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5-cell
In geometry , the 5-CELL is a four-dimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid
Platonic solid
), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base. The REGULAR 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}. CONTENTS * 1 Alternative names * 2 Geometry
Geometry
* 2.1 Construction * 2.2 Boerdijk–Coxeter helix * 2.3 Projections * 3 Irregular 5-cell
5-cell
* 4 Compound * 5 Related polytopes and honeycomb * 6 References * 7 External links ALTERNATIVE NAMES * Pentachoron * 4-simplex * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron) * Hyperpyramid, tetrahedral pyramidGEOMETRYThe 5-cell
5-cell
is self-dual , and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism . Its dihedral angle is cos−1(1/4), or approximately 75.52°
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Pentatope Number
A PENTATOPE NUMBER is a number in the fifth cell of any row of Pascal\'s triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. The first few numbers of this kind are : 1 , 5 , 15 , 35 , 70 , 126 , 210, 330, 495 , 715, 1001 , 1365 (sequence A000332 in the OEIS ) A pentatope with side length 5 contains 70 3-spheres. Each layer represents one of the first five tetrahedral numbers . For example the bottom (green) layer has 35 spheres in total. Pentatope numbers belong in the class of figurate numbers , which can be represented as regular, discrete geometric patterns. The formula for the nth pentatopic number is: ( n + 3 4 ) = n ( n + 1 ) ( n + 2 ) ( n + 3 ) 24 = n 4 4 ! . {displaystyle {n+3 choose 4}={frac {n(n+1)(n+2)(n+3)}{24}}={n^{overline {4}} over 4!}.} Two of every three pentatope numbers are also pentagonal numbers . To be precise, the (3k − 2)th pentatope number is always the ((3k2 − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k2 + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k2 + k)/2 in the formula for pentagonal numbers. (These expressions always give integers). The infinite sum of the reciprocals of all pentatopal numbers is 4 3 {displaystyle 4 over 3} . This can be derived using telescoping series
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Schlegel Diagram
In geometry , a SCHLEGEL DIAGRAM is a projection of a polytope from R d {displaystyle R^{d}} into R d 1 {displaystyle R^{d-1}} through a point beyond one of its facets or faces . The resulting entity is a polytopal subdivision of the facet in R d 1 {displaystyle R^{d-1}} that is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel , who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4-polytope
4-polytope
to 3-space , respectively. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes. CONTENTS * 1 Construction * 2 Examples * 3 See also * 4 References * 5 Further reading * 6 External links CONSTRUCTIONThe most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces
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Convex Regular 4-polytope
In mathematics , a REGULAR 4-POLYTOPE is a regular four-dimensional polytope . They are the four-dimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli
Ludwig Schläfli
in the mid-19th century, although the full set were not discovered until later. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. CONTENTS * 1 History * 2 Construction * 3 Regular convex 4-polytopes * 3.1 Properties * 3.2 Visualization * 4 Regular star (Schläfli–Hess) 4-polytopes * 4.1 Names * 4.2 Symmetry * 4.3 Properties * 5 See also * 6 References * 6.1 Citations * 6.2 Bibliography * 7 External links HISTORYThe convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli
Ludwig Schläfli
in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell , great stellated 120-cell , grand 600-cell , and great grand stellated 120-cell ). He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2). That excludes cells and vertex figures as {5,5/2} and {5/2,5}
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Schläfli Symbol
In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas. CONTENTS * 1 Description * 2 Cases * 2.1 Symmetry groups * 2.2 Regular polygons (plane) * 2.3 Regular polyhedra (3 dimensions) * 2.4 Regular 4-polytopes (4 dimensions) * 2.5 Regular n-polytopes (higher dimensions) * 2.6 Dual polytopes * 2.7 Prismatic polytopes * 3 Extension of Schläfli symbols * 3.1 Polygons and circle tilings * 3.2 Polyhedra and tilings * 3.2.1 Alternations, quarters and snubs * 3.2.2 Altered and holosnubbed * 3.3 Polychora and honeycombs * 3.3.1 Alternations, quarters and snubs * 3.3.2 Bifurcating families * 4 See also * 5 References * 6 Sources * 7 External links DESCRIPTIONThe Schläfli symbol
Schläfli symbol
is a recursive description, starting with {p} for a p-sided regular polygon that is convex . For example, {3} is an equilateral triangle , {4} is a square , {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices. For example, {5/2} is a pentagram
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Coxeter Diagram
