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The Einstein problem is a mathematical problem in discrete geometry. A geometric form is sought that can be joined together in rotated, shifted and/or mirrored copies without overlapping, so that the entire plane is completely covered (tiled) with it. In addition, this form should have the property that no tiling with a periodically recurring pattern - as on wallpaper - can be produced. The problem was considered unresolved for decades; In 2023, two solutions were proposed for the first time. The name of the problem is a humorous allusion to Albert Einstein and states that only one shape ("one stone") should tile the plane.[1]

Details of the problem

In mathematical terms, the Einstein problem is about whether there exists a single tile (prototile) that can tile the Euclidean plane with no additional rules to follow when assembling, but only in a non-periodic way. A prototile with this property is called "aperiodic". "Periodic" in this sense would be a tiling that repeats itself analogously to a wallpaper pattern (according to a crystallographic group), i.e. that can be shifted in two different (more precisely: linearly independent) directions in a straight line in such a way that the entire structure thereby is mapped to itself (translational symmetry). In a non-periodic tiling there are no two independent translational symmetries.

David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss were able to present such prototiles and the necessary evidence for the first time in March and May 2023. Some of these require computer support, and a peer review is pending.[2][3] While the tile presented in March must also be used mirrored for tiling, the tile presented in May solves the problem even under the stricter requirement that only plane rotations and translations are allowed.

The German term “Einstein” has become established in English for an aperiodic monotile. This pun on the words "a" and "stone", representing "a (single) tile", is attributed to the German mathematician Ludwig Danzer; there is no actual connection to Einstein's research.[4]

The problem can also be viewed as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that fills three-dimensional Euclidean space without gaps, but no space-filling through that polyhedron is isohedral.[5] Such anisohedral polyhedra were first presented by Karl Reinhardt in 1928.[6] In 1932, Heinrich Heesch found such a solution for the plain.

For the discovery of the so-called quasicrystals, which led to a Nobel Prize in chemistry in 2011, it was essential that important results on aperiodic prototiles had already been found in the 1970s (more on this in the following section.)

Solutions and previous solutions

The best-known example of aperiodic prototiles in the plane until March 2023 was the so-called Penrose tiling (1974), which, however, requires a set of two different prototiles. From this point on, the search for the aperiodic monotile began, which lasted almost 50 years.

In 2023, two solutions to the Einstein problem were found by Smith et al. suggested. Before it was unclear what such a tile could look like. The best approximations to the problem up until then either required additional tiling rules such as decorations, were incoherent, or had to accept overlaps or gaps in the tessellation. The Smith et al. The solutions found, on the other hand, are coherent, seamless and do not require additional tiling rules. In order to seamlessly tile the level with copies of the hat tile, an infinite number of mirrored (flipped) tiles must also be used. At the same time, the approach implies the existence of a family of slightly differently shaped monotiles with the same properties. This is visualized by an animation in the first article by M. Bischoff given under “Weblinks”. The authors around David Smith obtained their second solution to the Einstein problem from precisely this approach to further forms, an irregular non-convex 14-sided polygon from which the so-called. Specter tile obtained.

In 1988, Peter Schmitt discovered a polyhedron for the non-periodic, gapless tiling of the three-dimensional Euclidean space. While none of these space-fillings allow translation as a symmetry, some exhibit skew symmetry, which is defined as a combination of translation and rotation through an irrational multiple of circle number

PIIIIIIpi (Pi) can be understood such that no number of repeated operations ever result in a pure parallel shift. This construction was later extended by John Horton Conway and Ludwig Danzer to a convex aperiodic space filler, the Schmitt-Conway-Danzer tile (see figure). The presence of the skew symmetry led to a reassessment of the non-periodicity requirements.[7] Chaim Goodman-Strauss proposed to call a tiling strongly aperiodic if it does not admit an infinite cyclic group of Euclidean transformations as symmetries, and only call tiling sets that enforce strong aperiodicity strongly aperiodic, while other sets too weakly aperiodic are designated.[8]

In 1996, Petra Gummelt constructed a decorated decagonal prototile and showed that it can necessarily nonperiodically tile the plane if two types of overlaps between tile pairs are allowed (see figure).[9] Because of the invalid overlap rules, the Gummelt tile does not solve the problem.

Another approach from 2010 comes from Joshua Socolar and Joan Taylor.[10] The tiling of the Euclidean plane with the Socolar-Taylor tile requires merging rules that constrain the relative orientation of two tiles and refer to drawn decorations of the tiles. These rules apply to pairs of non-adjacent tiles. Alternatively, an undecorated but disjointed tile can be created with no merging rules (see figure). This variant of the Socolar-Taylor tile is composed of various prototiles in a fixed arrangement (19 in total) and is therefore no longer a closed topological disk by definition. This construction can in turn be extended to a space-filling, connected polyhedron without assembly rules. However, the space fillings that are possible with this are periodic in one direction, which is why the three-dimensional Socolar-Taylor tile is only weakly aperiodic.

