s about the Sun influences on the Earth magnetic fields both in a day and while the Earth is orbiting the Sun.

In the seasonal changes, the creative geomagnetic power increase and decrease because of the Sun radiation influence on the Earth magnetic fields daily and annually when the Earth axis leans away and towards the Sun throughout the season – and this qualitative changes goes both ways, “up and down” and of course it creates opposite seasonal phenomenon’s on the northern and the southern hemisphere at the same time. And the geomagnetic forces creates both the vegetable tops and the roots trough the Earth daily and seasonal movement.

In the Spring time you can observe the soil damping. It’s not only because of the Sun warming up the soil. Long before the Sun have a warming power the warmth of the geomagnetic force have slowly warmed up the soil deep within the ground. And if you cut a scratch or score in a tree stem some time before spring, you can observe how early the sap runs some time before the Sun have any greater warmth effect. This indicates an increasing geomagnetic  pressure from within the Earth up trough all vegetable matters. A geomagnetic pressure that increase and decrease in the seasons. Up and down – up and down. Just as the squirrel “Ratatosk” in the Norse Mythology.

Ratatosk is a squirrel running up and down on the Ash Yggdrasil world tree. Ratatosk can freely move between the worlds of ice in Niflheim and the world of fire in Muspelheim. Ratatosk brings the words from the Eagle in the top of Yggdrasil to the snake Nidhug below in the roots of Yggdrasil. Ratatosk talks with everybody in the 3 worlds or dimensions. Ratatosk brings news between all in the 3 worlds or dimensions. And there is a constantly fight between the Eagle in the crown of Yggdrasil and the Snake in the roots of Yggdrasil.

The squirrel Ratatosk is the specific symbol of the changing Geomagnetic Force itself. The story of Ratatosk is a fantastic precise construction of describing the creative process throughout the seasonal changes in the increasing and decreasing Geomagnetic creative force and as specific description of all vegetable growth.

RAGNAROK – IS NOT THE END OF THE WORLD

In a world described in circles, it’s NOT very likely that our Norse ancestors have a perception of a total end of the World! The story of Ragnarok is just a simple story of everything in life. Of Star Constellation figures are moving throughout the day and seasons. How everything grows and vanish. Of birth and dead – all the cyclic phenomenon’s we humans can experience in a lifetime as well in bigger cyclic periods beyond our life.

– With this description I provisionally conclude the story of the Creation in the Norse Mythology – I hope you now are open for this modern attempt of interpretation of the old story! For my own part, I’m sure one can find similar connections and explanations between old and modern facts in every cultural Story of Creation all over the World.

It’s just a matter of looking at the old stories and symbols in a new way and connect these to moderns scientific fact from Astronomy and Cosmology.

Link to an alternative Cosmology: http://www.cosmology-unified.net/Cosmology.Holistic.htm


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