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Decomino
A decomino, or 10-omino, is a polyomino of order 10, that is, a polygon in the plane made of 10 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 4,655 different ''free'' decominoes (the free decominoes comprise 195 with holes and 4,460 without holes). When reflections are considered distinct, there are 9,189 ''one-sided'' decominoes. When rotations are also considered distinct, there are 36,446 ''fixed'' decominoes. Symmetry The 4,655 free decominoes can be classified according to their symmetry groups: * 4,461 decominoes have no symmetry. Their symmetry group consists only of the identity mapping. * 90 decominoes have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. * 22 decominoes have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomino With Two Axes Of Symmetry
A decomino, or 10-omino, is a polyomino of order 10, that is, a polygon in the plane made of 10 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 4,655 different ''free'' decominoes (the free decominoes comprise 195 with holes and 4,460 without holes). When reflections are considered distinct, there are 9,189 ''one-sided'' decominoes. When rotations are also considered distinct, there are 36,446 ''fixed'' decominoes. Symmetry The 4,655 free decominoes can be classified according to their symmetry groups: * 4,461 decominoes have no symmetry. Their symmetry group consists only of the identity mapping. * 90 decominoes have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. * 22 decominoes have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two ele ... [...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 to 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, the study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free 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 to 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, the study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octomino
An octomino (or 8-omino) is a polyomino of order 8, that is, a polygon in the plane made of 8 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 369 different ''free'' octominoes. When reflections are considered distinct, there are 704 ''one-sided'' octominoes. When rotations are also considered distinct, there are 2,725 ''fixed'' octominoes. Symmetry The figure shows all possible free octominoes, coloured according to their symmetry groups: * 316 octominoes (coloured grey) have no symmetry. Their symmetry group consists only of the identity mapping. * 23 octominoes (coloured red) have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. :: * 5 octominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting oppo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Packing Problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a bin packing problem, you are given: * A ''container'', usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem. * A set of ''objects'', some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geomagic Square - 3x3 Decominoes
Geomagic is the professional engineering software brand of 3D Systems. The brand began when Geomagic Inc., a software company based in Morrisville, North Carolina, was acquired by 3D Systems in February 2013 and combined with that company's other software businesses (namely Rapidform acquired by 3D Systems in October 2012 and Albre in July 2011). Geomagic was founded in 1997 by Ping Fu and Herbert Edelsbrunner. Geomagic-branded software products are focused on computer-aided design, with an emphasis on 3D scanning and other non-traditional design methodologies, such as voxel-based modeling with haptic technology, haptic input. 3D Systems also markets 3D quality inspection software as well as 3D scanners under the Geomagic brand. Geomagic Products 3D Scanning Systems Geomagic Capture is an integrated system consisting of a blue LED structured-light 3D scanner and one of several pieces of application-specific software. The systems are marketed for use as scan-based design tools, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Perfect Self-tiling Tile Set Of Order 4
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguish it f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonomino
A nonomino (or enneomino or 9-omino) is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different ''free'' nonominoes. When reflections are considered distinct, there are 2,500 ''one-sided'' nonominoes. When rotations are also considered distinct, there are 9,910 ''fixed'' nonominoes. Symmetry The 1,285 free nonominoes can be classified according to their symmetry groups: * 1,196 nonominoes have no symmetry. Their symmetry group consists only of the identity mapping. * 38 nonominoes have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. * 26 nonominoes have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal and vertical reflection and 180-degree rotation), as the group of bitwise operation, bitwise exclusive or operations on two-bit binary values, or more abstract algebra, abstractly as , the Direct product of groups, direct product of two copies of the cyclic group of Order (group theory), order 2. It was named ''Vierergruppe'' (meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter V or as K4. The Klein four-group, with four elements, is the smallest group that is not a cyclic group. There is only one other group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric. In conclusion, a line of symmetry splits the shape in half and those halves should be identical. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). The symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids. Formal treatment Formally the rotational symmetry is symmetry with respect to some or all rotations in ''m''-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of ''E''+(''m'') (see Euclidean group). Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
