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Tiling an Equilateral Triangle

Michael Beeson

Abstract

Let $ABC$ be an equilateral triangle. For certain triangles $T$ (the "tile") and certain $N$, it is possible to cut $ABC$ into $N$ copies of $T$. It is known that only certain shapes of $T$ are possible, but until now very little was known about the possible values of $N$. Here we prove that for $N>3$, $N$ cannot be prime, and study more closely the possible tilings when the tile has a $π/3$ angle.

Tiling an Equilateral Triangle

Abstract

Let be an equilateral triangle. For certain triangles (the "tile") and certain , it is possible to cut into copies of . It is known that only certain shapes of are possible, but until now very little was known about the possible values of . Here we prove that for , cannot be prime, and study more closely the possible tilings when the tile has a angle.

Paper Structure

This paper contains 9 sections, 11 theorems, 37 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The coloring equation
  3. A tile with an angle $\gamma = \pi/3$ and $\alpha/\pi$ irrational
  4. An algorithm to decide if there is an $N$-tiling with $\gamma = \pi/3$
  5. Comparison with Laczkovich's results
  6. Laczkovich's graph $\Gamma_c$
  7. A tile $(\alpha,\beta,2\pi/3)$ with $\alpha/\pi$ irrational
  8. Necessary conditions when $\gamma = \pi/3$
  9. Appendix: Tilings found by Bryce Herdt

Key Result

Theorem \oldthetheorem

Suppose that triangle $ABC$ is tiled by the tile $(a,b,c)$ in such a way that (i) There is just one tile at $A$. (ii) At every boundary vertex an odd number of tiles meet. (iii) At every interior vertex an even number of tiles meet. (iv) The numbers of tiles at $B$ and $C$ are both even, or both odd holds, where $Y$ is the side of $ABC$ opposite $A$, and $X$ and $Z$ are the other two sides. The si

Figures (11)

  • Figure 1: A 3-tiling, a 6-tiling, and a 16-tiling
  • Figure 2: A 27-tiling due to Major MacMahon 1921, rediscovered 2011
  • Figure 3: $3m^2$ (hexagonal) tilings for $m=4$ and $m=5$
  • Figure 4: $N=10935$. The tile is $(3,5,7)$ and $\gamma = 2\pi/3$.
  • Figure 5: $N=1215$. The tile is $(3,5,7)$ and $\gamma = 2\pi/3$.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition 1: The directed graph $\Gamma_c$
  • Lemma 2: Laczkovich's Lemma 4.5
  • Lemma 3: Laczkovich's Lemma 4.6
  • Lemma 4
  • ...and 2 more