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Meagre sets and null sets

There are two competing notions for describing a subset of the real numbers as being ‘small’: a null set is a subset of the reals with Lebesgue measure zero; a meagre set is a countable union of nowhere-dense sets. Both … Continue reading

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Rigid heptagon linkage

In January 2000, Erich Friedman considered the problem of finding a rigid unit-distance graph G containing a regular heptagon as a subgraph. That is to say, the graph is immersed in the plane such that: every edge of G must … Continue reading

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Barreto-Naehrig curves and cryptographic pairings

There’s a very elegant cryptographic construction discovered by Barreto and Naehrig in a 2005 paper. It is beautiful from a pure mathematical perspective, but also has an impressive application: it was* part of the ingenious mechanism by which zCash supports … Continue reading

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Shallow trees with heavy leaves

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

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

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

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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