Category Archives: Uncategorized

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

Thomas Blok and David Madore have recently made significant progress on the problem of finding rational dodecahedra inscribed in the unit sphere, culminating in an infinite parametric family of solutions. In particular, Thomas began with the constrained version of the … Continue reading →