Related papers: Dodecahedral Structures with Mosseri-Sadoc Tiles
The adhesion approximation is a simple analytical model suggested for explanation of the major geometrical features of the observed structure in the galaxy distribution on scales from 1 to (a few)x100/h Mpc. It is based on Burgers' equation…
We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichm\"uller curves. Using an algorithm…
We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. The problem for non-cubic groups was studied in previous papers by D. Bochis and the second author…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The…
We propose an Euler transformation that transforms a given $d$-dimensional cell complex $K$ for $d=2,3$ into a new $d$-complex $\hat{K}$ in which every vertex is part of a uniform even number of edges. Hence every vertex in the graph…
We consider tessellations of the Euclidean $(d-1)$-sphere by $(d-2)$-dimensional great subspheres or, equivalently, tessellations of Euclidean $d$-space by hyperplanes through the origin; these we call conical tessellations. For random…
We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures.…
We classify the special families of dihedral folding tilings of the sphere derived from the M\"obius triangle $(2,3,4)$. Our study emerges from the study of isometric foldings in the Riemann sphere and meets at the juncture of the triangle…
A stable core-shell structure with Ih symmetry, Au12@Au30, has been investigated by first-principles calculations. It is composed of an icosahedron core and an icosidodecahedron shell. The stability of the core-shell Au42 structure is…
We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules…
We present a scheme to categorize the structure of different layered phosphorene allotropes by mapping their non-planar atomic structure onto a two-color 2D triangular tiling pattern. In the buckled structure of a phosphorene monolayer, we…
Tessellations of $R^3$ that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not facet-to-facet, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's…
In this paper we construct the quasi regular polyhedra and their duals which are the generalizations of the Archimedean and Catalan solids respectively. This work is an extension of two previous papers of ours which were based on the…
A periodic lattice in Euclidean 3-space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
The smallest maximum kissing-number Voronoi polyhedron of 3d spheres is the icosahedron and the tetrahedron is the smallest volume that can show up in Delaunay tessalation. No periodic lattice is consistent with either and hence these dense…
From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By…
Start with Gott (2019)'s envelope polyhedron (Squares-4 around a point): a unit cube missing its top and bottom faces. Stretch by a factor of 2 in the vertical direction so its sides become (2x1 unit) rectangles. This has 8 faces (4…
There are exactly eight edge-to-edge tilings of the sphere by congruent equilateral pentagons: three pentagonal subdivision tilings with 12, 24, 60 tiles; four earth map tilings with 16, 20, 24, 24 tiles; and one flip modification of the…
We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…