1.
Vertex configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
2.
Face configuration
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In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is one vertex type and therefore the vertex configuration fully defines the polyhedron. A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex, the notation a. b. c describes a vertex that has 3 faces around it, faces with a, b, and c sides. For example,3.5.3.5 indicates a vertex belonging to 4 faces, alternating triangles and this vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, the order is important, so 3.3.5.5 is different from 3.5.3.5. Repeated elements can be collected as exponents so this example is represented as 2. It has variously called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern. It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models, a vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. Different notations are used, sometimes with a comma and sometimes a period separator, the period operator is useful because it looks like a product and an exponent notation can be used. For example,3.5.3.5 is sometimes written as 2, the notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation means q p-gons around each vertex, so can be written as p. p. p. or pq. For example, an icosahedron is =3.3.3.3.3 or 35 and this notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron, the notation is ambiguous for chiral forms. For example, the cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration, the notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol, meaning it has 5 sides going around the centre twice, for example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of which expands to a vertex configuration 5/2. 5/2. 5/2. 5/2. 5/2 or combined as 5. The great stellated dodecahedron, has a vertex figure and configuration or 3
3.
Wythoff symbol
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In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra, a Wythoff symbol consists of three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators, with a slight extension, Wythoffs symbol can be applied to all uniform polyhedra. However, the methods do not lead to all uniform tilings in euclidean or hyperbolic space. In three dimensions, Wythoffs construction begins by choosing a point on the triangle. If the distance of this point from each of the sides is non-zero, a perpendicular line is then dropped between the generator point and every face that it does not lie on. The three numbers in Wythoffs symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, the triangle is also represented with the same numbers, written. In this notation the mirrors are labeled by the reflection-order of the opposite vertex, the p, q, r values are listed before the bar if the corresponding mirror is active. The one impossible symbol | p q r implies the point is on all mirrors. This unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, the resulting figure has rotational symmetry only. The generator point can either be on or off each mirror and this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. A node is circled if the point is not on the mirror. There are seven generator points with each set of p, q, r, | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isnt Wythoff-constructible, There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the many such patterns in the hyperbolic plane are also listed. The list of Schwarz triangles includes rational numbers, and determine the set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a domain, colored by even. Selected tilings created by the Wythoff construction are given below, for a more complete list, including cases where r ≠2, see List of uniform polyhedra by Schwarz triangle
4.
Truncated trihexagonal tiling
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In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex and it has Schläfli symbol of tr. There is only one uniform coloring of a trihexagonal tiling. A 2-uniform coloring has two colors of hexagons, 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares. The truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling, the first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a hexagon and surrounding triangles and square, in two different orientations. The Truncated trihexagonal tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing, circles can be alternatedly colored in this packing with an even number of sides of all the regular polygons of this tiling. The gap inside each hexagon allows for one circle, and each dodecagon allows for 7 circles, the kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree right triangles with 4,6, conway calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings and it can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. It is labeled V4.6.12 because each right triangle face has three types of vertices, one with 4 triangles, one with 6 triangles, and one with 12 triangles, the kisrhombille tiling triangles represent the fundamental domains of p6m, symmetry. There are a number of small index subgroups constructed from by mirror removal, creates *333 symmetry, shown as red mirror lines. The commutator subgroup is, which is 333 symmetry, a larger index 6 subgroup constructed as, also becomes, shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12. There are eight uniform tilings that can be based from the hexagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. This tiling can be considered a member of a sequence of patterns with vertex figure. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling
5.
Wallpaper group
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A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done
6.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
7.
Hexagonal tiling
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In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t, english mathematician Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees and it is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling, the hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, however, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, tubular graphene sheets have been synthesised, these are known as carbon nanotubes. They have many applications, due to their high tensile strength. Chicken wire consists of a lattice of wires. The hexagonal tiling appears in many crystals, in three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest known sphere packings in three dimensions, and are believed to be optimal, structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a tiling, all generated from reflective symmetry of Wythoff constructions. The represent the periodic repeat of one colored tile, counting hexagonal distances as h first, the 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling, in the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. The hexagons can be dissected into sets of 6 triangles and this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. This tiling is related to regular polyhedra with vertex figure n3. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6 and this tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry
8.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
9.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
10.
Isohedral figure
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
11.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
12.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. 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 set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
13.
Two-dimensional space
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In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons
14.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
15.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
16.
