Counting Truchet Tile Balls
This note explains how to count the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.
math.GM↗
arXiv subjects
Thomas Fernique
Publications and source records attributed to Thomas Fernique.
Telstar Balls Gone Wild
We describe an artistic project consisting of fabricating the 3532 different soccer balls that can be obtained by randomly assembling the 32 pieces of a classic Telstar soccer ball.
math.GM↗
How not to scramble a Rubik's cube
We model the scrambling of a Rubik's cube by a Markov chain and introduce a stopping time $T$ which is a quite natural candidate to be a strong uniform time. This may pave the way for estimating the number of moves required to scramble a cube. Unfortunately, we show that $T$ is not this is not strongly uniform.
math.PR↗
On Dense Tetrahedra in Binary Sphere Packings
This paper considers the density of tetrahedra arising in a specific decomposition of packings of unequal spheres in $\mathbb{R}^3$. It aims to extend a bound obtained in 2D in the 1960s by Florian. The focus is on packings of spheres of sizes $1$ and $\sqrt{2}-1$: the small sphere fits exactly into each octahedral hole of a hexagonal close packing of large spheres, yielding a conjecturally maximally dense packing (for these sizes). The paper slightly improves, by completely different means, the previous best upper bound on the density of such packings. The proof combines geometric insight with challenging interval arithmetic computations, which may be of independent interest.
math.MG↗
Square-Triangle Tilings: Lift & Flip to Sample?
We introduce an elementary transformation called flips on tilings by squares and triangles and conjecture that it connects any two tilings of the same region of the Euclidean plane.
cs.DM↗
Shield tilings
We provide a complete description of the edge-to-edge tilings with a regular triangle and a shield-shaped hexagon with no right angle. The case of a hexagon with a right angle is also briefly discussed.
math.CO↗
Packing unequal disks in the Euclidean plane
A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.
math.MG↗
Geometrical Penrose Tilings are characterized by their 1-atlas
Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such that the decoration match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by two different method: one based on the substitutive structure of the Penrose tilings and the other on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folk, no complete proof of this result has been published, to our knowledge.
cs.DM↗
Density of triangulated ternary disc packings
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 15 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 40 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.
cs.CG↗
Ammann Bars for Octagonal Tilings
Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tiles, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.
cs.DM↗
Purely local growth of a quasicrystal
Self-assembly is the process in which the components of a system, whether molecules, polymers, or macroscopic particles, are organized into ordered structures as a result of local interactions between the components themselves, without exterior guidance. In this paper, we speak about the self-assembly of aperiodic tilings. Aperiodic tilings serve as a mathematical model for quasicrystals - crystals that do not have any translational symmetry. Because of the specific atomic arrangement of these crystals, the question of how they grow remains open. In this paper, we state the theorem regarding purely local and deterministic growth of Golden-Octagonal tilings. Showing, contrary to the popular belief, that local growth of aperiodic tilings is possible.
cs.DM↗
Density of binary disc packings: lower and upper bounds
We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good idea of the exact maximum density. In particular, we get new intervals for $r$ which does not allow any packing more dense that the hexagonal packing of equal discs.
math.MG↗
Compact Packings are not always the Densest
We provide a counterexample to a conjecture by B. Connelly about density of circle packings
math.MG↗
Density of Binary Disc Packings: Playing with Stoichiometry
We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are rigorously determined via a computer-assisted proof. The density is maximal for the 1:1 stoichiometry: the large discs then form a square grid in each interstitial site of which a small disc nests. When there is an excess of large discs, the densest packings are made of a single phase which mixes the two types of discs in a chaotic way (it can be described by square-triangle tilings). When there is an excess of small discs, on the contrary, a phenomenon of phase separation appears: the large discs are involved in the densest 1:1 stoichiometry phases while the excess of small discs form compact hexagonal phases.
cs.DM↗
Density of Binary Disc Packings: The 9 Compact Packings
A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).
cs.DM↗
Compact packings of the plane with three sizes of discs
A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of size $1$. There are exactly $9$ values of $r$ which allow a compact packing by discs of sizes $1$ and $r$. We prove here that there are exactly $164$ pairs $(r,s)$ allowing a compact packing by discs of sizes $1$, $r$ and $s$.
cs.DM↗
Compact packings of space with three sizes of spheres
A sphere packing of the three-dimensional Euclidean space is compact if it has only tetrahedral holes, that is, any local maximum of the distance to the spheres is at equal distance to exactly four spheres. This papers describes all the compact packings with spheres of three different sizes. They are close-compact packings of unit spheres with holes filled in four different ways by smaller spheres.
math.MG↗
A Densest ternary circle packing in the plane
We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually tangent circles. Compact packings are believed to maximize the density when there are possible. We prove that it is indeed the case for these sizes. The proof should be generalizable to other sizes which allow compact packings and is a first step towards a general result.
cs.CG↗