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At the beginning of the series of papers we present systematic approach to exhaust the convex pentagonal tiles of edge-to-edge (EE) tilings. Our procedure is to solve the problem systematically step by step by restricting the candidates to some class. The first task is to classify both of convex pentagons and pentagonal tiling patterns. The classification… (More)

Let us consider an edge-to-edge and strongly balanced tiling of plane by pentagons. A node of valence s (≥3) in an edge-to-edge tiling is a point that is the common vertex of s tiles. Let W 1 be a finite closed disk satisfying the property that the average valence of nodes in W 1 is nearly equal to 10/3. Then, let T denote the union of the set of pentagons… (More)

We discovered the new tiling patterns each of which is composed of a single kind of convex pentagon. Moreover, we propose a new concept of classification method of the tessellating convex pentagons, which is the only unsolved case among the corresponding tessellating convex polygon problems.

We derived 14 types of tiling cases under a restricted condition in our previous report, which studied plane tilings with congruent convex pentagons. That condition is referred to as the category of the simplest set of node (vertex of edge-to-edge tiling) conditions when the tile is a convex pentagon with four equal-length edges. This paper shows the… (More)

Let C i (i = 1, ..., N) be the i-th open spherical cap of angular radius r and let M i be its center under the condition that none of the spherical caps contains the center of another one in its interior. We consider the upper bound, r N (not the lower bound!) of r of the case in which the whole spherical surface of a unit sphere is completely covered with… (More)

The coverage problem on the sphere is studied. We consider the sequential covering of identical spherical caps, so that none of them contains the center of another one. Set of the center points of such caps is said to form a Minkowski set. We give an algorithm of random sequential covering under the condition of Minkowski set. Problems of packing or… (More)

- Teruhisa Sugimoto
- Graphs and Combinatorics
- 2015

We introduce a plan toward a perfect list of convex pentagons that can tile the whole plane in edge-to-edge manner. Our strategy is based on Bagina's Proposition, and is direct and primitive: Generating all candidates of pentagonal tiles (several hundreds in number), classify them into the known 14 types, geometrically impossible cases, the cases that do… (More)

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