Self-affine quadrangles

TL;DR

This paper characterizes all 3-self-affine convex quadrangles, identifies families and examples, and discusses the existence of n-self-affine quadrangles for convex and non-convex cases, reducing the problem to the case n=4.

Contribution

It provides a complete classification of 3-self-affine convex quadrangles and explores the existence of n-self-affine quadrangles for convex and non-convex cases.

Findings

All convex quadrangles are n-self-affine for n ≥ 5.

Only trapezoids are 2-self-affine convex quadrangles.

There are n-self-affine non-convex quadrangles for all n ≥ 3.

Abstract

A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$. In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$.