Related papers: Aperiodic sets of three types of convex polygons

For a Euclidean building $X$ of type $A_{2}$, we classify the 0-dimensional subbuildings $A$ of $\partial_{T}X$ that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a…

In 2023, the quest for an aperiodic monotile was answered by the hat monotile. In this article, structures in this aperiodic tiling are discovered, which allow for a direct computation of the tiling, similar to well-known methods for the…

We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems -- we give an example of…

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…

A periodic surface is one that is invariant by a 2D lattice of translations. Deformation modes that stretch the lattice without stretching the surface are effective membrane modes. Deformation modes that bend the lattice without stretching…

A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible…

We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…

Periodic trees are combinatorial structures which are in bijection with cluster tilting objects in cluster categories of affine type $\tilde{A}_{n-1}$. The internal edges of the tree encode the $c$-vectors corresponding to the cluster…

This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar…

We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…

The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of…

Atkinson [2] found a sequence of three-dimensional hyperbolic polyhedra whose dihedral angles are $\pi /3$. In this paper, we construct another sequence of such polyhedra. We also determine the volumes of some of these polyhedra.

We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.

It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled) polyhedron based on a rectangle that meets every vertical line in a segment, has a grid unfolding: its surface may be unfolded to a single non-overlapping piece…

Suppose $f\in L^1(\mathbb{R}^d)$, $\Lambda\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles,…

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans…

We give a complete classification of edge-to-edge tilings of the sphere by regular polygons under a unified framework. Without assuming convexity of the tiles or polyhedrality of the underlying graph, our proof is independent of the…

We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences…

Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not…

In this paper it is proved that there exist periodic monohedral tilings and finite seeds of colored tiles, which force non-periodic coloring of the whole plane