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Golygon
A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 1990 ''Scientific American'' column (Smith). Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn angles other than 90°. Properties In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number ''n'' of sides must allow the solution of the system of equations :\pm 1 \pm 3 \pm \cdots \pm (n-1) = 0 :\pm 2 \pm 4 \pm \cdots \pm n = 0. It follows from this that ''n'' must be a multiple of 8. For example, in the figure we have -1 + 3 + 5 - 7 = 0 and 2 - 4 - 6 + 8 = 0. The number of golygons for a given permissible value of ''n'' may be computed efficiently using generati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lee Sallows
Lee Cecil Fletcher Sallows (born April 30, 1944) is a British electronics engineer known for his contributions to recreational mathematics. He is particularly noted as the inventor of golygons, self-enumerating sentences, and geomagic squares. Recreational mathematics Sallows is an expert on the theory of magic squares and has invented several variations on them, including alphamagic squares and geomagic squares. The latter invention caught the attention of mathematician Peter Cameron who has said that he believes that "an even deeper structure may lie hidden beyond geomagic squares" In "The lost theorem" published in 1997 he showed that every 3 × 3 magic square is associated with a unique parallelogram on the complex plane, a discovery that had escaped all previous researchers from ancient times down to the present day. In 2014 Sallows discovered a previously unnoticed result involving the medians of a triangle. A golygon is a polygon containing only right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spirolateral
In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M. ''Worm Paths'' Ch. 17 ''Knotted Doughnuts and Other Mathematical Entertainments'' New York: W. H. Freeman, pp. 205-221, 1986/ref> A ''simple spirolateral'' has only positive angles. A simple spiral approximates of a portion of an archimedean spiral. A ''general spirolateral'' allows positive and negative angles. A ''spirolateral'' which completes in one turn is a simple polygon, while requiring more than 1 turn is a star polygon and must be self-crossing. A simple spirolateral can be an equangular simple polygon with ''p'' vertices, or an equiangular star polygon with ''p'' vertices and ''q'' turns. Spirolaterals were invented and named by Frank C. Odds as a teenager in 1962, as ''square spirolaterals'' with 90° angles, drawn on gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spirolateral
In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M. ''Worm Paths'' Ch. 17 ''Knotted Doughnuts and Other Mathematical Entertainments'' New York: W. H. Freeman, pp. 205-221, 1986/ref> A ''simple spirolateral'' has only positive angles. A simple spiral approximates of a portion of an archimedean spiral. A ''general spirolateral'' allows positive and negative angles. A ''spirolateral'' which completes in one turn is a simple polygon, while requiring more than 1 turn is a star polygon and must be self-crossing. A simple spirolateral can be an equangular simple polygon with ''p'' vertices, or an equiangular star polygon with ''p'' vertices and ''q'' turns. Spirolaterals were invented and named by Frank C. Odds as a teenager in 1962, as ''square spirolaterals'' with 90° angles, drawn on gr ... [...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 num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagon Golygon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. Properties of the general octagon The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).Dao Thanh Oai (2015), "Equilater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926-1934) and ''National Mathematics Magazine'' (1934-1945.) Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Monthl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed. Special case examples include the following: * All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle. * Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ..., that has no solutions in positive integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Criterion
In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try the Conway Criterion!' by Doris Schattschneider Mathematics Magazine Vol. 53, No. 4 (Sep, 1980), pp. 224-233 The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that: * the boundary part from A to B is congruent to the boundary part from E to D by a translation T where T(A) = E and T(B) = D. * each of the boundary parts BC, CD, EF, and FA is centrosymmetric—that is, each one is congruent to itself when rotated by 180-degrees around its midpoint. * some of the six points may coincide but at least three of them must be distinct. Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only 180-degree rotations. The Conway criter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (excep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
