Category Archives: Uncategorized

Every finite phoenix has period 2

A phoenix is an oscillator in Conway’s Life where every cell dies in every generation. The smallest example is Phoenix 1, which oscillates with period 2 and has a constant population of 12: All known finite phoenices have period 2, … Continue reading

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Miscellaneous discoveries

Soon after the previous post announcing the discovery of an aperiodic monotile by Smith, Myers, Kaplan, and Goodman-Strauss, the same authors published a second aperiodic monotile which has the property that all of the tiles are of the same orientation: … Continue reading

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Aperiodic monotile

David Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss have discovered an aperiodic monotile: a polygon that tiles the plane by rotations and reflections, but cannot tile the plane periodically. Any tiling induced by the monotile is scalemic: the majority … Continue reading

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The Osmiumlocks Prime

A couple of years ago I described a prime p which possesses various properties that renders it useful for computing number-theoretic transforms over the field . Specifically, we have: where the first of these equalities uses the identity that: where rad(k) … Continue reading

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The ordered partial partition polytope

In the tensor rank paper we introduced a new family of axis-aligned n-dimensional polytopes, one for each positive integer n. The vertices are naturally identified with ordered partial partitions (OPPs) of {1, …, n}, and the edges correspond to converting … Continue reading

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Tensor rank paper

Robin Houston, Nathaniel Johnston, and I have established some new bounds on the tensor rank of the determinant over various fields. The paper is now available as an arXiv preprint and contains the following results: A new formula for the … Continue reading

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Matrix multiplication update

At the end of the recent post on a combinatorial proof of Houston’s identity, I ended with the following paragraph: This may seem paradoxical, but there’s an analogous situation in fast matrix multiplication: the best known upper bound for the … Continue reading

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Updates and errata

In the Treefoil article, I erroneously described John Rickard’s length-24 cycle in as being the ‘uniquely minimal’ example of a cycle whose three axis-parallel projections are all trees (see here for a more detailed history on this problem). Dan Simms … Continue reading

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A combinatorial proof of Houston’s identity

Robin Houston recently discovered a rather interesting formula for the determinant of an n-by-n matrix. In particular, the formula improves upon the best known upper bound for the tensor rank of the determinant (viewed as a multilinear map which takes … Continue reading

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Tetrational machines

A pair of people called Pavel have independently developed remarkable automata that last record-breakingly long before halting. In both cases, the number of timesteps that it takes for each automaton to halt is so large that it cannot be written … Continue reading

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