TL;DR
This paper explores the mathematical problem of simultaneous tiling, where an object tiles space with multiple translation sets, including the famous Steinhaus problem involving rotations of the integer lattice.
Contribution
It discusses the challenges and properties of simultaneous tiling, focusing on the Steinhaus problem and related issues in tiling theory.
Findings
Analysis of conditions for simultaneous tiling
Insights into the Steinhaus problem
Identification of open problems in tiling theory
Abstract
We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a set simultaneously tiling the plane with all rotates of the integer lattice as translation sets.
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Taxonomy
TopicsQuasicrystal Structures and Properties · Cellular Automata and Applications · Art, Technology, and Culture