Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and the Stark-Heegner theorem on imaginary quadratic number fields of class number one; see for a survey of properties.
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Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternion that is nowadays often called an octonion, and showing that they were what we now call associative, which may have been the invention of the concept.
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In particular, by using the fact that the automorphism group of the split octonions is the split algebraic group G 2, one sees the correspondence of isomorphism classes of octonion " F "-algebras with isomorphism classes of G 2-torsors over " F ".
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For examples, ( a ) the Dirac equation in physics ( the equation of motion of a free spin 1 / 2 particle, like e . g . an electron or a proton ) can be expressed on native split-octonion arithmetic, ( b ) the supersymmetric quantum mechanics has an octonionic extension.
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