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Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces. [1]

102 relations: Alum, Antiprism, Ball (mathematics), Bipyramid, Buckminster Fuller, Cantilever, Cartesian coordinate system, Centered octahedral number, Conformal map, Convex uniform honeycomb, Coordinate system, Coordination complex, Coxeter element, Coxeter–Dynkin diagram, Cross-polytope, Crystal, Cube, Cuboctahedron, Diamond, Disdyakis dodecahedron, Dual polyhedron, Equilateral triangle, Faceting, Fluorite, Fundamental domain, Geometry, Golden ratio, Harold Scott MacDonald Coxeter, Hexagonal prism, Hexany, Honeycomb (geometry), Hyperbolic space, Hypercube, Hyperoctahedral group, Hyperplane, Hypersimplex, Icosahedron, Icosidodecahedron, Iron, Jahn–Teller effect, Johnson solid, K-vertex-connected graph, Kamacite, Kite (geometry), List of spherical symmetry groups, Maximal independent set, Meteorite, Net (polyhedron), Nickel, Octahedral molecular geometry, ..., Octahedral symmetry, Octahedrite, Ohm, Orbifold notation, Order (group theory), Orthogonality, Orthographic projection, Pentagonal bipyramid, Platonic solid, Polyhedron, Polytope compound, Projection (linear algebra), Pyramid (geometry), Radius, Rectification (geometry), Regular Polytopes (book), Resistor, Role-playing game, Schönhardt polyhedron, Schläfli symbol, Simplicial polytope, Snub disphenoid, Space frame, Sphere, Spherical polyhedron, Square bifrustum, Stellated octahedron, Stellation, Stereographic projection, Subgroup, Symmetry group, Tangent, Taxicab geometry, Tesseract, Tetragonal trapezohedron, Tetrahedral symmetry, Tetrahedral-octahedral honeycomb, Tetrahedron, Tetrahemihexahedron, Triakis octahedron, Triangle, Truncated octahedron, Truncated tetrahedron, Turán graph, Uniform coloring, Vertex arrangement, Vertex configuration, Volume, Well-covered graph, Widmanstätten pattern, Wythoff construction, Wythoff symbol. Expand index (52 more) »

Alum

Alum is both a specific chemical compound and a class of chemical compounds.

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Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

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Ball (mathematics)

In mathematics, a ball is the space inside a sphere.

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Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

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Buckminster Fuller

Richard Buckminster "Bucky" Fuller (July 12, 1895 – July 1, 1983) was an American architect, systems theorist, author, designer, and inventor.

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Cantilever

A cantilever is a rigid structural element, such as a beam or a plate, anchored at only one end to a (usually vertical) support from which it is protruding.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

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Centered octahedral number

A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin.

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Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

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Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.

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Coordination complex

In chemistry, a coordination complex or metal complex consists of a central atom or ion, which is usually metallic and is called the coordination centre, and a surrounding array of bound molecules or ions, that are in turn known as ligands or complexing agents.

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Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

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Coxeter–Dynkin 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).

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Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

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Crystal

A crystal or crystalline solid is a solid material whose constituents, such as atoms, molecules or ions, are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.

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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

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Diamond

In mineralogy, diamond (or; from the ancient Greek ἀδάμας – adámas "unbreakable") is a metastable allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice.

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Disdyakis dodecahedron

In geometry, a disdyakis dodecahedron, or hexakis octahedron or kisrhombic dodecahedron), is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It superficially resembles an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron.

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Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other.

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Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

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Faceting

Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.

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Fluorite

Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2.

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Fundamental domain

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

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Geometry

Geometry (from the 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.

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Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base.

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Hexany

In music theory, the hexany is a six-note just intonation structure, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube.

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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.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n.

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Hyperoctahedral group

In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope.

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Hyperplane

In geometry a hyperplane is a subspace of one dimension less than its ambient space.

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Hypersimplex

In polyhedral combinatorics, a hypersimplex, Δd,k, is a convex polytope that generalizes the simplex.

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Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

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Icosidodecahedron

In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces.

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Iron

Iron is a chemical element with symbol Fe (from ferrum) and atomic number 26.

