# Glossary

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Glossary for terms related to polytopes. Also see the Multidimensional glossary, PolyGloss, Hedrondude's glossary and Stella polyhedral glossary.

## A

• Abstract polytope n
• A simplified version of a polytope that disregards the constraints of solid geometry, only caring about which elements are connected to one another.
• Alternation n
• An operation that discards alternate vertices of a polytope; half of the vertices are replaced with vertex figures, and the facets are alternated in turn.
• Antiprism n
• An antiprism is a polytope formed by lacing a base polytope and its dual.
• The word might also refer to other similar constructions, such as alterprisms or alternations of prisms.
• Archimedean solid n
• A convex, uniform, finite polyhedron that is Wythoffian, is not a prism or antiprism, and is also not regular.
• Can also refer to such polytopes in higher dimensions.
• Atop prep
• A common word to refer to two bases of a monostratic polytope: A atop B. For example, a triangular cupola is a triangle atop hexagon.
• Army n
• A set of polytopes with the same vertices.

## C

• Cantellation/Cantellate
• An operation done to a polytope. Can be thought of as "expand the faces outwards and connect them with new squares." Only applicable to polytopes of three or more dimensions.
• Cell n
• A three-dimensional element of a polytope.
• Central symmetry/Central inversion symmetry n
• A symmetry that reflects a polytope across its center, can be thought of as multiplying all coordinates by –1. In odd dimensions, this symmetry is a reflection, in even dimensions it is a rotation.
• The radius of a sphere whose surface contains the vertices of the given polytope.
• A polytope whose vertices lie on a hypersphere.
• A polytope for which a line drawn between two points on its surface always goes through the polytope. Simply speaking, a convex polytope has no spikes, dents, or holes.
• Company n
• A set of polytopes with the same vertices, edges, and faces. Different polytopes can be in the same company in 4 dimensions or higher.
• Cupola (plural cupolae or cupolas) n
• A lace prism of a polytope atop its expansion. In 4D or higher, it sometimes also refer to similar lace prisms, such as a polytope atop its truncation.

## D

• Density n
• Dimension n
• The number of dimensions of a space is the number of coordinates it takes to uniquely identify a point in that space.
• Every polytope has a dual associated with it. The polytope's facets correspond to its dual's vertices, the polytope's ridges correspond to its dual's edges, and so on.

## E

• Edge n
• A one-dimensional element of a polytope.
• Element n
• A part of a polytope, for example a vertex, edge, face, etc.. Elements of polytopes are also polytopes.
• A polytope all of whose edges are the same length.
• Expansion/Expand
• An operation done to a polytope, in an arbitrary number of dimensions. Can be thought of as "move the facets outwards, and connect the facets with new prisms."
• Coincides with truncation in 2D, cantellation in 3D, and runcination in 4D.

## F

• Face n
• A two-dimensional element of a polytope.
• Facet n
• One of the elements of a polytope that has the highest dimension. For a polyhedron, the facets are the faces.
• Fastegium (also spelled fastigium) n
• A special case of a wedge that is a polytope atop its prism. They are sub-symmetric variants of a triangle-P duoprism. A triangular prism can be considered a dyad fastegium.
• Figure n
• The arrangement of higher-dimensional elements around a specific element. Generalisation of vertex figure to elements of any dimension.
• Having coincident elements or compound figures.
• Flag n
• A series of elements of a polytope containing a vertex, edge... all the way up to a facet, such that all of the elements contain or are contained by one another.

## G

• See Aggrandisement.
• Greatening/Great
• An operation that extends the faces while keeping them in the same planes, for example making the great dodecahedron from the dodecahedron.

## H

• Hemipolytope n
• A polytope containing facets passing through the center. Hemipolytopes have no well-defined dual in Euclidean space, though it exists in projective space.
• Honeycomb n
• A tessellation, possibly specifically one of rank 4, depending on author.
• Hypercube n
• One of the three infinite families of regular spherical polytopes. Its facets are hypercubes and its vertex figures are simplexes. Examples include the square, cube, and tesseract.

## I

• The radius of a sphere that is tangent to the facets of a given polytope.
• Interior angle n
• The fraction of the neighbourhood of a point that is in the interior of the polytope.

## K

• Kepler-Poinsot solid/polyhedron n
• A regular, non-convex, finite polyhedron. There are 4 such polyhedra.

## M

• A polytope whose vertices lie on two parallel hyperplanes.

## N

• Nullitope n
• A (−1)-dimensional element of a polytope. Not often useful to consider on its own.

