Tuesday, June 6, 2023

A new geometric shape of non-repetition tiles

An aperiodic monotile, sometimes called an “einstein”, is a figure that is a plane tile, but never periodic. – UNIVERSITY OF WATERLOO

MADRID, March 24

Four mathematicians from the universities of York, Cambridge, Waterloo and Arkansas have found a 2D geometric shape that does not repeat when tiled.

David Smith, Joseph Samuel Myers, Craig Kaplan, and Chaim Goodman-Strauss wrote an article describing how they discovered the unique shape and the possible uses for it. His full article is available on the arXiv preprint server.

When people cover floors, they tend to use simple geometric shapes that lend themselves to repeating patterns, such as squares or triangles. Sometimes, however, people want patterns that don’t repeat themselves but challenge if the same types of shapes are used. In this new effort, the research team discovered one geometric shape that, if used in mosaics, would not produce repeatable patterns, the University of Waterloo reports.

During their mission, the explorers noticed that the tiles were to be joined together in such a way that there were no slips or gaps. A tile that does not have repeating patterns is known as an aperiodic tile and is usually created through the use of multiple tile shapes. For many years mathematicians have studied the idea of ​​creating shapes that could create an infinite variety of patterns when tiled.

One of the first attempts resulted from a set of 20,426 tiles. This was followed by the development of Penrose Mosaics, introduced in 1974, which come in two differently shaped diamonds. Since then, mathematicians have continued to search for what is the “einstein” form, the only form that can be used by itself for periodic tables.

Notably, the name of the stone comes from German, not a noble scientist. In this new effort, the research group claims that Einstein has discovered an illusory pattern and that it has been mathematically proven, reports Phys.org.

The figure has 13 sides and is simply referred to as a “bias hat”. They found it first limiting the possibilities of using a computer and then researched it to make it smaller by hand. After they had a good possibility, they proved it using the conjunction software, and later proved that the shape was aperiodic using an argument from geometric incommensurability. The researchers conclude by suggesting that the most likely application is in the arts.

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