Weblinks

• Manon Bischoff: Hobby-Mathematiker findet die lang ersehnte Einstein-Kachel, spektrum.de, 29. März 2023
• Manon Bischoff: Vampir-Kachel löst den »Einstein« ab, spektrum.de, 31. Mai 2023
• Spezielle Website zu "A chiral aperiodic monotile", Mai 2023
• Blog von David Smith, 24. April 2016 until today

FIGURES FIGURES FIGURES FIGURES FIGURES

LEAD: Section of a tessellation with the so-called Specter tile, which was presented for the first time in May 2023. Specter means ghost in English. Ausschnitt einer Parkettierung mit der sogenannten Spectre-Kachel, welche im Mai 2023 erstmals vorgestellt wurde. Spectre bedeutet im Englischen Gespenst. File:Spectre aperiodic monotile small patch.svg

Section of a tiling with the hat tile, presented in March 2023. The blue tiles all have the same shape, the yellow ones are mirror images of it. Ausschnitt einer Parkettierung mit der Hut-Kachel, vorgestellt im März 2023. Die blauen Kacheln haben alle die gleiche Form, die gelben sind dazu spiegelbildlich. File:Aperiodic monotile smith 2023.svg

A Specter tile (right) is obtained by edge-modifying a 14-sided polygon tile (left). The tile shape on the right excludes solutions that contain mirrored tiles. In the case of the polygon (left), reflections would have to be explicitly forbidden, as they could occur. Eine Spectre-Kachel (rechts) erhält man durch Kantenmodifizierung einer 14-seitigen Polygon-Kachel (links). Die rechte Kachelform schließt Lösungen aus, die gespiegelte Kacheln enthalten. Beim Polygon (links) müsste man Spiegelungen explizit verbieten, da sie auftreten könnten. File:Aperiodische Monokacheln Mai 2023.png

In this variant of the Specter tile, the edges have been shaped slightly differently. There is a certain degree of freedom in their exact shape, provided that all edges - as with puzzle pieces - fit together and do not intersect. Bei dieser Variante der Spectre-Kachel wurden die Kanten etwas anders geformt. Bei deren genauer Form besteht eine gewisse Freiheit, sofern alle Kanten - wie bei Puzzleteilen - zueinander passen und sich nicht schneiden. File:Spectre aperiodic monotile single.svg

The three-dimensional Schmitt-Conway-Danzer tile. Die dreidimensionale Schmitt-Conway-Danzer-Kachel. File:SCD tile.svg

Gummelt's decorated decagonal prototile (left) with decomposition into Penrose tiles (kite and arrow) by dashed lines and possible overlaps (right). Gummelts dekorierte zehneckige Protokachel (links) mit Zerlegung in Penrose-Kacheln (Drachen und Pfeil) durch gestrichelte Linien und mögliche Überlappungen (rechts). File:Gummelt decagon.svg

The disjointed Socolar-Taylor tile solves the Einstein problem only with limitations, but was considered the first good approximation of an aperiodic monotile until 2023. Die unzusammenhängende Socolar-Taylor-Kachel löst das Einstein-Problem nur mit Einschränkungen, galt aber bis 2023 als erste gute Approximation einer aperiodischen Monokachel. File:Socolar-Taylor tile.svg

PREVIOUS VERSION

An aperiodic monotile is a single tile that tiles the plane with the additional property that it does not contain arbitrarily large periodic regions or patches.

History

An einstein (German: ein Stein, one stone) is an aperiodic tiling that uses only a single shape.

In 2023, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss posted a preprint proving the existence of a tile which when considered with its mirror image form an aperiodic prototile set. The tile, a "hat" formed from eight copies of a 60°–90°–120°–90° kite, glued edge-to-edge, can be generalized to an infinite family of tiles with the same aperiodic property.

The existence of a strongly aperiodic tile set for the Euclidean plane consisting of one connected tile without matching rules is an unsolved problem.

External links

Category:Aperiodic tilings

Private notes by editor

an aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

The "hat" aperiodic monotile resolves the question of whether a single shape can force aperiodicity in the plane. However, all tilings by the hat require reflections; that is, they must incorporate both left- and right-handed hats. Mathematically, this leaves open the question of whether a single shape can force aperiodicity using only translations and rotations. (It also complicates the practical application of the hat in some decorative contexts, where extra work would be needed to manufacture both a shape and its reflection.)

Aperiodic tiling

Penrose tiling

Einstein problem

List of aperiodic sets of tiles

• Two algorithms for randomly generating aperiodic tilings Simon Tatham, 2023-04-10

A 13-sided shape called ‘the hat’ forms a pattern that never repeats

NYT

https://www.popularmechanics.com/science/math/a43402074/mathematicians-discover-new-13-sided-shape/ Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things It will tile a plane without ever repeating. Tim NewcombBY TIM NEWCOMBPUBLISHED: MAR 24, 2023 Popular Mechanics

xxxx At Long Last, Mathematicians Have Found a Shape With a Pattern That Never Repeats Experts have searched for decades for a polygon that only makes non-repeating patterns. But no one knew it was possible until now Will Sullivan Smithsonian Magazine March 29, 2023

xxxx A hobbyist in the U.K. has come up with a new 13-sided shape called 'the hat' March 31, 20235:56 AM ET NPR Mornin gEdition