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Jahn–Teller effect

The Jahn–Teller effect, sometimes also known as Jahn–Teller distortion, describes the geometrical distortion of molecules and ions that is associated with certain electron configurations.

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Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism.

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K-vertex-connected graph

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed.

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Kamacite

Kamacite is an alloy of iron and nickel, which is found on earth only in meteorites.

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Kite (geometry)

In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

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List of spherical symmetry groups

Spherical symmetry groups are also called point groups in three dimensions; however, this article is limited to the finite symmetries.

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Maximal independent set

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set.

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Meteorite

A meteorite is a solid piece of debris from a source such as an asteroid or a comet, which originates in outer space and survives its impact with the Earth's surface.

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Net (polyhedron)

In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.

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Nickel

Nickel is a chemical element with symbol Ni and atomic number 28.

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Octahedral molecular geometry

In chemistry, octahedral molecular geometry describes the shape of compounds wherein six atoms or groups of atoms or ligands are symmetrically arranged around a central atom, defining the vertices of an octahedron.

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Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.

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Octahedrite

Octahedrites are the most common structural class of iron meteorites.

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Ohm

The ohm (symbol: Ω) is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm.

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Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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Orthogonality

In mathematics, orthogonality is the relation of two lines at right angles to one another (perpendicularity), and the generalization of this relation into n dimensions; and to a variety of mathematical relations thought of as describing non-overlapping, uncorrelated, or independent objects of some kind.

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Orthographic projection

Orthographic projection (or orthogonal projection) is a means of representing a three-dimensional object in two dimensions.

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Pentagonal bipyramid

In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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Polyhedron

In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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Polytope compound

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre.

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Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

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Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.

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Radius

In classical geometry, the radius of a circle or sphere is the length of a line segment from its center to its perimeter.

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Rectification (geometry)

In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

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Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter.

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Resistor

A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element.

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Role-playing game

A role-playing game (RPG and sometimes roleplaying game) is a game in which players assume the roles of characters in a fictional setting.

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Schönhardt polyhedron

In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices.

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Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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Simplicial polytope

In geometry, a simplicial polytope is a polytope whose facets are all simplices.

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Snub disphenoid

In geometry, the snub disphenoid or dodecadeltahedron is a dodecahedron and one of the Johnson solids (J84).

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Space frame

In architecture and structural engineering, a space frame or space structure is a truss-like, lightweight rigid structure constructed from interlocking struts in a geometric pattern.

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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).

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Spherical polyhedron

In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons.

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Square bifrustum

The square bifrustum or square truncated bipyramid is the second in an infinite series of bifrustum polyhedra.

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Stellated octahedron

The stellated octahedron, or stella octangula, is the only stellation of the octahedron.

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Stellation

In geometry, stellation is the process of extending a polygon (in two dimensions), polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure.

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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Subgroup

In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Symmetry group

In abstract algebra, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

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Taxicab geometry

Taxicab geometry, considered by Hermann Minkowski in 19th century Germany, is a form of geometry in which the usual distance function of metric or Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.

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Tesseract

In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square.

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Tetragonal trapezohedron

The tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual to the antiprisms.

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Tetrahedral symmetry

A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

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Tetrahedral-octahedral honeycomb

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex.

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Tetrahemihexahedron

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4.

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Triakis octahedron

In geometry, a triakis octahedron (or kisoctahedron) is an Archimedean dual solid, or a Catalan solid.

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Triangle

A triangle is a polygon with three edges and three vertices.

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Truncated octahedron

In geometry, the truncated octahedron is an Archimedean solid.

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Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid.

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Turán graph

No description.

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Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive.

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Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions.

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Vertex configuration

In geometry, a vertex configuration by Walter Steurer, Sofia Deloudi, (2009) p.18-20, p.51-53 edited by David E. Laughlin, (2014) p. 16-20 is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex.

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Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

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Well-covered graph

In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover.

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Widmanstätten pattern

Widmanstätten patterns, also called Thomson structures, are historically figures of long nickel-iron crystals, found in the octahedrite iron meteorites and some pallasites.

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Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

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Wythoff symbol

In geometry, the Wythoff symbol was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra.

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References

[1] https://en.wikipedia.org/wiki/Octahedron

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