Author Archives: apgoucher

Shallow trees with heavy leaves

Posted on December 13, 2020 by apgoucher

There are two very different state-of-the-art chess engines: Stockfish and Leela Chess Zero. Stockfish searches many more positions (100 000 000 per second) and evaluates them using computationally cheap heuristics. The tree search methodology is a refinement of alpha-beta pruning. … Continue reading →

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Let the circumcentre be your origin

Posted on December 3, 2020 by apgoucher

Suppose we have two vectors, u and v, in a Euclidean vector space. If we wanted to somehow quantify the proximity of these two vectors, there are two particularly appealing choices: the squared distance, |u − v|²; the inner product, … Continue reading →

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An attempt to understand the Monster group

Posted on November 29, 2020 by apgoucher

The Monster group is very large, very complicated, and very mysterious. According to the Classification of Finite Simple Groups that was completed last century, the Monster group is the largest of only 26 finite simple groups that do not fit … Continue reading →

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The exceptional Jordan algebra

Posted on October 28, 2020 by apgoucher

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

Posted on October 17, 2020 by apgoucher

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

Posted on October 10, 2020 by apgoucher

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

Posted on October 1, 2020 by apgoucher

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

Posted on September 29, 2020 by apgoucher

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

Posted on August 19, 2020 by apgoucher

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

Posted on August 5, 2020 by apgoucher

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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