Ikuro Sato

Atoms for Parallelohedra

A parallelohedron is a convex polyhedron which tiles 3-dimensional space by translations only. A polyhedron σ is said to be an atom for the set Π of parallelohedra if for each parallelohedron P in Π, there exists an affine-stretching transformation A: ℝ3 → ℝ3 such that A(P) is the union of a finite number of copies of σ. In this paper, we will present two different atoms for the parallelohedra, and determine the number of these atoms used to make up each parallelohedron. We will also show an arrangement of the parallelohedra in lattice-like order and introduce the notion of indecomposability.

On Reversibility among Parallelohedra

Given two convex polyhedra α and β, we say that α and β are a reversible pair if α has a dissection into a finite number of pieces which can be rearranged to form β in such a way that no face of the dissection of α includes any part of an edge of α, no face of the dissection of β includes any part of an edge of β, the pieces are hinged on some of their edges so that the pieces of the dissection are connected as in a tree-structure, all of the exterior surface of α is in the interior of β, and all of the exterior surface of β comes from the interior of α. Let \(\mathfrak{P}_{i}\) denote one of the five families of parallelohedra (see Section 2 for the corresponding definitions). In this paper, it is shown that given an arbitrary canonical parallelohedron P, there exists a canonical parallelohedron \(Q \in \mathfrak{P}_{i}\) such that the pair P and Q is reversible for each \(\mathfrak{P}_{i}\).

Determination of all tessellation polyhedra with regular polygonal faces

A polyhedron (3-dimensional polytope) is defined as a tessellation polyhedron if it possesses a net of which congruent copies can be used to tile the plane. In this paper we determine all convex polyhedra with regular polygonal faces which are tessellation polyhedra.

On the diagonal weights of inscribed polytopes

Jin Akiyama received a D.Sc. from Tokyo University of Science for his work in graph theory and combinatorics. He is now the director of the Research Center for Science and Math Education at Tokyo University of Science and also serves as the founding editor of the journal of Graphs and Combinatorics. He is interested in graph theory, discrete and computational geometry, and also in mathematics education.

A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes

In this paper, an n-dimensional polytope is called Wythoffian if it is derived by the Wythoff construction from an n-dimensional regular polytope whose finite reflection group belongs to An, Bn, Cn, F4, G2, H3, H4 or I2(p). Based on combinatorial and topological arguments, we give a matrix-form recursive algorithm that calculates the number of k-faces (k = 0, 1, . . . , n) of all the Wythoffian-n-polytopes using Schläfli-Wythoff symbols. The correctness of the algorithm is reconfirmed by the method of exhaustion using a computer.

Tessellabilities, Reversibilities, and Decomposabilities of Polytopes

In this talk, we discuss tessellabilities, reversibilities, and decomposabilities of polygons, polyhedra, and polytopes, where by the word “tessellability”, we mean the capability of the polytope to tessellate. Although these three concepts seem quite different, but there is a strong connection linking them. These connections will be shown when we consider the lattices of tilings in ℝ2 and tessellations in ℝ3, which can be regarded as discrete metric spaces. Many old and new results together with various research problems will be presented.