In geometry , a COXETER–DYNKIN DIAGRAM (or COXETER DIAGRAM, COXETER GRAPH) is a graph with numerically labeled edges (called BRANCHES) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras
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3-face
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. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope
4-polytope
. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4-POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex
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Tetrahedron
In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), 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 and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets . For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces
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2-face
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. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope . With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol
Schläfli symbol
POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4-POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex
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Edge (geometry)
In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higher-dimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz\'s theorem as being exactly the 3-vertex-connected planar graphs . NUMBER OF EDGES IN A POLYHEDRONAny convex polyhedron 's surface has Euler characteristic V E + F = 2 , {displaystyle V-E+F=2,} where V is the number of vertices , E is the number of edges, and F is the number of faces
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place. OF A POLYTOPEA vertex is a corner point of a polygon , polyhedron , or other higher-dimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called "convex " if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles ) ; otherwise, it is called "concave" or "reflex"
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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. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings , or space-filling tessellation with polytope cells . AS A FLAT SLICEMake a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This is perhaps the most common approach, and the most easily understood
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Petrie Polygon
In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive side (but no n) belongs to one of the facets . The PETRIE POLYGON of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces . For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon
Petrie polygon
becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane
Coxeter plane
of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group
Coxeter group
. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes. CONTENTS * 1 History * 2 The Petrie polygons of the regular polyhedra * 3 The Petrie polygon
Petrie polygon
of regular polychora (4-polytopes) * 4 The Petrie polygon
Petrie polygon
projections of regular and uniform polytopes * 5 Notes * 6 References * 7 External links HISTORY The Petrie polygon
Petrie polygon
for a cube is a skew hexagon passing through 6 of 8 vertices
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Pentagon
In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and angle ) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon) is called a pentagram . CONTENTS* 1 Regular pentagons * 1.1 Derivation of the area formula * 1.2 Inradius * 1.3 Chords from the circumscribed circle to the vertices * 1.4 Construction of a regular pentagon * 1.4.1 Richmond\'s method * 1.4.2 Carlyle circles * 1.4.3 Using trigonometry and the Pythagorean Theorem * 1.4.3.1 The construction * 1.4.4 † Proof that cos 36° = 1 + 5 4 {displaystyle {tfrac {1+{sqrt {5}}}{4}}} * 1.4.5 Side length is given * 1.4.5.1 The golden ratio * 1.5 Euclid\'s method * 1.5.1 Simply using a protractor (not a classical construction) * 1.6 Physical methods * 1.7 Symmetry * 2 Equilateral pentagons * 3 Cyclic pentagons * 4 General convex pentagons * 5 Graphs * 6 Examples of pentagons * 6.1 Plants * 6.2 Animals * 6.3 Artificial * 7 Pentagons in tiling * 8 Pentagons in polyhedra * 9 See also * 10 In-line notes and references * 11 External links REGULAR PENTAGONSA regular pentagon has Schläfli symbol
Schläfli symbol
{5} and interior angles are 108°
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Coxeter Group
In mathematics , a COXETER GROUP, named after H. S. M. Coxeter , is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors ). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 ). Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes , and the Weyl groups of simple Lie algebras . Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane , and the Weyl groups of infinite-dimensional Kac–Moody algebras . Standard references include (Humphreys 1992 ) and (Davis 2007 )
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Dual Polytope
In geometry , any polyhedron is associated with a second DUAL figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. 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