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Vertex Figure
In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g
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Triangular Prism
In geometry , a TRIANGULAR PRISM is a three-sided prism ; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides . A RIGHT TRIANGULAR PRISM has rectangular sides, otherwise it is oblique. A UNIFORM TRIANGULAR PRISM is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms . All cross-sections parallel to the base faces are the same triangle
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Isosceles Triangle
In geometry , an ISOSCELES TRIANGLE is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case . By the isosceles triangle theorem , the two angles opposite the equal sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem
Steiner–Lehmus theorem
, every triangle with two angle bisectors of equal length is isosceles
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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices _A_, _B_, and _C_ is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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Square (geometry)
In geometry , a SQUARE is a regular quadrilateral , which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles ). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted {displaystyle square } ABCD
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Vertex Configuration
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 only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.) 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 pentagons . This vertex configuration defines the vertex-transitive icosidodecahedron . The notation is cyclic and therefore is equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5
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Great Icosahedron
In geometry , the GREAT ICOSAHEDRON is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra ), with Schläfli symbol {3,5/2} and Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. CONTENTS * 1 Images * 2 As a snub * 3 Related polyhedra * 4 References * 5 External links IMAGES TRANSPARENT MODEL DENSITY STELLATION DIAGRAM NET A transparent model of the great icosahedron (See also Animation ) It has a density of 7, as shown in this cross-section. It is a stellation of the icosahedron, counted by Wenninger as model and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter . × 12 Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron
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Pentagram
A PENTAGRAM (sometimes known as a PENTALPHA or PENTANGLE or a STAR PENTAGON ) is the shape of a five-pointed star drawn with five straight strokes. Pentagrams were used symbolically in ancient Greece and Babylonia
Babylonia
, and are used today as a symbol of faith by many Wiccans , akin to the use of the cross by Christians and the Star of David
Star of David
by Jews. The pentagram has magical associations, and many people who practice Neopagan faiths wear jewelry incorporating the symbol. Christians once more commonly used the pentagram to represent the five wounds of Jesus . The pentagram has associations with Freemasonry
Freemasonry
and is also utilized by other belief systems. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line"
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Star Polygon
In geometry , a STAR POLYGON is a type of non-convex polygon . Only the REGULAR STAR POLYGONS have been studied in any depth; star polygons in general appear not to have been formally defined. Branko Grünbaum
Branko Grünbaum
identified two primary definitions used by Kepler
Kepler
, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons . The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") 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 formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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Polyhedron
In geometry , a POLYHEDRON (plural POLYHEDRA or POLYHEDRONS) 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- (stem of πολύς, "many") + -hedron (form of ἕδρα, "base" or "seat"). 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
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Polytope
In elementary geometry , a POLYTOPE is a geometric object with "flat" sides. It is a generalisation in any number of dimensions, of the three-dimensional polyhedron . Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or N-POLYTOPE. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations , decompositions or tilings of curved manifolds including spherical polyhedra , and set-theoretic abstract polytopes . Polytopes in more than three dimensions were first discovered by Ludwig Schläfli
Ludwig Schläfli

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Coxeter
HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London
London
but spent most of his adult life in Canada
Canada
. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal
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Tessellation
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. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and Semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. 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 small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons
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Honeycomb (geometry)
In geometry , a HONEYCOMB is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces , such as hyperbolic honeycombs . Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles , as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell
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Cell (geometry)
In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes , the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). CONTENTS* 1 Polygonal face * 1.1 Number of polygonal faces of a polyhedron * 2 k-face * 2.1 Cell or 3-face
3-face
* 2.2 Facet or (n-1)-face * 2.3 Ridge or (n-2)-face * 2.4 Peak or (n-3)-face * 3 See also * 4 References * 5 External links POLYGONAL FACEIn elementary geometry, a FACE is a polygon on the boundary of a polyhedron . Other names for a polygonal face include SIDE of a polyhedron, and TILE of a Euclidean plane tessellation . For example, any of the six squares that bound a cube is a face of the cube
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