
Review of Short Phrases and Links 
This Review contains major "Vertex" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A vertex is a point where four or more edges meet.
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 A vertex is a position along with other information such as color, normal vector and texture coordinates.
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 A vertex is the intersection of 2lines, or in general nhyperplanes in LP problems with ndecision variables.
 A vertex (basic element) is simply drawn as a node or a dot.
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 A vertex is identified and assigned a reference.
 In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex).
 In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e.
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 In geometry a henagon (or monogon) is a polygon with one edge and one vertex.
 In a uniform polyhedron, every face is required to be a regular polygon, and every vertex is required to be identical, but the faces need not be identical.
 A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex.
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 Vertex description A way of describing a uniform polyhedron, by giving the sequence of face types meeting around a vertex.
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 This allows vertex figures to have negative angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat.
 There are no equilateral triangles in a geodesic dome, although the differences in the edges and vertex is not always immediately visible.
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 For example, the regular hexagon divides into 6 equilateral triangles and is the vertex figure for the regular triangular tiling.
 The regular polyhedron with three equilateral triangles at every vertex has four faces altogether and is called a regular tetrahedron.
 Similarly when four faces meet at each vertex, p.q.r.s, if one number is odd its neighbors must be equal.
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 FIG. 6 shows grid Type F, a diamond lattice composed of one type of vertex having tetrahedral symmetry.
 Turning back to FIG. 3, it is seen that a distance "n" from point 7 to the upper vertex 16 or 16' is equal to "h" minus "k".
 FIG. 6 shows an example of a vertex being moved in order to properly image the entire object from the scanning view.
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 Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron.
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 Of these, the center and four vertices in a tetrahedral relation have four edges per vertex and the alternating four vertices have eight edges per vertex.
 All four edges are identical, and the "geometry" at each vertex is the same.
 It shares the vertex and edge arrangement, as well as its pentagonal and pentagrammic faces, with the rhombidodecadodecahedron.
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 Thus a triangulation can be viewed as a collection of triangular faces, such that two faces either have an empty intersection or share an edge or a vertex.
 A polyhedron having 20 triangular faces with five edges meeting at each vertex.
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 It is composed of 20 triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence.
 The vertex figure of a triangular prism is an isosceles triangle.
 Its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.
 Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.
 The small stellated dodecahedron has five faces at each vertex in a pentagonal pattern and looks like a dodecahedron with pentagonal pyramids on each face.
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 In the case of the icosahedron, five faces meet at each vertex.
 Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex.
 Consider a cube where one face is replaced by a square pyramid: this is convex and the defects at each vertex are each positive.
 The face lattice of a square pyramid, drawn as a Hasse diagram; each face in the lattice is labeled by its vertex set.
 For example 3.5.3.5, represents the icosidodecahedron which alternates two triangle s and two pentagon s around each vertex.
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 Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.
 The icosidodecahedron, vertex configuration 3.5.3.5.
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 It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schl fli symbol 5,3.
 It has pentagonal faces, and 3 pentagons around each vertex.
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 A simple polygon has exactly one internal angle per vertex.
 For any { p, q }, the total angular measure around each vertex is q internal angles of the regular polygon { p }.
 The internal angles of the polygons meeting at a vertex must add to 360 degrees.
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 Archimedean solids have regular polygons as faces, and have symmetries which send any vertex to any other.
 A common heuristic for the Archimedean solids is that the arrangement of faces surrounding each vertex must be the same for all vertices.
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 The polygons meet in a the same pattern around each vertex, so the Archimedean solids are vertex transitive.
 A number of other uniform polyhedra which have tetragonal vertices (four faces meeting at each vertex) have duals containing irregular quadrilateral faces.
 Uniform polyhedra have regular faces meeting in the same manner at every vertex.
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 Uniform polyhedra are vertex uniform and every face is a regular polygon.They are either regular, quasiregular, or semiregular but not necessarily convex.
 The dimensions of a cube are the lengths of the three edges which meet at any vertex.
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 The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex.
 It is written as {p, q}, where p is the number of sides of each face, and q is the number of faces meeting at each vertex.
