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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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Closed-form numbers

Recall that the logarithm function (defined on the punctured complex plane) is multi-valued, 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 Barnes-Wall lattices

For any nonnegative integer n, there exists the highly symmetric Barnes-Wall 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 n-dimensional polytope P subsumes a regular n-dimensional polytope Q if the vertex-set of Q is geometrically similar to a subset of the vertex-set of P. For instance, the dodecahedron subsumes a cube (the convex hull … Continue reading

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Relocation

In June of this year, I read Paul Graham’s essay on names. The author (whom you may know from his book On Lisp) begins with the following advice: If you have a US startup called X and you don’t have … Continue reading

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Associative universal gates

The Boolean function NAND is famously universal, in that any Boolean function on n inputs and m outputs can be implemented as a circuit composed entirely of NAND gates. For example, the exclusive-or operation, A XOR B, can be written … Continue reading

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Another two rational dodecahedra

Since finding one rational dodecahedron inscribed in the unit sphere, I decided to port the search program to CUDA so that it can run on a GPU and thereby search a larger space in a reasonable amount of time. Firstly, … Continue reading

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Banach-Tarski and the Axiom of Choice

Tomasz Kania and I recently coauthored a paper about Banach spaces. The paper makes extensive use of the axiom of choice, involving a transfinite induction in the proof of Theorem B as well as several appeals to the fact that … Continue reading

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Rational dodecahedron inscribed in unit sphere

Moritz Firsching asked in 2016 whether there exists a dodecahedron, combinatorially equivalent to a regular dodecahedron, with rational vertices lying on the unit sphere. The difficulty arises from the combination of three constraints: The twelve pentagonal faces must all be … Continue reading

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That group of order 348364800

In nested lattices, we talked about the E8 lattice and its order-696729600 group of origin-preserving symmetries. In minimalistic quantum computation, we saw that this group of 8-by-8 real orthogonal matrices is generated by a set of matrices which are easily … Continue reading

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