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Category Archives: Uncategorized
A curious construction of the Mathieu group M11
Previously, we discussed which regular polytopes have vertexsets that occur as proper subsets of the vertexset of another regular polytope in the same dimension. In particular, when there is a Hadamard matrix of order 4k, then then the (4k−1)dimensional simplex … Continue reading
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Assorted topics
This is a digest of things that have happened this month, but which are individually too small to each warrant a separate cp4space post. Firstly, there have been a couple of exciting results in the field of combinatorics: The ErdősFaberLovász … Continue reading
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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 nowheredense sets. Both … Continue reading
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Rigid heptagon linkage
In January 2000, Erich Friedman considered the problem of finding a rigid unitdistance 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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BarretoNaehrig 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 stateoftheart 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 alphabeta 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 − v²; 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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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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