Conway polyhedron notation
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In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation defined by Kepler, the basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example tC represents a cube, and taC, parsed as t, is a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like a cube is an octahedron. Applied in a series, these allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology, while exact geometry is not constrained, the seed polyhedra are the Platonic solids, represented by the first letter of their name, the prisms for n-gonal forms, antiprisms, cupolae and pyramids. Any polyhedron can serve as a seed, as long as the operations can be executed on it, for example regular-faced Johnson solids can be referenced as Jn, for n=1.92. In general, it is difficult to predict the appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice becomes the same as the operation, aa=e, while a truncation after ambo produces bevel. There has been no general theory describing what polyhedra can be generated in by any set of operators, instead all results have been discovered empirically. Elements are given from the seed to the new forms, assuming seed is a polyhedron, An example image is given for each operation. The basic operations are sufficient to generate the reflective uniform polyhedra, some basic operations can be made as composites of others. Special forms The kis operator has a variation, kn, which only adds pyramids to n-sided faces, the truncate operator has a variation, tn, which only truncates order-n vertices. The operators are applied like functions from right to left, for example, a cuboctahedron is an ambo cube, i. e. t = aC, and a truncated cuboctahedron is t = t = taC. Chirality operator r – reflect – makes the image of the seed. Alternately an overline can be used for picking the other chiral form, the operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices. The first row generates the Archimedean solids and the row the Catalan solids. Comparing each new polyhedron with the cube, each operation can be visually understood, the truncated icosahedron, tI or zD, which is Goldberg polyhedron G, creates more polyhedra which are neither vertex nor face-transitive
17.
Square tiling
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In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of, meaning it has 4 squares around every vertex, the internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane, the other two are the triangular tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a square tiling, naming the colors by indices on the 4 squares around a vertex,1111,1112,1112,1122,1123,1123,1212,1213,1234. Cases have simple reflection symmetry, and glide reflection symmetry, three can be seen in the same symmetry domain as reduced colorings, 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. This tiling is related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane. Like the uniform there are eight uniform tilings that can be based from the regular square tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. However treating faces identically, there are only three distinct forms, square tiling, truncated square tiling, snub square tiling. Other quadrilateral tilings can be made with topologically equivalent to the square tiling, isohedral tilings have identical faces and vertex-transitivity, there are 17 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two colinear edges. Symmetry given assumes all faces are the same color, the square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing, the packing density is π/4=78. 54% coverage. There are 4 uniform colorings of the circle packings, there are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices, Regular apeirogons pr are contrained by, 1/p + 2/q + 1/r =1. Edges have p vertices, and vertex figures are r-gonal, checkerboard List of regular polytopes List of uniform tilings Square lattice Tilings of regular polygons Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p.296, Table II, Regular honeycombs Klitzing, 2D Euclidean tilings o4o4x - squat - O1. The Geometrical Foundation of Natural Structure, A Source Book of Design, p36 Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Square Grid
18.
Uniform coloring
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In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different symmetries can be expressed on the geometric figure with the faces following different uniform color patterns. A uniform coloring can be specified by listing the different colors with indices around a vertex figure, in addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive. A related term is Archimedean color requires one vertex figure coloring repeated in a periodic arrangement, a more general term are k-Archimedean colorings which count k distinctly colored vertex figures. For example this Archimedean coloring of a tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring, Grünbaum, Branko, Shephard. CS1 maint, Multiple names, authors list Uniform and Archimedean colorings, pp. 102–107 Weisstein, Uniform Tessellations on the Euclid plane Tessellations of the Plane David Baileys World of Tessellations k-uniform tilings n-uniform tilings
19.
Archimedean coloring
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In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different symmetries can be expressed on the geometric figure with the faces following different uniform color patterns. A uniform coloring can be specified by listing the different colors with indices around a vertex figure, in addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive. A related term is Archimedean color requires one vertex figure coloring repeated in a periodic arrangement, a more general term are k-Archimedean colorings which count k distinctly colored vertex figures. For example this Archimedean coloring of a tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring, Grünbaum, Branko, Shephard. CS1 maint, Multiple names, authors list Uniform and Archimedean colorings, pp. 102–107 Weisstein, Uniform Tessellations on the Euclid plane Tessellations of the Plane David Baileys World of Tessellations k-uniform tilings n-uniform tilings
20.