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 Obviously there need to be at least 3 faces meeting at each vertex to get a polyhedron.
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 Shape operators are computed at each vertex of the mesh from finite differences of vertex normals.
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 These normals are themselves interpolated along polygon edges from vertex normals that are computed, if necessary, just as in Gouraud shading.
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 Next, the converter computes new vertex normals for polygons that don't have any normals and that have a nonzero smoothing group assigned to them.
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 Lighting, which is computed per vertex, works with other rendering capabilities, including texture mapping, transparency, and Gouraud shading.
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 The second shading model, Gouraud shading, computes an intensity for each vertex and then interpolates the computed intensities across the polygons.
 Phong shading is more computationally expensive than Gouraud shading since the reflection model must be computed at each pixel instead of at each vertex.
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 Half way there (i.e., at rectification) one achieves a cuboctahedron which is a polywell shape (i.e., having an even number of faces at each vertex).
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 The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of the Cuboctahedron and Icosidodecahedron.
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 These symmetries show in the Cayley graph in the fact that the cuboctahedron has rotational and mirror symmetry around an axis through a vertex.
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 Uniform colorings and symmetry The cube has 3 uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.
 For example, 3.4.3.4 is the cuboctahedron with alternating triangular and square faces around each vertex.
 It shares the vertex and edge arrangement, as well as its square faces, with the uniform great rhombicosidodecahedron.
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 For example 4.8.8 means one square and two octagons on a vertex.
 It is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex.
 For example, the Truncated Cube (see the image at the right) has one triangle and two octagons around each vertex, so it is notated as (3,8,8).
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 For the vertex with eight faces, we get a new octagon face formed of six triangles.
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 A Johnson solid is a convex polyhedron each face of which is a regular polygon which is not vertex uniform.
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 Each vertex figure has an angle defect, and a convex polyhedron will have a combined angle defect of 720 degrees.
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 Since this sum is greater than 360 degrees, this symbol is not the vertex symbol of a convex polyhedron.
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 For instance, at any vertex of a cube there are three angles of, so the angle defect is.
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 You can visualize the angle defect by cutting along an edge at that vertex, and then flattening out a neighborhood of the vertex into the plane.
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 This is like angle defect: in fact, the total curvature of a region of a polyhedron containing exactly one vertex is the angle defect at that vertex.
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 The dodecahedron is a regular polyhedron with SchlÃ¤fli symbol {5,3}, having 3 pentagons around each vertex.
 The dodecahedron, with SchlÃ¤fli symbol {5,3}, faces are pentagons, vertex figures are triangles.
 The icosahedron, with SchlÃ¤fli symbol {3,5}, faces are triangles, vertex figures are pentagons.
 There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon.
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 Choose a point A on the circle that will serve as one vertex of the pentagon.
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 Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle.
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 The pentagonal antiprism is the polyhedron with three triangles and one pentagon around each vertex.
 Triangle (v) is a "vertex" triangle, and is adjacent to three triangles.
 At each vertex, three triangles and one of the chosen polygons meet.
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 In this Gouraud shading, the lighting calculation is done per vertex and interpolated across each fragment (pixel).
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 The colors for each vertex are taken from the c value, and interpolated across the polygon.
 Then, the position is interpolated across polygons to which the vertex is attached.
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 Texture coordinates are specified at each vertex of a given triangle, and these coordinates are interpolated using an extended Bresenham's line algorithm.
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 Since no vertex can be created by mesh conversion, triangles can inherit texture coordinates from tetragons.
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 Texture coordinates are assigned to each vertex of a polygon.
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 Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6) 2.
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 Fig. 7. Wrapping of (4.8.8.4.8.8) tilings of to form the pcu net with six hyperbolic squares per vertex, {4,6}.
 Dual to this is a tiling of the hyperbolic plane by equilateral triangles with 7 triangles meeting at each vertex.
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 At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares.
 This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4 n).
 The internal angle at each vertex of a regular octagon is 135 ° and the sum of all the internal angles is 1080 °.
