|
Voderberg Tiling
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician (1911-1945). It is a monohedral tiling: it consists of only one shape that tessellates the plane with congruent copies of itself. In this case, the prototile is an elongated irregular nonagon, or nine-sided figure. The most interesting feature of this polygon is the fact that two copies of it can fully enclose a third one. E.g., the lowest purple nonagon is enclosed by two yellow ones, all three of identical shape. Before Voderberg's discovery, mathematicians had questioned whether this could be possible. Because it has no translational symmetries, the Voderberg tiling is technically non-periodic, even though it exhibits an obvious repeating pattern. This tiling was the first spiral tiling to be devised, preceding later work by Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Spiral ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009. # a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. # a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a . The first definition describes a |
Mathematical Tiling
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and '' semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An '' aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A '' tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prototile
In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in . It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined. A tessellation is said to be monohedral if it has exactly one prototile. Aperiodicity A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. It is unknown whethe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonagon
In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogone'' and in English from the 17th century. The name ''enneagon'' comes from Greek ''enneagonon'' (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common than "nonagon". Regular nonagon A '' regular nonagon'' is represented by Schläfli symbol and has internal angles of 140°. The area of a regular nonagon of side length ''a'' is given by :A = \fraca^2\cot\frac=(9/2)ar = 9r^2\tan(\pi/9) :::= (9/2)R^2\sin(2\pi/9)\simeq6.18182\,a^2, where the radius ''r'' of the inscribed circle of the regular nonagon is :r=(a/2)\cot(\pi/9) and where ''R'' is the radius of its circumscribed circle: :R = \sqrt=r\sec(\pi/9). Construction Although a regular nonagon is not constructible with com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translational Symmetries
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation. Analogously an operator on functions is said to be translationally invariant with respect to a translation operator T_\delta if the result after applying doesn't change if the argument function is translated. More precisely it must hold that \forall \delta \ A f = A (T_\delta f). Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law. Translational symmetry of an object means that a particular translation does not change the object. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tiling
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem< ...
[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geoffrey C
Geoffrey, Geoffroy, Geoff, etc., may refer to: People * Geoffrey (name), including a list of people with the name * Geoffroy (surname), including a list of people with the name * Geoffrey of Monmouth (c. 1095–c. 1155), clergyman and one of the major figures in the development of British history * Geoffrey I of Anjou (died 987) * Geoffrey II of Anjou (died 1060) * Geoffrey III of Anjou (died 1096) * Geoffrey IV of Anjou (died 1106) * Geoffrey V, Count of Anjou (1113–1151), father of King Henry II of England * Geoffrey II, Duke of Brittany (1158–1186), one of Henry II's sons * Geoffrey, Archbishop of York (c. 1152–1212) * Geoffroy du Breuil of Vigeois, 12th century French chronicler * Geoffroy de Charney (died 1314), Preceptor of the Knights Templar * Geoffroy IV de la Tour Landry (c. 1320–1391), French nobleman and writer * Geoffrey the Baker (died c. 1360), English historian and chronicler * Geoffroy (musician) (born 1987), Canadian singer, songwriter and multi-instrumenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An ''aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern. A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Prototile")
