A 720° rotation is needed to return where we started (see e.g. MTW Ch 41.5):pic.twitter.com/Zz5p0hd18c
𝐄𝐮𝐥𝐞𝐫'𝐬 𝐈𝐝𝐞𝐧𝐭𝐢𝐭𝐲 tells us how to rotate a complex number 𝑧 around the origin.
The 𝐏𝐚𝐮𝐥𝐢 𝐈𝐝𝐞𝐧𝐭𝐢𝐭𝐲 tells us how to rotate a 2 component 𝒗𝒆𝒄𝒕𝒐𝒓 of complex numbers, called a spinor, around an axis θ̂.
The {σᵢ} are known as the Pauli matrices .pic.twitter.com/XJ0qbJMhDI
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The coolest visualization I ever saw of it was on a mobius strib
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Geometric algebra let's us obtain coordinate free expressions that do this! There's a nice way to rotate vectors and sub spaces by an arbitrary plane and radian using the exponential of "a+b e1 /\ e" and the geometric product.
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Sorry I misspoke. Let v1 and v2 span the plane you want to rotate in an let v3 be a blade you want to rotate by an angle t. Let * be the geometric product. For shortness say B = -(v1/\v2)/(v1/\v2)^2 Then the precise formula is exp(-B t/2) * v3 * exp(B t/2).
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The Pauli matrices are (of course) very important in quantum mechanics. The Pauli Identity is analogous to Euler's formula (as you point out Berger) extended to quaternions.
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