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Moduli Spaces of Pentagonal Subdivision Tilings

Jinjin Liang, Erxiao Wang, Min Yan

Abstract

Pentagonal subdivision gives three families of edge-to-edge tilings of the sphere by congruent pentagons. Each family forms a two dimensional moduli. We describe the moduli in detail.

Moduli Spaces of Pentagonal Subdivision Tilings

Abstract

Pentagonal subdivision gives three families of edge-to-edge tilings of the sphere by congruent pentagons. Each family forms a two dimensional moduli. We describe the moduli in detail.

Paper Structure

This paper contains 12 sections, 4 theorems, 87 equations, 22 figures, 3 tables.

Table of Contents

  1. Moduli Space Region
  2. Boundary of Moduli Space
  3. Stereographic Projection
  4. Determination of Boundary
  5. Formula for Boundary
  6. Picture of Moduli Space
  7. Features of Moduli Space
  8. Companion
  9. Location of Boundary
  10. Upper Bound of Edge Length
  11. Reduction
  12. Size of Moduli Space

Key Result

Theorem 1

The moduli space of pentagonal subdivision tilings is given by the locations of the anchor point in an open region of the sphere bounded by two arcs and three curves in Figure moduliM.

Figures (22)

  • Figure 1: Pentagonal subdivision of a regular triangle.
  • Figure 2: Pentagonal subdivision tilings with $12$, $24$, $60$ tiles.
  • Figure 3: $A$- and $B$-projections of $18$ regions for the tetrahedron.
  • Figure 4: The pentagon is simple when $V$ is in $\Omega_1,\Omega_2,\Omega_3,\Omega_7$.
  • Figure 5: Pentagonal subdivision tilings when $V$ is in $\Omega_2,\Omega_3$, or part of $\Omega_8$.
  • ...and 17 more figures

Theorems & Definitions (4)

  • Theorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem