{"id":251122,"date":"2023-04-04T16:24:46","date_gmt":"2023-04-04T14:24:46","guid":{"rendered":"https:\/\/climatescience.press\/?p=251122"},"modified":"2023-04-04T16:24:47","modified_gmt":"2023-04-04T14:24:47","slug":"mathematicians-discover-an-elusive-einstein-tile-with-fibonacci-undertones","status":"publish","type":"post","link":"https:\/\/climatescience.press\/?p=251122","title":{"rendered":"Mathematicians discover an elusive \u2018einstein\u2019 tile, with Fibonacci undertones"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

<\/p>\n\n\n\n

From Tallbloke’s Talkshop<\/a><\/p>\n\n\n\n

April 3, 2023 by\u00a0oldbrew<\/strong><\/p>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Image credit: D. SMITH, J.S. MYERS, C.S. KAPLAN AND C. GOODMAN-STRAUSS (CC BY 4.0)]<\/p>\n\n\n\n

<\/p>\n\n\n\n

Something different here: \u2018A 13-sided shape called \u2018the hat\u2019 forms a pattern that never repeats\u2019, they say. We note an extra feature: each \u2018hat\u2019 contains 8 \u2018kites\u2019, and 13 and 8 are Fibonacci numbers<\/a>.<\/em>
\u2013 \u2013 \u2013
After half a century, mathematicians succeed in finding an \u2018einstein,\u2019 a shape that forms a tiled pattern that never repeats, says Science News<\/a><\/em>.<\/p>\n\n\n\n

A 13-sided shape known as \u201cthe hat\u201d has mathematicians tipping their caps.<\/p>\n\n\n\n

It\u2019s the first true example of an \u201ceinstein,\u201d a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.<\/p>\n\n\n\n

\u201cEverybody is astonished and is delighted, both,\u201d says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. Mathematicians had been searching for such a shape for half a century. \u201cIt wasn\u2019t even clear that such a thing could exist,\u201d Senechal says.<\/p>\n\n\n\n

Although the name \u201ceinstein\u201d conjures up the iconic physicist, it comes from the German ein Stein, meaning \u201cone stone,\u201d referring to the single tile. The einstein sits in a weird purgatory between order and disorder. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can\u2019t form a pattern that repeats.<\/p>\n\n\n\n

With a periodic pattern, it\u2019s possible to shift the tiles over and have them match up perfectly with their previous arrangement. An infinite checkerboard, for example, looks just the same if you slide the rows over by two.<\/p>\n\n\n\n

While it\u2019s possible to arrange other single tiles in patterns that are not periodic, the hat is special because there\u2019s no way it can create a periodic pattern.<\/p>\n\n\n\n

Identified by David Smith, a nonprofessional mathematician who describes himself as an \u201cimaginative tinkerer of shapes,\u201d and reported in a paper posted online March 20 at arXiv.org, the hat is a polykite \u2014 a bunch of smaller kite shapes stuck together.<\/p>\n\n\n\n

That\u2019s a type of shape that hadn\u2019t been studied closely in the search for einsteins, says Chaim Goodman-Strauss of the National Museum of Mathematics in New York City, one of a group of trained mathematicians and computer scientists Smith teamed up with to study the hat.<\/p>\n\n\n\n

It\u2019s a surprisingly simple polygon.<\/p>\n\n\n\n

Full article\u00a0here<\/a>\u00a0[includes brief video demo.]<\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

\n