Triangle Tiling II: Nonexistence theorems

TL;DR

This paper characterizes specific nonexistence and existence conditions for tiling a triangle with congruent tiles, focusing on cases where the tile and the triangle are not similar, and introduces new tiling families.

Contribution

It identifies only two families of tilings under certain conditions and introduces the concept of 'triquadratic tilings' with detailed algebraic and geometric analysis.

Findings

Only two tiling families exist under the specified conditions.

Introduction of 'triquadratic tilings' in special angle cases.

Application of algebraic number theory and linear algebra techniques.

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 . We wish to understand possible tilings by completely characterizing the triples (ABC, T, N) such that ABC can be N -tiled by T. In particular, this understanding should enable us to specify for which N there exists a tile T and a triangle ABC that is N-tiled by T; or given N, determine which tiles and triangles can be used for N-tilings; or given ABC, to determine which tiles and N can be used to N-tile ABC. This is the second of four papers on this subject. In the first paper, we dealt with the case when ABC is similar to T, and the case when T is a right triangle. In this paper, we assume that ABC is not similar to T, and T is not a right triangle, and furthermore that if T has a 120 degree angle, then T is isosceles. Under those hypotheses, there are only two families of tilings. There are tilings of an equilateral triangle ABC by an isosceles tile with base angles 30 degrees, and there are newly-discovered "triquadratic tilings", which are treated in the third paper in this series. These arise in the special case that 3\alpha + 2\beta = \pi or 3\beta+ 2\alpha = \pi with \alpha not a rational multiple of \pi, where \alpha and \beta are the two smallest angles of the tile, and the sides of the tile have rational ratios. The case when the tile has a 120 degree angle and is not isosceles is taken up in a subsequent paper. We use techniques from linear algebra and elementary field theory, and in one case we use some algebraic number theory. We use some counting arguments and some elementary geometry and trigonometry.