An isohedron is a 3-dimensional polyhedron all faces of which are equivalent under symmetries of the polyhedron. Many well known polyhedra are isohedra; among them are the Platonic solids, the polars of Archimedean polyhedra, and a variety of polyhedra important in crystallography. Less well known are isohedra with nonconvex faces. We establish that such polyhedra must be starshaped and hence o...

We give a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to Kepler’s small stellated dodecahedron. Starting with Threlfall’s description, we introduce several other flat conformal geometries on this surface which are related to holomorphic 1-forms. They allow us to show that the Jacobian is isogeneous to a fourfold product of a single elliptic curve whos...

The dodecahedral conjecture states that in a packing of unit spheres in R3, the Voronoi cell of minimum possible volume is a regular dodecahedron with inradius one. The conjecture was first stated by L. Fejes Tóth in 1943, and was finally proved by Hales and McLaughlin over 50 years later using techniques developed by Hales for his proof of the Kepler conjecture. In 1964, Fejes Tóth described a...

In a 1979 paper by Coxeter and Huybers [4] , simple formulas were given to construct the snub cube and the snub dodecahedron from the octahedron and the icosahedron respectively , the vertices of the octahedron and icosahedron being the intersection points of certain pairs of edges of the two snub polyhedra . In a later series of papers , Hohenberg [6 , 7 , 8] studied the intersection points of...

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of poly...

We give a fairly simple proof of a result announced in [2] that there are only seven non-extendible finite (r, q)-polycycles: the tetrahedron without a face, the cube without a face, the octahedron without a face, the dodecahedron without a face, the icosahedron without a face, the split-vertex octahedron and the split-vertex icosahedron.

The cluster formation in Three Dimensional Wireless Sensor Networks (3D-WSN) give rise to overlapping of signals due to spherical sensing range which leads to information redundancy in the network. To address this problem, we develop a sensing algorithm for 3D-WSN based on dodecahedron topology which we call Three Dimensional Distributed Clustering (3D-DC) algorithm. Using 3D-DC algorithm in 3D...