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Teruhisa Sugimoto

Publications and source records attributed to Teruhisa Sugimoto.

17 recordsLinked to original sources
arXiv2026

Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.

math.MG↗
arXiv2025

Converting non-periodic tilings with Tile(1, 1) into tilings with three types of pentagons, I

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the resulting non-periodic tilings with three types of pentagons exhibit two patterns. Note that, during the conversion process in this manuscript, the rhombus is not subdivided into smaller similar rhombuses.

math.MG↗
arXiv2025

Aperiodic sets of three types of convex polygons

Sets of three types of convex pentagons that are aperiodic with no matching conditions on the edges are created from a chiral aperiodic monotile Tile(1, 1). This method divides the interior of Tile(1,1) into five convex polygons with five or more edges, and we have so far identified four methods.

math.MG↗
arXiv2023

Converting tilings with multiple types of rhombuses to pentagonal tilings

The results involving rotationally symmetric tilings with multiple types of rhombuses, discovered by Penrose, Ammann, Beenker, or Socolar, are converted to tilings with multiple types of pentagons are presented. The pentagons can be convex or concave, and can be degenerated into a trapezoid or parallelogram. If the pentagons are convex, they belong to the Type 2 family.

math.MG↗
arXiv2022

Pentagons and rhombuses that can form rotationally symmetric tilings

In this study, various rotationally symmetric tilings that can be formed using pentagons that are related to rhombus are discussed. The pentagons can be convex or concave and can be degenerated into a trapezoid. If the pentagons are convex, they belong to the Type 2 family. Because the properties of pentagons correspond to those of rhombuses, the study also explains the correspondence between pentagons and various rhombic tilings.

math.MG↗
arXiv2022

Convex pentagons and convex hexagons that can form rotationally symmetric tilings

In this study, the properties of convex hexagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the convex hexagons are equilateral convex parallelohexagons, convex pentagons generated by bisecting the hexagons can form rotationally symmetric non-edge-to-edge tilings. In addition, under certain circumstances, tiling-like patterns with an equilateral convex polygonal hole at the center can be formed using these convex hexagons or pentagons.

math.MG↗
arXiv2022

Convex pentagons and concave octagons that can form rotationally symmetric tilings

In this study, the properties of convex pentagons that can form rotationally symmetric edge-to-edge tilings are discussed. Because the rotationally symmetric tilings are formed by concave octagons that are generated by two convex pentagons connected through a line symmetry, they are considered to be equivalent to rotationally symmetric tilings with concave octagons. In addition, under certain circumstances, tiling-like patterns with a regular polygonal hole at the center can be formed using these convex pentagons.

math.MG↗
arXiv2018

Properties of Convex Pentagonal Tiles for Periodic Tiling

A convex pentagonal tile is a convex pentagon that admits a monohedral tiling. We show that a convex pentagonal tile that admits a periodic tiling has a property in which the sum of three internal angles of the pentagon is equal to 360°.

math.MG↗
arXiv2018

Convex Pentagons with Positive Heesch Number

We found convex pentagons whose Heesch number is equal to one, and which admit an edge-to-edge corona. In this manuscript, we present a new classification of these convex pentagons.

math.MG↗
arXiv2018

Convex Pentagon Tilings and Heptiamonds, I

In 1995, Marjorie Rice discovered an interesting tiling by using convex pentagons. The authors discovered novel properties of the tilings of convex pentagon which Rice used in the discovery. As a result, many new convex pentagon tilings (tessellations) were found. The convex pentagon tilings are related to tilings by heptiamonds.

math.MG↗
arXiv2018

Convex Pentagon Tilings and Heptiamonds, II

In the previous manuscript, new tilings (tessellations) were presented using convex pentagonal tiles belonging to Type 1 and Type 5. The convex pentagon tilings are related to heptiamond tilings. Later, the authors found Johannes Hindriks' site that summarized the research results of heptiamond tilings (tessellations). In this manuscript, the results of converting the heptiamond tilings of Hindriks to convex pentagon tilings are introduced. As a result, many new convex pentagon tilings are presented.

math.MG↗
arXiv2017

Convex Polygons for Aperiodic Tiling

If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we prove that a convex polygon that can generate an edge-to-edge monohedral tiling must be able to generate a periodic tiling.

math.MG↗
arXiv2016

Properties of Strongly Balanced Tilings by Convex Polygons

Every normal periodic tiling is a strongly balanced tiling. The properties of periodic tilings by convex polygons are rearranged from the knowledge of strongly balanced tilings. From the results, we show the properties of representative periodic tilings by a convex pentagonal tile.

math.MG↗
arXiv2015

Exact value of Tammes problem for N=10

Let $C_{i}$ ($\,i=1,\ldots ,N\,$) 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 $N$ congruent open spherical caps under the condition, sequentially for $i=2,\ldots ,N-1\,$, that $M_{i}$ is set on the perimeter of $C_{i-1}$, and that each area of the set $(\cup _{ν=1}^{i-1}C_{ν})\cap C_{i}$ becomes maximum. In this paper, for $N = 10$, we found out that the solutions of our sequential covering and the solutions of the Tammes problem were strictly correspondent. Especially, we succeeded to obtain the exact value $r_{10}$ for $N = 10$.

math.MG↗