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Hexomino
A hexomino (or 6-omino) is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different ''free'' hexominoes. When reflections are considered distinct, there are 60 ''one-sided'' hexominoes. When rotations are also considered distinct, there are 216 ''fixed'' hexominoes. Symmetry The figure above shows all 35 possible free hexominoes, coloured according to their symmetry groups: * The twenty grey hexominoes have no symmetry. Their symmetry group consists only of the identity mapping. * The six red hexominoes have an axis of mirror symmetry parallel to the gridlines. Their symmetry group has two elements, the identity and a reflection in a line parallel to the sides of the squares. * The two green hexominoes have an axis of mirror symmetry at 45° to the gridlin ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
35 (number)
35 (thirty-five) is the natural number following 34 and preceding 36. In mathematics 35 is the sum of the first five triangular numbers, making it a tetrahedral number. 35 is the number of ways that three things can be selected from a set of seven unique things, also known as the " combination of seven things taken three at a time". 35 is a centered cube number, a centered tetrahedral number, a pentagonal number, and a pentatope number. 35 is a highly cototient number, since there are more solutions to the equation x - \varphi (x) = 35 than there are for any other integers below it except 1. There are 35 free hexominoes, the polyominoes made from six squares. Since the greatest prime factor of 35^ + 1 = 1226 is 613, which is more than 35 twice, 35 is a Størmer number. 35 is the tenth discrete semiprime (5 \times 7) and the first with 5 as the lowest non-unitary factor. The aliquot sum of 35 is 13, this being the second composite number with such an aliquot sum; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
216 (number)
216 (two hundred ndsixteen) is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato. In mathematics 216 is the cube of 6, and the sum of three cubes:216=6^3=3^3+4^3+5^3. It is the smallest cube that can be represented as a sum of three positive cubes, making it the first nontrivial example for Euler's sum of powers conjecture. It is, moreover, the smallest number that can be represented as a sum of any number of distinct positive cubes in more than one way. It is a highly powerful number: the product 3\times 3 of the exponents in its prime factorization 216 = 2^3\times 3^3 is larger than the product of exponents of any smaller number. Because there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number. Although it is not a semiprime, the three closest numbers on either side of it are, making it the middle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
60 (number)
60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called ''threescore'' in older literature ('' kopa'' in Slavic, ''Schock'' in Germanic). In mathematics * 60 is a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect number. * It is the smallest number divisible by the numbers 1 to 6: there is no smaller number divisible by the numbers 1 to 5. * It is the smallest number with exactly 12 divisors. * It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. * It is the smallest number that is the sum of two odd primes in six ways.Wells, D. '' The Pengui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
All 35 Free Hexominoes
All or ALL may refer to: Language * All, an indefinite pronoun in English * All, one of the English determiners * Allar language (ISO 639-3 code) * Allative case (abbreviated ALL) Music * All (band), an American punk rock band * ''All'' (All album), 1999 * ''All'' (Descendents album) or the title song, 1987 * ''All'' (Horace Silver album) or the title song, 1972 * ''All'' (Yann Tiersen album), 2019 * "All" (song), by Patricia Bredin, representing the UK at Eurovision 1957 * " All (I Ever Want)", a song by Alexander Klaws, 2005 * "All", a song by Collective Soul from '' Hints Allegations and Things Left Unsaid'', 1994 Science and mathematics * ALL (complexity), the class of all decision problems in computability and complexity theory * Acute lymphoblastic leukemia * Anterolateral ligament Sports * American Lacrosse League * Arena Lacrosse League, Canada * Australian Lacrosse League Other uses * All, Missouri, a community in the United States * All, a brand of Sun Product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is 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 exclusive or operations on two-bit binary values, or more abstractly as , the direct product of two copies of the cyclic group of 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 of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups. The smallest non-abelian group is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3- zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spheric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (polyhedron)
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel. Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The 11 Cubic Nets
''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Checkerboard
A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of checkered pattern on which checkers (also known as English draughts) is played. Most commonly, it consists of 64 squares (8×8) of alternating dark and light color, typically green and buff (official tournaments), black and red (consumer commercial), or black and white (printed diagrams). An 8×8 checkerboard is used to play many other games, including chess, whereby it is known as a chessboard. Other rectangular square-tiled boards are also often called checkerboards. Games and puzzles using checkerboards Martin Gardner featured puzzles based on checkerboards in his November 1962 Mathematical Games column in Scientific American. A square checkerboard with an alternating pattern is used for games including: * Amazons * Chapayev * Chess and some of its variants (see chessboard) * Czech draughts * Draughts, also known as checkers * Fox games * Frisian draughts * Gou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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