Exact Solution of an Octagonal Random Tiling Model

We consider the two-dimensional random tiling model introduced by Cockayne, i.e. the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eight-fold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing {\em exact} values of the entropy and elastic constants.Comment: 4 pages,3 Postscript figures, uses revte

Tiling the plane with equilateral triangles

Let T be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then T is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar. The same result has been obtained independently by Richter and Wirth

Lozenge tilings of a halved hexagon with an array of triangles removed from the boundary

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi recently extended this tiling enumeration to a halved hexagon with a triangle removed from the boundary. In this paper we prove a generalization of the results of Proctor and Rohatgi by enumerating lozenge tilings of a halved hexagon in which an array of adjacent triangles has been removed from the boundary.Comment: 28 pages. Third version: fixed several typo

Tilings with noncongruent triangles

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that any tiling of a convex polygon with more than three sides with finitely many triangles contains a pair of triangles that share a full side. © 2018 Elsevier Lt

Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation

The paper considers a type of radial pentagon-based tiling consisting of two shapes: triangle and rectangle. The ob tained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II 5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to the one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons, we use exactly equi lateral triangles and squares, thanks to the trigonometric prop erties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial sec tion of a regular pentagon in 5 parts. The squares are project ed into rectangles whose ratio is: a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles form a ra dial tiling consisting of 5 sectors of the plane, where the pat terns of the established tiles are repeated locally periodically. However, with 5-fold rotation of the pattern, the tiling itself is non-periodic. The various tiling solutions that can be obtained in this way may serve as inspiration for the geometric design, especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the likehttps://smartart-conference.rs/sr/%d1%82%d0%b5%d0%bc%d0%b0%d1%82%d1%81%d0%ba%d0%b8-%d0%b7%d0%b1%d0%be%d1%80%d0%bd%d0%b8%d0%ba-2021/ http://doi.fil.bg.ac.rs/pdf/eb_ser/smartart/2022-2/smartart-2022-2-ch19.pd

Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly the Delone cells tiling the same root lattice projects onto two isosceles Robinson triangles which also lead to Penrose tilings with kites and darts. We point out that the Coxeter element of order h=n+1 and the dihedral subgroup of order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the Coxeter plane. After setting the general scheme we give examples leading to tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with rhombic and triangular tilings of the plane which are useful in modelling the quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4 fold symmetry.Comment: 22 pages, 17 figure