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on which he is whirled by winds through the air;

In the greek wikipedia he is turning around tied with snakes on a burning wheel in Tartarus, which is a punishment place entered through some caves .

In Perseus one finds the following quote in an article on Apollodorus

According to some, the wheel of Ixion was fiery (Scholiast on Eur. Ph. 1185); according to the Vatican Mythographer it was entwined with snakes. The fiery aspect of the wheel is supported by vase paintings. From this and other evidence Mr. A. B. Cook argues that the flaming wheel launched through the air is a mythical expression for the Sun, and that Ixion himself “typifies a whole series of human Ixions who in bygone ages were done to death as effete embodiments of the sungod.”

This is some possible symbolism as the sun wheel is found in several mythologies.

Why this punishment? Greek mythology is full of horrible punishments for hubris, see what happened to Prometheus.

In the book Zeus: A Study in Ancient Religion, Volume 3, Part 2, A,B.Cook I screen captured the amphora depicting the torture of Ixion

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The light source is definitely sun like, reflections in the water. The wheel is on fire, but not with the strong light of the sun. It could be the artist is thinking "close enough to the sun to burn to a crisp forever". As the modern wish "may you burn in hell" . It is consistent with the Ikaros myth ( the closer to the sun the more heat) .