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Chaim Goodman-Strauss

Publications and source records attributed to Chaim Goodman-Strauss.

13 recordsLinked to original sources
arXiv2026

The {\em 4DLO} and other tubing models of $S^3$ symmetry

The {\em Four-dimensional Light Orchestra} or {\em 4DLO} was an interactive sculpture at the National Museum of Mathematics (MoMath) from November 20, 2025 through January 2026, illustrating various sub-symmetries of the 24-cell with colored lights. This was part of a larger sequence of tubing sculptures aiming to bring to life a few lines of tables appearing in~\cite{conwayandsmith}, reprinted in~\cite{sot}, and further illuminated in~\cite{rastanawi}. Best of all museum patrons could manipulate {\em 4DLO}'s lighting by singing and making funny noises into a microphone, and they did so with gusto. Here we describe some of the technical aspects of this sculpture and its context.

math.HO↗
arXiv2024

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

math.CO↗
arXiv2024

Remark on Pascal's Triangle

Through a series of elementary exercises, we explain the fractal structure of Pascal's triangle when written modulo $p$ using an 1852 theorem due to Kummer: A prime $p$ divides $\dfrac {n!}{i!j!} $ if and only if there is a carry in the addition $i+j=n$ when written in base $p$.

math.HO↗
arXiv2024

A Woven Klein Quartic

We describe a new method of weaving a model of the Klein quartic, a highly symmetric, but abstract genus-3 surface akin to a platonic polyhedron, with negatively-curved geometry. The Klein quartic cannot be realized in its fully symmetric form in three-dimensional space, but this model exhibits the most rigid symmetry that is possible. With remarkably little time and material you can have a Klein quartic of your own!

math.CO↗
arXiv2023

A chiral aperiodic monotile

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.

math.CO↗
arXiv2017

Lots of Aperiodic Sets of Tiles

Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational applications, but its study and use has been hindered by a lack of easily generated examples. Through readily generalized, robust techniques for controlling hierarchical structure, we increase the catalogue of explicit constructions of aperiodic sets of tiles hundreds-fold, in lots, easily assembled and configured from atomic subsets of 211 tiles, enforcing 25,380 distinct "domino" substitution tiling systems. Among these, we notice three non-periodic, non-unique decomposition substitution tiling systems.

math.CO↗
arXiv2016

The Trilobite and Crab: a full explanation

The trilobite and crab are among the very simplest aperiodic sets of tiles known: two tiles in eight translation classes. Yet the proof that they are an aperiodic set is surprisingly complex.

math.CO↗
arXiv2016

Tessellations

A survey of tessellations, which appeared in Italian in La Matematica vol 3 in 2010. The Tiling Lemma appears for the first time.

math.HO↗
arXiv2015

Strongly aperiodic subshifts on surface groups

We give strongly aperiodic subshifts of finite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.

math.GR↗
arXiv2002

Cubic Polyhedra

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal surfaces (under an appropriate smoothing flow, keeping their symmetries). Here we give a complete classification of the cubic polyhedra. Among these are five new infinite uniform polyhedra and an uncountable collection of new infinite semi-regular polyhedra. We also consider the somewhat larger class of all discrete minimal surfaces in the cubic lattice.

math.MG↗