Vertex arrangement
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In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes, for example, a square vertex arrangement is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same vertex arrangement if they share the same 0-skeleton, the same set of vertices can be connected by edges in different ways. For example, the pentagon and pentagram have the same vertex arrangement, a vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a vertex arrangement. Infinite tilings can also share common vertex arrangements, for example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces. Polyhedra can also share an edge arrangement while differing in their faces, for example, of the ten nonconvex regular Schläfli-Hess polychora, there are only 7 unique face arrangements. Synonyms for special cases include company for a 2-regiment and army for a 0-regiment, n-skeleton - a set of elements of dimension n and lower in a higher polytope. Vertex figure - A local arrangement of faces in a polyhedron around a single vertex, archived from the original on 4 February 2007. Archived from the original on 4 February 2007, archived from the original on 4 February 2007
21.
Simplectic honeycomb
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In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the A ~ n affine Coxeter group symmetry. It is given a Schläfli symbol, and is represented by a Coxeter-Dynkin diagram as a graph of n+1 nodes with one node ringed. It is composed of facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes x + y +, ∈ Z, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of a honeycomb is an expanded n-simplex. In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles, in 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph, with 5-cell, in 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph, filling space by 6-simplex, rectified 6-simplex, for 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall short on higher dimensions. For 4 to 8 dimensions, the numbers are 20,30,42,56, and 72 spheres, while the greatest solutions are 24,40,72,126. Norman Johnson Uniform Polytopes, Manuscript Coxeter, H. S. M, Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III
22.
Circle packing
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This article describes the packing of circles on surfaces. For the related article on circle packing with an intersection graph. In geometry, circle packing is the study of the arrangement of circles on a surface such that no overlapping occurs. The associated packing density, η, of an arrangement is the proportion of the covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, while the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. Packing densities of concave shapes such as star polygons can be arbitrarily small, the density of this arrangement is η h = π23 ≈0.9069. Axel Thue provided the first proof that this was optimal in 1890, showing that the lattice is the densest of all possible circle packings. However, his proof was considered by some to be incomplete, the first rigorous proof is attributed to László Fejes Tóth in 1940. At the other extreme, very low density arrangements of rigidly packed circles have been identified, there are 11 circle packings based on the 11 uniform tilings of the plane. In these packings, every circle can be mapped to every other circle by reflections and rotations, the hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing, the snub hexagonal tiling has two mirror-image forms. A related problem is to determine the arrangement of identically interacting points that are constrained to lie within a given surface. The Thomson problem deals with the lowest energy distribution of electric charges on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere and this is analogous to distributing non-point charges on a sphere. Packing circles in simple bounded shapes is a type of problem in recreational mathematics. The influence of the walls is important, and hexagonal packing is generally not optimal for small numbers of circles. There are also a range of problems which permit the sizes of the circles to be non-uniform, one such extension is to find the maximum possible density of a system with two specific sizes of circle. Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in contact with two other circles
23.
Kissing number problem
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In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, other names for kissing number that have been used are Newton number, and contact number. In general, the number problem seeks the maximum possible kissing number for n-dimensional spheres in -dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space, finding the kissing number when centers of spheres are confined to a line or a plane is trivial. Proving a solution to the case, despite being easy to conceptualise and model in the physical world. Solutions in higher dimensions are more challenging, and only a handful of cases have been solved exactly. For others investigations have determined upper and lower bounds, but not exact solutions. In one dimension, the number is 2, In two dimensions, the kissing number is 6, Proof, Consider a circle with center C that is touched by circles with centers C1. These rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°, assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, say C C1 and C C2, are separated by an angle of less than 60°, the segments C Ci have the same length – 2r – for all i. Therefore the triangle C C1 C2 is isosceles, and its third side – C1 C2 – has a length of less than 2r. Therefore the circles 1 and 2 intersect – a contradiction, in three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of left over. This was the subject of a disagreement between mathematicians Isaac Newton and David Gregory. Newton correctly thought that the limit was 12, Gregory thought that a 13th could fit, some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by Reinhold Hoppe, but the first correct proof did not appear until 1953. The twelve neighbors of the sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size. A coordination number of 12 is found in a cubic close-packed or a hexagonal close-packed structure, in four dimensions, it was known for some time that the answer is either 24 or 25. It is easy to produce a packing of 24 spheres around a central sphere, as in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was even less clear
24.
