Dissection of a rectangle into rectangles with given side ratios

TL;DR

This paper characterizes all rectangles that can be tiled by a set of given rectangles with quadratic irrational side ratios, providing a comprehensive understanding of such tilings and their constraints.

Contribution

It offers a complete classification of rectangles tileable by similar rectangles with quadratic irrational ratios, extending previous partial results.

Findings

Provides explicit conditions for tiling possibilities

Classifies all rectangles that can be tiled by given quadratic irrational rectangles

Extends tiling theory to include quadratic irrational side ratios

Abstract

Given a collection of N rectangles such that the side ratio of each one is a quadratic irrationality, we find all rectangles which can be tiled by rectangles similar to one of the given ones. It means that each possible shape can be used several times or not used at all, so that the number of rectangles in the tiling is not necessarily equal to N.