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Polyominoid
In geometry, a polyominoid (or minoid for short) is a set of equal squares in 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the planar polyominoids. The surface of a cube is an example of a ''hexominoid,'' or 6-cell polyominoid, and many other polycubes have polyominoids as their boundaries. Polyominoids appear to have been first proposed by Richard A. Epstein. Classification 90-degree connections are called ''hard''; 180-degree connections are called ''soft''. This is because, in manufacturing a model of the polyominoid, a hard connection would be easier to realize than a soft one. (archive o [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyform
In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as detailed in the table below. For example, a square basic polygon results in the well-known polyominoes. Construction rules The rules for joining the polygons together may vary, and must therefore be stated for each distinct type of polyform. Generally, however, the following rules apply: #Two basic polygons may be joined only along a common edge, and must share the entirety of that edge. #No two basic polygons may overlap. #A polyform must be connected (that is, all one piece; see connected graph, connected space). Configurations of disconnected basic polygons do not qualify as polyforms. #The mirror image of an asymmetric polyform is not co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called '' 3-manifolds''. The term may also refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D domain), a '' solid figure''. Technically, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the pair formed by a -dimensional Euclidean space and a Cartesian coordinate system. When , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics, it serve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in '' Fairy Chess Review'' between the years 1937 and 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 " Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with 1, unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polycube
image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricubes into a 3×3×3 cube A polycube is a solid figure formed by joining one or more equal cube (geometry), cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes. Enumerating polycubes image:AGK-pentacube.png, A Chirality (mathematics), chiral pentacube Like polyominoes, polycubes can be enumerated in two ways, depending on whether Chirality (mathematics), chiral pairs of polycubes (those equivalent by Reflection symmetry, mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Arnold Epstein
Richard Arnold Epstein (March 5, 1927 in Los Angeles, California – July 5, 2016), also known under the pseudonym E. P. Stein, was an American game theorist. Education Epstein obtained his Bachelor of Arts degree from the University of California, Los Angeles, in 1948. He then studied at the University of California Berkeley. He received his doctorate in physics, on the Born formalization of isochromatic lines, in 1961, from the University of Barcelona. Career He then shifted from spectroscopy to space communications, and worked for eighteen years as an electronics and communications engineer for various U.S. space and missile programs. He was variously employed by Parsons-Aerojet Company at Cape Canaveral, Glenn L. Martin Company, TRW Space Technology Laboratories, the Jet Propulsion Laboratory, and Hughes Aircraft Space Systems Division. Epstein has numerous technical publications in the areas of probability theory, statistics, game theory, and space communications. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polystick
In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regular square grid. A triangular polystick is a connected subset of a regular triangular grid. Polysticks are classified according to how many line segments they contain. The name "polystick" seems to have been first coined by Brian R. Barwell. The names "polytrig" and "polytwigs"David Goodger, "An Introduction to Polytwigs (Hexagonal-Grid Polysticks)," (2015), http://puzzler.sourceforge.net/docs/polytwigs-intro.html has been proposed by David Goodger to simplify the phrases "triangular-grid polysticks" and "hexagonal-grid polysticks," respectively. Colin F. Brown has used an earlier term "polycules" for the hexagonal-grid polysticks due to their appearance resembling the spicules of sea sponges. There is no standard term for line segments built on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in '' Fairy Chess Review'' between the years 1937 and 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 " Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polycube
image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricubes into a 3×3×3 cube A polycube is a solid figure formed by joining one or more equal cube (geometry), cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes. Enumerating polycubes image:AGK-pentacube.png, A Chirality (mathematics), chiral pentacube Like polyominoes, polycubes can be enumerated in two ways, depending on whether Chirality (mathematics), chiral pairs of polycubes (those equivalent by Reflection symmetry, mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
