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Aperiodic sets of three types of convex polygons

Teruhisa Sugimoto

Abstract

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.

Aperiodic sets of three types of convex polygons

Abstract

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.
Paper Structure (16 sections, 39 figures, 1 table)

This paper contains 16 sections, 39 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dividing Tile$(1, 1)$ into ASP comprising convex polygons
  3. Method 1 and Method 2
  4. Method 3
  5. Method 4
  6. Conclusion
  7. Convex pentagonal monotiles
  8. Convex hexagonal monotiles
  9. Periodic tiling with $P_{1}$, $P_{2}$, and reflected $P_{2}$
  10. ASPs comprising four or five types of convex polygons
  11. Preparation
  12. Dividing Tile$(1, 1)$ into convex pentagons and convex hexagons
  13. ASPs satisfying the restriction on the number of types of congruent diagrams created from Tile$(1, 1)$
  14. Conditions for treating anterior and posterior sides as the same type
  15. Supplement 1
  16. ...and 1 more sections

Figures (39)

  • Figure 1: Tiling (d) generated by an aperiodic set of prototiles created by Ammann, which used three convex polygons with no matching condition on the edges. As shown in (c), this is a recomposition by Penrose tiles (rhombuses) and is obtained by markings of using the points $J$, $K$, and $L$ as shown in (a) and (b). These markings are completely determined by the choice of the point $J$, as we must have GL = DJ, FL = CJ =EK, and HK = BJ. For aperiodicity, the point $J$ must be chosen so that DJ, CJ, BJ, and KL are of different lengths G_and_S_1987Sugimoto_2017.
  • Figure 2: Tile$(1, 1)$, and periodic tiling with anterior and posterior sides of Tile$(1, 1)$.
  • Figure 3: Non-periodic tiling generated by using only one side of Tile$(1, 1)$.
  • Figure 4: Cases in Method 1.
  • Figure 5: Cases in Method 2.
  • ...and 34 more figures