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Einstein Tile
In plane discrete geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellation, tessellate space but only in a aperiodic tiling, nonperiodic way. Such a shape is called an einstein, a word play on ''ein Stein'', German for "one stone". Several variants of the problem, depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, were solved beginning in the 1990s. The strictest version of the problem was solved in 2023, after an initial discovery in 2022. The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. Such anisohedral tiling, anisohedral tiles were found by Karl Reinhardt (mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Monotile Tiling
A periodic function, also called a periodic waveform (or simply periodic wave), is a Function (mathematics), function that repeats its values at regular intervals or period (physics), periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit Frequency, periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig S
Craig may refer to: People and fictional characters *Craig (surname), including a list of people and fictional characters * Craig (given name), including a list of people and fictional characters * Clan Craig, a Scottish clan Places United States *Craig, Alaska, a city * Craig, Colorado, a city * Craig, Iowa, a city * Craig, Missouri, a city * Craig, Montana, an unincorporated place * Craig, Nebraska, a village * Craig, Ohio, an unincorporated community * Craig County, Oklahoma * Craig County, Virginia * Craig Township, Switzerland County, Indiana * Craig Township, Burt County, Nebraska * Mount Craig (Colorado) * Mount Craig (North Carolina) * Craig Mountain, Oregon *Craig Field (airport), a public airport near Selma, Alabama, formerly: **Craig Air Force Base, a former United States Air Force base * Craig Hospital, a neurorehabilitation and research hospital in Englewood, Colorado, United States * Fort Craig, a United States Army fort in New Mexico * The Craig School, an independ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deltoidal Trihexagonal Tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two Square (geometry), squares, and one hexagon on each vertex (geometry), vertex. It has Schläfli symbol of rr. John Horton Conway, John Conway calls it a rhombihexadeltille.Conway, 2008, p288 table It can be considered a Cantellation, cantellated by Norman Johnson (mathematician), Norman Johnson's terminology or an Expansion (geometry), expanded hexagonal tiling by Alicia Boole Stott's operational language. There are three List of regular polytopes#Euclidean tilings, regular and eight List of uniform tilings, semiregular tilings in the plane. Uniform colorings There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.) With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kite (geometry)
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word ''deltoid'' may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals.See H. S. M. Coxeter's review of in : "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid." A kite may also be called a dart, particularly if it is not convex. Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the right kites, with two opposite right angles; the rhombus, rhombi, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Smith (amateur Mathematician)
David Smith is an amateur mathematician and retired print technician from Bridlington, Yorkshire, England, who is best known for his discoveries related to aperiodic monotiles that helped to solve the einstein problem. Einstein tile Initial discovery Smith discovered a 13-sided polygon in November 2022 whilst using a software package called ''PolyForm Puzzle Solver'' to experiment with different shapes. After further experimentation using cardboard cut-outs, he realised that the shape appeared to tessellate but seemingly without ever achieving a regular pattern. Contacting experts Smith contacted Craig S. Kaplan from the University of Waterloo to alert him to this potential discovery of an aperiodic monotile. They nicknamed the newly discovered shape "the hat", because of its resemblance to a fedora. Kaplan proceeded to further inspect the polykite shape. During this time, Smith informed Kaplan that he had discovered yet another shape, which he nicknamed "the turtle", that appea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socolar–Taylor Tile
The Socolar–Taylor tile is a single non-connected tessellation, tile which is aperiodic on the Euclidean plane, meaning that it admits only aperiodic tiling, non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed.. It is the first known example of a single aperiodic tile, or "einstein problem, einstein". The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. One of their papers shows a realization of the tile as a connected set. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a simply connected space, simply connected set. This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile. Taylor and Socolar remark that the 3D monotile ape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of each point (mathematics), point. It is an affine space, which includes in particular the concept of parallel lines. It has also measurement, metrical properties induced by a Euclidean distance, distance, which allows to define circles, and angle, angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a ''Cartesian plane''. The set \mathbb^2 of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called ''the'' Euclidean plane or ''standard Euclidean plane'', since every Euclidean plane is isomorphic to it. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Motion
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation. In dimension three, every rigid motion can be decomposed as the composition of a rotation and a translation, and is thus sometimes called a rototranslation. In di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Cyclic Group
In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group \Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/''n''Z, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaim Goodman-Strauss
Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathematician for the National Museum of Mathematics. He is co-author with John H. Conway and Heidi Burgiel of '' The Symmetries of Things'', a comprehensive book surveying the mathematical theory of patterns. Education and career Goodman-Strauss received both his B.S. (1988) and Ph.D. (1994) in mathematics from the University of Texas at Austin.Chaim Goodman-Strauss The College Board His doctoral advisor was John Edwin Luecke. He joined the faculty at the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
