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 The exceptional Jordan algebra  Complex Projective 4Space on Pappian and Desarguesian planes
 Una Construcción Nueva: Definición – Mengenlehre on Von Neumann universe
 jasonhise64 on The BarnesWall lattices
 The BarnesWall lattices  Complex Projective 4Space on Minimalistic quantum computation
 The BarnesWall lattices  Complex Projective 4Space on Subsumptions of regular polytopes
Author Archives: apgoucher
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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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 exclusiveor 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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BanachTarski 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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Fastgrowing functions revisited
There have been many exciting results proved by members of the Googology wiki, a website concerned with fastgrowing functions. Some of the highlights include: Wythagoras’s construction of an 18state Turing machine which takes more than Graham’s number of steps to … Continue reading
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