33 relations: Abstract polytope, Cubic honeycomb, Dual polyhedron, Face (geometry), Facet (geometry), Geometry, Graph theory, Harold Scott MacDonald Coxeter, Honeycomb (geometry), Hyperplane, Isogonal figure, Isosceles triangle, List of regular polytopes and compounds, Octahedron, Pentagram, Polygon, Polyhedron, Polytope, Regular 4-polytope, Regular polyhedron, Rhombicuboctahedron, Schläfli symbol, Schlegel diagram, Square, Square pyramid, Star polygon, Tessellation, Triangle, Truncated cube, Uniform polytope, Vertex (geometry), Vertex configuration, 4-polytope.
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc.
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The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.
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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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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.
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In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
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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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In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
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Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.
In geometry a hyperplane is a subspace of one dimension less than its ambient space.
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In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same.
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In geometry, an isosceles triangle is a triangle that has two sides of equal length.
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This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces.
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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.
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In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.
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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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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope.
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope.
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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.
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In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
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In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^ through a point beyond one of its facets or faces.
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or right angles).
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In geometry, a square pyramid is a pyramid having a square base.
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In geometry, a star polygon (not to be confused with a star-shaped polygon) is a concave polygon.
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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.
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A triangle is a polygon with three edges and three vertices.
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In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid.
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
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In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.
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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.
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.
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