 The angle defect at a vertex of a polygon is defined to be minus the sum of the angles at the corners of the faces at that vertex.
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 Then an elementary calculation of angles shows that the sum of the exterior angles of the polygon is equal to the sum of the face angles at the vertex.
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 Pyramids: These have any regular polygon for a base and isosceles triangles with a common vertex as the sides.
 The sum of the face angles around the common vertex is 60 + 90 + 108 + 108 = 366 degrees.
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 Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.
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 As the polyhedron being shown is made up of hexagons and squares, the angles among the three edges at the vertex varies.
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 There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.
 This model has two hexagons and two triangles meeting at each vertex, and all faces lie parallel to faces of an octahedron.
 The hexagon can form a regular tessellate the plane with a SchlÃ¤fli symbol {6,3}, having 3 hexagons around every vertex.
 There are one triangle, two squares, and one hexagon on each vertex.
 It is the dual tessellation of the great rhombitrihexagonal tiling which has one square and one hexagon and one dodecagon at each vertex.
 A rhombicosahedron shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms.
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 It is vertextransitive with 8 tetrahedra and 6 octahedra around each vertex.
 The 4dimensional crosspolytope, with SchlÃ¤fli symbol {3,3,4}, faces are tetrahedra, vertex figures are octahedra.
 The 24cell, with Schläfli symbol {3,4,3}, faces are octahedra, vertex figures are cubes.
 Yes, there are polytopes P with square faces, joined three per vertex (that is, there are polytopes of type {4,3}).
 Johnson's explanation in Polytopes  abstract and real uses the terms "body", edge" and "vertex" in an abstract context.
 It builds polytopes by a vertex figure and reflection, so only initially knows vertices and edges.
 Rectification is to cut off the vertices of a polytope such that the cutting hyperplane lies on the midpoints of the edges meeting at the vertex.
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 The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.
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 Each polytope is constructed from (n1) simplex and (n1) orthoplex facets, each has a vertex figure as the previous form.
 The Archimedean solid (shown at right) in which three squares and a triangle meet at each vertex is the rhombicuboctahedron.
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 For example, in III every vertex is the intersection of three triangles and three squares.
 For example, in Circle Limit III every vertex belongs to three triangles and three squares.
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 The faces of The Platonic Solids are congruent regular polygons, all meeting in the same way around each vertex.
 Every vertex star consists of three squares and a triangle, all using the same dihedral angles, and so all of the vertex stars are congruent.
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 We show that all the vertices are congruent by showing that the same number of faces around each vertex is the same for all vertices.
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 A regular octahedron is a Platonic solid composed of eight equilateral triangle s, four of which meet at each vertex.
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 The Platonic solids, which are either selfdual or dual with another Platonic solid, are vertex, edge, and facetransitive isogonal, isotoxal, and isohedral.
 By similar reasoning, a Platonic solid could only have three squares or three pentagons meeting at each vertex.
 Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
 Since each vertex is shared by three faces, there are 12*53 = 20 edges on the dodecahedron.
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 Proof. Each vertex of a Platonic solid is incident with at least three faces.
Categories
 Science > Mathematics > Geometry > Polygons
 Polyhedron
 Faces
 Science > Mathematics > Geometry > Vertices
 Science > Mathematics > Geometry > Edges
Subcategories
Related Keywords
* Angle
* Angles
* Arrangement
* Coordinates
* Cube
* Dodecahedron
* Edge
* Edges
* Faces
* Facets
* Figure
* Graph
* Icosahedron
* Midpoint
* Number
* Octahedron
* Pixel Shaders
* Point
* Polygon
* Polygons
* Polyhedra
* Polyhedron
* Polyhedron Vertex
* Set
* Shaders
* Side
* Sides
* Square
* Squares
* Tessellation
* Tetrahedra
* Tetrahedron
* ThreeDimensional Solid
* Three Meeting
* Triangle
* Triangles
* VertexTransitive
* Vertex Angles
* Vertex Arrangement
* Vertex Configuration
* Vertex Coordinates
* Vertex Figure
* Vertex Figures
* Vertex Shader
* Vertex Shaders
* Vertices

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