Voronoi diagram
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In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points is specified beforehand, and for each seed there is a region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells, the Voronoi diagram of a set of points is dual to its Delaunay triangulation. It is named after Georgy Voronoi, and is called a Voronoi tessellation, a Voronoi decomposition. Voronoi diagrams have practical and theoretical applications to a number of fields, mainly in science and technology. They are also known as Thiessen polygons, in the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. Each such cell is obtained from the intersection of half-spaces, the line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices are the points equidistant to three sites, let X be a metric space with distance function d. Let K be a set of indices and let k ∈ K be a tuple of nonempty subsets in the space X. In other words, if d = inf denotes the distance between the point x and the subset A, then R k = The Voronoi diagram is simply the tuple of cells k ∈ K. In principle some of the sites can intersect and even coincide, in addition, infinitely many sites are allowed in the definition, but again, in many cases only finitely many sites are considered. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram, however, in general the Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the definition in usual terms. Each Voronoi polygon R k is associated with a generator point P k, let X be the set of all points in the Euclidean space. Let P1 be a point that generates its Voronoi region R1, P2 that generates R2, and P3 that generates R3, and so on. Then, as expressed by Tran et al all locations in the Voronoi polygon are closer to the point of that polygon than any other generator point in the Voronoi diagram in Euclidian plane. As a simple illustration, consider a group of shops in a city, suppose we want to estimate the number of customers of a given shop. With all else being equal, it is reasonable to assume that customers choose their preferred shop simply by distance considerations, they will go to the shop located nearest to them
25.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
26.
Isosceles triangle
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
27.
Right triangle
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A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a triangle is the basis for trigonometry. The side opposite the angle is called the hypotenuse. The sides adjacent to the angle are called legs. Side a may be identified as the adjacent to angle B and opposed to angle A, while side b is the side adjacent to angle A. If the lengths of all three sides of a triangle are integers, the triangle is said to be a Pythagorean triangle. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the then the other is height. As a formula the area T is T =12 a b where a and b are the legs of the triangle and this formula only applies to right triangles. From this, The altitude to the hypotenuse is the mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the hypotenuse, in equations, f 2 = d e, b 2 = c e, a 2 = c d where a, b, c, d, e, f are as shown in the diagram. Moreover, the altitude to the hypotenuse is related to the legs of the triangle by 1 a 2 +1 b 2 =1 f 2. For solutions of this equation in integer values of a, b, f, the altitude from either leg coincides with the other leg. Since these intersect at the vertex, the right triangles orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that, In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs. This can be stated in equation form as a 2 + b 2 = c 2 where c is the length of the hypotenuse, Pythagorean triples are integer values of a, b, c satisfying this equation. The radius of the incircle of a triangle with legs a and b. The radius of the circumcircle is half the length of the hypotenuse, thus the sum of the circumradius and the inradius is half the sum of the legs, R + r = a + b 2
28.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
29.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
30.
Pyramid (geometry)
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
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Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
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Icosahedron
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In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedra or icosahedrons. There are many kinds of icosahedra, with some being more symmetrical than others, the best known is the Platonic, convex regular icosahedron. There are two objects, one convex and one concave, that can both be called regular icosahedra, each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. The term regular icosahedron generally refers to the variety, while the nonconvex form is called a great icosahedron. Its dual polyhedron is the dodecahedron having three regular pentagonal faces around each vertex. The great icosahedron is one of the four regular star Kepler-Poinsot polyhedra, like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges and its dual polyhedron is the great stellated dodecahedron, having three regular star pentagonal faces around each vertex. Stellation is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron and it is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure. In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 58 such stellations of the regular icosahedron, of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them, other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are referred to as such. A regular icosahedron can be distorted or marked up as a lower symmetry, and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron. This can be seen as a truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently, pyritohedral symmetry has the symbol, with order 24. Tetrahedral symmetry has the symbol, +, with order 12 and these lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles. These symmetries offer Coxeter diagrams, and respectively, each representing the lower symmetry to the regular icosahedron, the coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form. These coordinates represent the truncated octahedron with alternated vertices deleted and this construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector, where ϕ is the golden ratio
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Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
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Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
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Hyperbolic space
