Left: the sun or sun god (Ra), arising out of a mound (pyramid), that arises out of the flood (Nun or Nu). Middle: a depiction of “Phoenix”, the name ascribed to the latter by Herodotus (c.450BC), in a Mosaic from Antioch (Antakya) (c.300AD), coming out of the primeval mound, surrounded by the chaotic waters of Nu. [1] Right: the modern idea of the phoenix is a “fire bird”.
In religio-mythology, Phoenix, called the "Bennu" (or benu bird), signified by the hieroglyphic:Bennu bird, by the Egyptians, is the name given by Herodotus (c.450BC) to the mythical bird, i.e. heron, hawk, or something resembling an eagle with red and gold feathers, according to Egyptian mythology, that carried the sun, sun disc, or sun god, depending on interpretation, balanced on its head each day, during its journey through the sky. [2]

In 3100BC, in Egyptian mythology, the predominate model of the origin of things, Heliopolis creation myth in particular, asserted that originally everything was a flood (called Nun or Nu), out of which a land mound arose, from which the sun (or sun god) burst forth carried by a bird on its head.

In c.435BC, Herodotus, in his The Histories (pg. 110-11), in his chapter on Egypt, in recounting the various animals he learned of and seen with his own eyes, e.g. crocodile, hippopotamus, otter, and eel, the gives the following account of the sun god Ra, which he was told was a sacred bird that rarely appears:

“There is also another sacred bird, called the phoenix, which I have never seen except in a picture; for it seldom makes its appearance among them, only once in five hundred years, as the Heliopolitans affirm: they say that it comes on the death of its sire. If he is like the picture, he is of the following size and description: The plumage of his wings is partly golden-colored, and partly red; in outline and size he is very like an eagle. They say that he has the following contrivance, which in my opinion is not credible. They say that he comes from Arabia, and brings the body of his father [god name] to the temple of the sun, having enclosed him in myrrh, and there buries him in the temple. He brings him in this manner: first he molds an egg of myrrh as large as he is able to carry; then he tries to carry it, and when he has made the experiment, he hollows out the egg, and puts his parent into it, and stops up with some more myrrh the hole through which he had introduced the body, so when his father is put inside, the weight is the same as before: then, having covered it over, he carries him to the temple of the sun in Egypt. This they say is done by this bird.”

Here, to note, Herodotus questionably believes that this "phoenix" was a real animal, that existed in Egypt, and not, as is correct, the story of the birth of sun, according to Egyptian mythology.

1. Ellis, Ralph. (2004). Eden in Egypt: Adam and Eve were Pharaoh Akhenaton and Nefertiti (Phoenix, pg. 169). Edfu Books, 2010.
2. Budge, Wallis. (1904). The Gods of the Egyptians, Volume Two (pg. 96). Dover, 1969.

Further reading
● Dolloff, Norman H. (1975). Heat Death and the Phoenix: Entropy, Order, and the Future of Man (figure 1.3, pg. 19; free energy, 27+ pgs; Gibbs, 9+ pgs; god; 3+ pgs; social, 9+ pgs). Exposition Press.

External links
Phoenix (mythology) – Wikipedia.

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