(This complication is one of the most difficult aspects of Listing's law to understand, but it follows directly from the non-commutative laws of physical rotation, which specify that one rotation followed by a second rotation does not yield the same result as these same rotations performed in the inverse order.)
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His 1902 paper in Mathematische Annalen recounts William Kingdon Clifford's construction of his 2n dimensional algebra with n − 1 anti-commuting square roots of −1.