Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. 

Dirac spinor 

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Intricate Jigsaw Puzzle with Missing Piece

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Day of the Dead Woman with Golden Pistols

Dead wears many faces 

Living Literacy

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Chinese terracotta army of the first emperor, excavation site

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Dragon and Phoenix in Colorful Splash Art

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