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In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. When embedded to a Euclidean space, every point of a space is a saddle point. Hyperbolic n-space, denoted Hn, is the symmetric, simply connected. Hyperbolic space is a space exhibiting hyperbolic geometry and it is the negative-curvature analogue of the n-sphere. Although hyperbolic space Hn is diffeomorphic to Rn, its negative-curvature metric gives it very different geometric properties, hyperbolic 2-space, H2, is also called the hyperbolic plane. Instead, the postulate is replaced by the following alternative, Given any line L and point P not on L. It is then a theorem that there are many such lines through P. This axiom still does not uniquely characterize the hyperbolic plane up to isometry, there is a constant, the curvature K <0. However, it does uniquely characterize it up to homothety, meaning up to bijections which only change the notion of distance by an overall constant, by choosing an appropriate length scale, one can thus assume, without loss of generality, that K = −1. Models of hyperbolic spaces that can be embedded in a flat spaces may be constructed, in particular, the existence of model spaces implies that the parallel postulate is logically independent of the other axioms of Euclidean geometry. There are several important models of space, the Klein model, the hyperboloid model, the Poincaré ball model. These all model the same geometry in the sense that any two of them can be related by a transformation that preserves all the properties of the space. The hyperboloid model realizes hyperbolic space as a hyperboloid in Rn+1 =, the hyperboloid is the locus Hn of points whose coordinates satisfy x 02 − x 12 − ⋯ − x n 2 =1, x 0 >0. In this model a line is the curve formed by the intersection of Hn with a plane through the origin in Rn+1, the hyperboloid model is closely related to the geometry of Minkowski space. The space Rn+1, equipped with the bilinear form B, is an -dimensional Minkowski space Rn,1, one can associate a distance on the hyperboloid model by defining the distance between two points x and y on H to be d = arcosh B. This function satisfies the axioms of a metric space and it is preserved by the action of the Lorentz group on Rn,1. Hence the Lorentz group acts as a group preserving isometry on Hn. An alternative model of geometry is on a certain domain in projective space
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Dihedron
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A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons, a regular dihedron is the dihedron formed by two regular polygons, which may be described by the Schläfli symbol. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere, the dual of a n-gonal dihedron is the n-gonal hosohedron, where n digon faces share two vertices. A dihedron can be considered a degenerate prism consisting of two n-sided polygons connected back-to-back, so that the object has no depth. The polygons must be congruent, but glued in such a way one is the mirror image of the other. This characterization holds also for the distances on the surface of a dihedron, as a spherical tiling, a dihedron can exist as nondegenerate form, with two n-sided faces covering the sphere, each face being a hemisphere, and vertices around a great circle. The regular polyhedron is self-dual, and is both a hosohedron and a dihedron, in the limit the dihedron becomes an apeirogonal dihedron as a 2-dimensional tessellation, A regular ditope is an n-dimensional analogue of a dihedron, with Schläfli symbol. It has two facets, which share all ridges, in common, polyhedron Polytope Weisstein, Eric W. Dihedron
37.
Order-7 triangular tiling
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In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of. The symmetry group of the tiling is the group. The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, the dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces. It is related to two star-tilings by the vertex arrangement, the order-7 heptagrammic tiling, and heptagrammic-order heptagonal tiling. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbol. From a Wythoff construction there are eight uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. Weisstein, Eric W. Poincaré hyperbolic disk, Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
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Order-8 triangular tiling
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In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of, having eight regular triangles around each vertex, the half symmetry = can be shown with alternating two colors of triangles, From symmetry, there are 15 small index subgroups by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, is the subgroup of. A larger subgroup is constructed, index 8, as with gyration points removed, the symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, from a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings. Drawing the tiles colored as red on the faces, yellow at the original vertices. Weisstein, Eric W. Poincaré hyperbolic disk, Hyperbolic and Spherical Tiling Gallery KaleidoTile 3, Educational software to create spherical, planar and hyperbolic tilings Hyperbolic Planar Tessellations, Don Hatch
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Infinite-order triangular tiling
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In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of. All vertices are ideal, located at infinity and seen on the boundary of the Poincaré hyperbolic disk projection, a lower symmetry form has alternating colors, and represented by cyclic symbol. The tiling also represents the fundamental domains of the *∞∞∞ symmetry and this tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Chapter 10, the Beauty of Geometry, Twelve Essays. Weisstein, Eric W. Poincaré hyperbolic disk
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Catalan solid
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In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, the Catalan solids are all convex. They are face-transitive but not vertex-transitive and this is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons, however, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra, additionally, two of the Catalan solids are edge-transitive, the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids, just as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral, the pentagonal icositetrahedron and these each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids, the Catalan solids, along with their dual Archimedean solids, can be grouped by their symmetry, tetrahedral, octahedral, and icosahedral. There are 6 forms per symmetry, while the self-symmetric tetrahedral group only has three forms and two of those are duplicated with octahedral symmetry. J. lÉcole Polytechnique 41, 1-71,1865, alan Holden Shapes, Space, and Symmetry. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Williams, the Geometrical Foundation of Natural Structure, A Source Book of Design. California, University of California Press Berkeley, chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Catalan Solids. Archived from the original on 4 February 2007, Archimedean duals – at Virtual Reality Polyhedra Interactive Catalan Solid in Java
41.
