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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. [1]

## Abstract polytope

In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc.

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

## 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).

## Cubic honeycomb

The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.

## Dual polyhedron

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.

## Face (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.

## Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

## 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.

## Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Harold Scott MacDonald Coxeter

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

## 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.

## Hyperplane

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

## Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

## List of regular polytopes and compounds

This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.

## Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

## Polygon

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 polygonal chain or circuit.

## 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.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

## Regular polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.

## Rhombicuboctahedron

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces.

## Schläfli symbol

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

## Schlegel diagram

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.

## Square

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.

## Square pyramid

In geometry, a square pyramid is a pyramid having a square base.

## Star polygon

In geometry, a star polygon is a type of non-convex polygon.

## 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.

## Truncated cube

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid.

## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

## Vertex (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

## Vertex configuration

In geometry, a vertex configuration by Walter Steurer, Sofia Deloudi, (2009) pp.

## 4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.

## References

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