Soon after getting resleeved Takashi gets an infinity 8 dragon tattoo. This symbol is also shown in the intro to Altered Carbon. What is the importance of this symbol? What did it come from and what does it mean?
The image is "Ouroborous"
The symbol is called Ouroboros, which denotes from the Greek word 'Oura' (tail), and Boros (eating). It's cyclical, and represents an infinite processs, which is of course analogous of the constant life, death, and rebirth process that takes place in Altered Carbon.
The ouroboros or uroborus is an ancient symbol depicting a serpent or dragon eating its own tail. Originating in ancient Egyptian iconography, the ouroboros entered western tradition via Greek magical tradition and was adopted as a symbol in Gnosticism and Hermeticism, and most notably in alchemy. Via medieval alchemical tradition, the symbol entered Renaissance magic and modern symbolism, often taken to symbolize introspection, the eternal return or cyclicality, especially in the sense of something constantly re-creating itself. It also represents the infinite cycle of nature's endless creation and destruction, life and death.