Triakis tetrahedron
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In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron and it can be seen as a tetrahedron with triangular pyramids added to each face, that is, it is the Kleetope of the tetrahedron. This interpretation is expressed in the name, the length of the shorter edges is 3/5 that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area 5/3√11, a triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will be coplanar and this can be seen by adding the 6 edges of tetrahedron inside of a cube. This chiral figure is one of thirteen stellations allowed by Millers rules, the triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry, the Geometrical Foundation of Natural Structure, A Source Book of Design
42.
Tetrakis hexahedron
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In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the octahedron, an Archimedean solid. It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron, the tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. Naturally occurring formations of tetrahexahedra are observed in copper and fluorite systems, polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers. The tetrakis hexahedron appears as one of the simplest examples in building theory, consider the Riemannian symmetric space associated to the group SL4. Its Tits boundary has the structure of a building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices can be obtained by taking the radial projection of a tetrakis hexahedron, with Td, tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere and it can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point. The edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles and these edges can also be seen as a compound of 3 orthogonal square hosohedrons. If we denote the length of the base cube by a. The inclination of each face of the pyramid versus the cube face is arctan. One edge of the triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length. This yields an altitude of √5a/4 in the triangle and its area is √5a/8, and the internal angles are arccos and the complementary 180° −2 arccos. The volume of the pyramid is a3/12, so the volume of the six pyramids. It can be seen as a cube with square pyramids covering each square face and it is a polyhedra in a sequence defined by the face configuration V4.6. 2n. With an even number of faces at every vertex, these polyhedra, each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3, n mirrors at each triangle face vertex. Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face Catalan solid, the Geometrical Foundation of Natural Structure, A Source Book of Design
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Pentakis dodecahedron
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In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face, that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name, there are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include, The usual Catalan pentakis dodecahedron, a convex hexecontahedron with sixty isosceles triangular faces illustrated in the sidebar figure and it is a Catalan solid, dual to the truncated icosahedron, an Archimedean solid. As the heights of the pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi. As the height is raised further, the shape becomes non-convex, other more non-convex geometric variants include, The small stellated dodecahedron. Great pentakis dodecahedron Wenningers third stellation of icosahedron, if one affixes pentagrammic pyramids into Wenningers third stellation of icosahedron one obtains the great icosahedron. The pentakis dodecahedron in a model of buckminsterfullerene, each surface segment represents a carbon atom, equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom. The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses and these have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron. The pentakis dodecahedron has three positions, two on vertices, and one on a midedge, The Spaceship Earth structure at Walt Disney Worlds Epcot is a derivative of a pentakis dodecahedron. The model for a campus arts workshop designed by Jeffrey Lindsay was actually a hemispherical pentakis dodecahedron https, in Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron and these Domes also appear whenever the player transforms on a dome in the Hypno Ray level. Some Geodomes in which play on are Pentakis Dodecahedra. The Geometrical Foundation of Natural Structure, A Source Book of Design, the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Pentakis dodecahedron at MathWorld. Pentakis Dodecahedron – Interactive Polyhedron Model
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Truncated order-7 triangular tiling
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In geometry, the Order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a similar to a conventional soccer ball with heptagons in place of pentagons. It has Schläfli symbol of t and this tiling is called a hyperbolic soccerball for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a surface can be constructed in 3-space. The dual tiling is called a heptakis heptagonal tiling, named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point. This hyperbolic tiling is related as a part of sequence of uniform truncated polyhedra with vertex configurations. From a Wythoff construction there are eight uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the faces, yellow at the original vertices. The Beauty of Geometry, Twelve Essays, weisstein, Eric W. Poincaré hyperbolic disk