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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] |
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] |
Polyiamond
A polyiamond (also polyamond or simply iamond, or sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word ''polyiamond'' is a back-formation from ''diamond'', because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial 'di-' looks like a Greek prefix meaning 'two-' (though ''diamond'' actually derives from Greek '' ἀδάμας'' - also the basis for the word "adamant"). The name was suggested by recreational mathematics writer Thomas H. O'Beirne in ''New Scientist'' 1961 number 1, page 164. Counting The basic combinatorial question is, How many different polyiamonds exist with a given number of cells? Like polyominoes, polyiamonds may be either free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation. One-sided polyiamonds distinguish reflections. The number of free ''n''-iamonds for ''n'' = 1, 2, 3, ... is: :1, 1, 1, 3 ... [...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] |
Pseudo-polyomino
A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a Shape, plane geometric figure formed by joining one or more equal squares edge-to-edge or corner-to-corner at 90°. It is a polyform with Square (geometry), square cells. The polyominoes are a subset of the polykings. The name "polyking" refers to the king (chess), king in chess. The ''n''-kings are the ''n''-square shapes which could be occupied by a king on an infinite chessboard in the course of legal moves. Solomon W. Golomb, Golomb uses the term ''pseudo-polyomino'' referring to kingwise-connected sets of squares. Enumeration of polykings Free, one-sided, and fixed polykings There are three common ways of distinguishing polyominoes and polykings for enumeration: *''free'' polykings are distinct when none is a rigid transformation (translation (geometry), translation, rotation, reflection (mathematics), reflection or glide reflection) of another (pieces that can be picked up and flipped over). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domino (mathematics)
In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one ''free'' domino. Since it has reflection symmetry, it is also the only ''one-sided'' domino (with reflections considered distinct). When rotations are also considered distinct, there are two ''fixed'' dominoes: The second one can be created by rotating the one above by 90°. In a wider sense, the term ''domino'' is sometimes understood to mean a tile of any shape. Packing and tiling Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×''n'' rectangle with dominoes is F_n, the ''n''th Fibonacci number. Domino tilings figure in several celebrated problems, including the Aztec diamond problem in which large diamond-shaped regions have a number of tilings equal to a power of two, with most tilings appearing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blokus Trigon
''Blokus'' ( ) is an abstract strategy game, abstract strategy board game for two to four players, where players try to score points by occupying most of the board with pieces of their colour. The board is a square regular grid and the pieces are polyominoes. It was designed by French mathematician Bernard Tavitian and first released in 2000 by Sekkoïa, a French company. It has won several awards, including the Mensa Select award and the 2004 Teacher's Choice Award. In 2009, the game was sold to Mattel. Gameplay The game is played on a square board divided into 20 rows and 20 columns, for a total of 400 squares. There are a total of 84 game tiles, organized into 21 shapes in each of the four colors: blue, yellow, red, and green. The 21 shapes are based on Polyomino#Enumeration of polyominoes, free polyominoes of one to five squares (one monomino, one Domino (mathematics), domino, two trominoes/triominoes, five tetrominoes, and 12 pentominoes). These shapes are the first in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling Uniform Coloring 1
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, now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deltille
In geometry, the triangular tiling or triangular tessellation is one of the three Euclidean tilings by convex regular polygons#Regular tilings, regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of English mathematician John Horton Conway, John Conway called it a deltille, named from the triangular shape of the Greek letter Delta (letter), delta (Δ). The triangular tiling can also be called a kishextille by a Conway kis operator, kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of List of regular polytopes#Euclidean tilings, three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a triangular ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Triangular Tiling 111111
A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools, and by inmates in prisons. In some countries, some other officials also wear uniforms in their duties; such is the case of the Commissioned Corps of the United States Public Health Service or the French prefects. For some organizations, such as police, it may be illegal for non-members to wear the uniform. Etymology From the Latin ''unus'' (meaning one), and ''forma'' (meaning form). Variants Corporate and work uniforms Workers sometimes wear uniforms or corporate clothing of one nature or another. Workers required to wear a uniform may include retail workers, bank and post-office workers, public-security and health-care workers, blue-collar employees, personal trainers in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
