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Tilings of an Isosceles Triangle

Michael Beeson

Abstract

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, and prove that N must be even. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or even squarefree.

Tilings of an Isosceles Triangle

Abstract

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, and prove that N must be even. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or even squarefree.

Paper Structure

This paper contains 32 sections, 52 theorems, 110 equations, 25 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Definitions and notation
  4. History
  5. Laczkovich
  6. Some number-theoretic facts
  7. Tilings of an isosceles $ABC$ by a right-angled tile: examples
  8. Laczkovich's graphs $\Gamma_a$
  9. Tilings of an isosceles $ABC$ by a right-angled tile: theory
  10. Possible values of $N$ in tilings with commensurable angles
  11. Tilings with $\gamma = 2\alpha$ and $(a,b,c)$ commensurable
  12. No tilings with $\gamma = 2\alpha$ and $(a,b,c)$ incommensurable
  13. Stars and centers
  14. The tile is rational
  15. On the number of tiles required when $\gamma = 2\alpha$
  16. ...and 17 more sections

Key Result

Theorem \oldthetheorem

Suppose $ABC$ is $N$-tiled by tile $T$ similar to $ABC$. If $N$ is not a square, then $T$ and $ABC$ are right triangles. Then either (i) $N$ is three times a square and $T$ is a 30-60-90 triangle, or (ii) $N$ is a sum of squares $e^2 + f^2$, the right angle of $ABC$ is split by the tiling, and the a

Figures (25)

  • Figure 1: A quadratic tiling of an arbitrary triangle
  • Figure 2: Biquadratic tilings with $N = 13 = 3^2 + 2^2$ and $N=74 = 5^2 + 7^2$
  • Figure 3: Two 12-tilings
  • Figure 4: Two 27-tilings
  • Figure 5: A 54-tiling; $N/2$ is three times a square. Tile is 30-60-90.
  • ...and 20 more figures

Theorems & Definitions (54)

  • Theorem \oldthetheorem: Snover et. al. snover1991
  • Theorem \oldthetheorem: Laczkovich laczkovich1995
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem: Pythagorean triangles
  • Lemma \oldthetheorem
  • Definition \oldthetheorem: The directed graph $\Gamma_a$
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 44 more