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

TL;DR

Problem: extend rotationally symmetric tilings using a convex pentagon that belongs to both Type 1 and Type 7 families via $P(T1 \cap T7)$. Approach: construct edge-to-edge tilings from Octa-units and Hexa-units under concentration relations (e.g., $4A = 360°$) to realize $C_{8}$, $C_{4}$, and $C_{2}$ symmetry, including spiral patterns. Contributions: define Octa-unit, Hexa-unit, and Hexadeca-4oc-unit; generate representative Type 1 and Type 7 tilings and a variety of rotationally symmetric tilings (spiral and non-spiral) with central units; connect to Dailey's and AraKi tilings. Significance: broadens the repertoire of convex-pentagon tilings with rotational symmetry and provides constructive methods for tilings featuring central holes and symmetry centers.

Abstract

Rotationally symmetric tilings by a convex pentagonal tile belonging to both the Type 1 and Type 7 families are introduced. Among them are spiral tilings with two- and four-fold rotational symmetry. Those rotationally symmetric tilings are connected edge-to-edge and have no axis of reflection symmetry.

Figures (16)

• Figure 1: Convex pentagonal tile $P(T1 \cap T7)$ that belongs to both the Type 1 and Type 7 families
• Figure 2: Examples of variations of Type 1 tilings by a convex pentagonal tile $P(T1 \cap T7)$
• Figure 3: Representative tiling of Type 7 and example of other tiling by a convex pentagonal tile $P(T1 \cap T7)$
• Figure 4: Octa-unit and Hexa-unit formed by two pieces of $P(T1 \cap T7)$
• Figure 5: Eight-fold rotationally symmetric tiling with a regular convex octagonal hole at the center by a convex pentagonal tile $P(T1 \cap T7)$