## O

• OBSA n
• Short for Official Bowers-Style Acronym. Abbreviation for polytope names.
• Omnitruncation/Omnitruncate
• The operation of truncating every element of a polytope such that each flag of the origin polytope corresponds to a vertex of the new polytope.
• A uniform polytope with a number of vertices equal to its symmetry order.
• Operation n
• A change made to a polytope that results in another polytope.
• A polytope that is circumscribable and equilateral.
• Orthoplex n
• One of the three infinite families of regular spherical polytopes. Its facets are simplices and its vertex figures are orthoplexes. Examples include the square, octahedron, and hexadecachoron.

## P

• Polychoron (plural polychora) n
• A four-dimensional polytope.
• Polygon n
• A two-dimensional polytope.
• Polyhedron (plural polyhedra) n
• A three-dimensional polytope.
• Polytope n
• A type of geometrical figure that generalizes the idea of "flat" shapes to higher dimensions.
• Peak n
• One of the elements of a polytope that has the third-highest dimension. For a polyhedron, the peaks are the vertices.
• Prism n
• A polytope formed as the prism product of a given polytope (the base) and a dyad. It can be thought of as the base extruded into the next dimension.
• Pyramid n
• A polytope constructed by tapering a given polytope (the base) to a point (the apex) along a new dimension. The facets of a pyramid are precisely the base and the pyramids of all of the base's facets.

## R

• Rank n
• The intrinsic property of a polytope that distinguishes polygons, polyhedra, polychora, and others. This is similar to, but is different from, dimension.
• Rectification/Rectify/Rectate
• An operation done to a polytope. Can be thought of as "cut away beneath the vertices until the cuts reach one another in the middle of the edges." Leaves new facets where the vertices once were, and new vertices where the edges once were.
• Regiment n
• A set of polytopes with the same vertices and edges. Different polytopes can be in the same regiment in 3 dimensions or higher.
• A polytope that is transitive on its flags.
• Ridge n
• One of the elements of a polytope that has the second-highest dimension. For a polyhedron, the ridges are the edges.
• Runcination/Runcinate
• An operation done to a polytope. Can be thought of as "expand the cells outwards and connect them with new polygonal prisms." Only applicable to polytopes of four or more dimensions.

## S

• A less restrictive version of uniform. A scaliform polytope must be transitive upon its vertices and have one edge length, but its facets do not need to be uniform. This allows for polytopes such as the orbiform Johnson solids to be used in their construction.
• Segmentotope n
• A polytope which is monostratic and orbiform. Pyramids, prisms, antiprisms, and cupolae are segmentotopes.
• A polytope that is transitive upon its vertices and has semi-uniform facets. All uniform polytopes are semi-uniform.
• Simplex (plural simplices or simplexes) n
• The simplest non-degenerate polytope in every dimension and one of the three infinite families of regular spherical polytopes. Its facets and vertex figures are simplices. Examples include the triangle, tetrahedron, and pentachoron.
• A snub element of a uniform polytope is one whose vertices are not equivalent under the symmetry of the whole polytope. A snub polytope is one that contains snub elements. An example is the square antiprism, whose triangles' base vertices are not equivalent to their apex vertices.
• Often used as a synonym for an operation involving alternation, but the specific operation varies:
• Alternating the omnitruncate
• Alternating the truncate
• Alternating the polytope itself
• Space n
• The surroundings in which a polytope exists. Can be spherical, Euclidean (flat), or hyperbolic.
• Stellation n
• An operation done to a polytope that extends the facets outward but keeps them connected to one another.
• It can also refer specifically to the operation that extends the edges while keeping them in the same lines, for example making the small stellated dodecahedron from the dodecahedron.
• Symmetry n
• An isometry (i.e. translation, rotation, or reflection) that maps an object (usually a polytope) onto itself while keeping its appearance exactly the same. The square, for instance, can be rotated three different ways or reflected about any of four axes.
• The symmetry order is the number of symmetries that an object has, including the "identity" (that is, making no change to the object).

## T

• Teron (plural tera or terons) n
• A four-dimensional element of a polytope.
• Tiling n
• A tessellation, possibly specifically one of rank 3, depending on author.
• Truncation/Truncate
• An operation done to a polytope. Can be thought of either as "cut slightly beneath the vertices, leaving new facets behind in the shape of the vertex figure" or "expand the edges outwards and connect them with new edges." Only applicable to polytopes of two or more dimensions.

## U

• A polytope that is transitive upon its vertices, has one edge length, and has uniform facets. Regular polygons are defined to be uniform.

## V

• Vertex (plural vertices) n
• A zero-dimensional element of a polytope.
• Vertex figure n
• A special polytope that represents which facets come together at the vertex of a given polytope.

## W

• Wedge n
• A monostratic polytope, one of whose bases is sub-dimensional (lies in a (n-2)-space for a n-polytope) but greater than a point. Often refers to the specific case of a triangular prism variant.