Convex pentagons and convex hexagons that can form rotationally symmetric tilings

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.

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.

同时属于 Type 1 与 Type 7 族的凸五边形旋转对称铺砌

Rotationally symmetric tilings with convex pentagons belonging to both the Type 1 and Type 7 families

研究发现同时归属 Type 1 与 Type 7 族的凸五边形可构造边对边连接、含二重与四重旋转对称但无反射对称轴的螺旋铺砌。

Cyclic pentagons and hexagons with integer sides, diagonals and areas

In this paper we obtain cyclic pentagons and hexagons with rational sides, diagonals and area all of which are expressed in terms of rational functions of several arbitrary rational parameters. On suitable scaling, we obtain cyclic pentagons and hexagons whose sides, diagonals and area are all given by integers.

具有凸五边形和六边形的旋转对称铺砌

Rotationally Symmetric Tilings with Convex Pentagons and Hexagons

本文研究具有旋转对称性的平面铺砌，引入一类特定的凸五边形，证明对于任意对称类型 $\mathbf{C}_{n}$ 或 $\mathbf{D}_{n}$ 均存在由该类五边形生成的单面铺砌，且 $n>1$ 时为 $n$ 臂螺旋铺砌，该结论同样适用于凸六边形。

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.