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Monthly Archives: October 2020
The exceptional Jordan algebra
In the early 1930s, Pascual Jordan attempted to formalise the algebraic properties of Hermitian matrices. In particular: Hermitian matrices form a real vector space: we can add and subtract Hermitian matrices, and multiply them by real scalars. That is to … Continue reading
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Closedform numbers
Recall that the logarithm function (defined on the punctured complex plane) is multivalued, with the different solutions to exp(x) = y differing by integer multiples of 2πi. James Propp remarked that the values of the form i^i^i are a … Continue reading
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The BarnesWall lattices
For any nonnegative integer n, there exists the highly symmetric BarnesWall lattice in dimension . In low dimensions, these are (up to scaling and rotation) familiar lattices: For n = 0, this is just the integer lattice, . For n … Continue reading
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Subsumptions of regular polytopes
We say that a regular ndimensional polytope P subsumes a regular ndimensional polytope Q if the vertexset of Q is geometrically similar to a subset of the vertexset of P. For instance, the dodecahedron subsumes a cube (the convex hull … Continue reading
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