# 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[edit | edit source]

**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.

- A simplified version of a polytope that disregards the constraints of solid geometry, only caring about which

**Aggrandisement/Grand**- An operation that extends the
**cells**while keeping them in the same 3-planes, for example making the grand hecatonicosachoron from the hecatonicosachoron. *See also***stellation**.

- An operation that extends the

**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.

- An operation that discards alternate vertices of a polytope; half of the vertices are replaced with

**Antifastegium***n*- A special case of a
**wedge**that is a polytope**atop**its**prism**in a gyrated orientation. Examples include the triangular antifastegium, square antifastegium, etc.

- A special case of a

**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**.

- An antiprism is a polytope formed by

**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.

- A

**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.

- A common word to refer to two bases of a

**Army***n*- A set of polytopes with the same
**vertices**. *See also***regiment**,**company**.

- A set of polytopes with the same

## B[edit | edit source]

**Biformic***adj*- A polytope or compound that has two different types of vertices, but they coincide in space. An example is the compound of 20 tetrahemihexahedra.
*See also***fissary**.

## C[edit | edit source]

**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**.

- An operation done to a polytope. Can be thought of as "expand the

**Cell***n*- A three-dimensional
**element**of a polytope.

- A three-dimensional

**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.

- A

**Chirality/Chiral**- A polytope is
**chiral**if it is not equal to its mirror image. Examples include the snub cube and the gyroelongated square bicupola.

- A polytope is

**Circumradius**(plural*circumradii*)*n*- The radius of a sphere whose surface contains the
**vertices**of the given polytope.

- The radius of a sphere whose surface contains the

**Circumscribable***adj*- A polytope whose vertices lie on a hypersphere.

**Convex***adj*- 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. *See also:***army**,**regiment**.

- A set of polytopes with the same

**Coxeter-Dynkin diagram***n*- Specialized graph that represents a polytope by its
**symmetries**.

- Specialized graph that represents a polytope by its

**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**.

- A

## D[edit | edit source]

**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.

- The number of dimensions of a

**Dual***n, adj*- 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.

- Every

## E[edit | edit source]

**Edge***n*- A one-dimensional
**element**of a polytope.

- A one-dimensional

**Element***n*- A part of a polytope, for example a
**vertex**,**edge**,**face**, etc.. Elements of polytopes are also polytopes.

- A part of a polytope, for example a

**Equilateral***adj*- 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.

- An operation done to a polytope, in an arbitrary number of dimensions. Can be thought of as "move the

## F[edit | edit source]

**Face***n*- A two-dimensional
**element**of a polytope.

- A two-dimensional

**Facet***n*- One of the
**elements**of a polytope that has the highest**dimension**. For a**polyhedron**, the facets are the**faces**.

- One of the

**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.

- A special case of a

**Figure***n*- The arrangement of higher-dimensional
**elements**around a specific element. Generalisation of**vertex figure**to elements of any dimension.

- The arrangement of higher-dimensional

**Fissary***adj*- Having coincident
**elements**or compound**figures**.

- Having coincident

**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.

- A series of

## G[edit | edit source]

**Grand***adj*- See
.**Aggrandisement**

- See

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

- An operation that extends the

## H[edit | edit source]

**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.

- A

## I[edit | edit source]

**Inradius**(plural*inradii*)*n*- The radius of a sphere that is tangent to the
**facets**of a given polytope.

- The radius of a sphere that is tangent to the

**Interior angle***n*- The fraction of the neighbourhood of a point that is in the interior of the polytope.

## J[edit | edit source]

**Johnson solid***n*- A non-
**uniform, convex, regular**-faced**polyhedron**. There are 92 Johnson solids. *See also Convex regular-faced polytope.*

- A non-

## K[edit | edit source]

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

- A

## L[edit | edit source]

**Level of complexity***n*- A quantitative measure of the complexity of a model of a polytope.

## M[edit | edit source]

**Monostratic***adj*- A polytope whose
**vertices**lie on two parallel hyperplanes.

- A polytope whose

## N[edit | edit source]

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

- A (−1)-dimensional

## O[edit | edit source]

**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.

- The operation of truncating every element of a polytope such that each

**Operation***n*- A change made to a polytope that results in another polytope.

**Orbiform***adj*- A polytope that is
**circumscribable**and**equilateral**.

- A polytope that is

**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.

- One of the three infinite families of regular spherical polytopes. Its

## P[edit | edit source]

**Platonic solid***n*- A
**regular, convex**, finite**polyhedron**.

- A

**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**.

- One of the

**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.

- A polytope formed as the

**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.

- A polytope constructed by tapering a given polytope (the base) to a

## Q[edit | edit source]

## R[edit | edit source]

**Rank***n*- The intrinsic property of a
**polytope**that distinguishes**polygons**,**polyhedra**,**polychora**, and others. This is similar to, but is different from,**dimension**.

- The intrinsic property of a

**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.

- An operation done to a polytope. Can be thought of as "cut away beneath the

**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. *See also:***army**,**company**.

- A set of polytopes with the same

**Regular***adj*- A polytope that is transitive on its
**flags**.

- A polytope that is transitive on its

**Ridge***n*- One of the
**elements**of a polytope that has the second-highest**dimension**. For a**polyhedron**, the ridges are the**edges**.

- One of the

**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**.

- An operation done to a polytope. Can be thought of as "expand the

## S[edit | edit source]

**Scaliform***adj*- 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.

- A less restrictive version of

**Schläfli-Hess polychoron***n*- A
**regular**, non-**convex**, finite**polychoron**. There are 10 such polychora.

- A

**Segmentotope***n*- A polytope which is
**monostratic**and**orbiform**.**Pyramids**,**prisms**,**antiprisms**, and**cupolae**are segmentotopes.

- A polytope which is

**Semi-uniform***adj*- A polytope that is transitive upon its
**vertices**and has semi-uniform**facets**. All uniform polytopes are semi-uniform.

- A polytope that is transitive upon its

**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.

- The simplest non-degenerate polytope in every dimension and one of the three infinite families of regular spherical polytopes. Its

**Snub***adj*- 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

- Alternating the

- A

**Space***n*- The surroundings in which a polytope exists. Can be
**spherical, Euclidean**(flat), or**hyperbolic**.

- The surroundings in which a polytope exists. Can be

**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.*See also***greatening**,**aggrandisement**.

- An operation done to a polytope that extends the

**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[edit | edit source]

**Teron**(plural*tera*or*terons*)*n*- A four-dimensional
**element**of a polytope.

- A four-dimensional

**Tessellation***n*- A polytope that fills a
**space**.

- A polytope that fills a

**Tiling***n*- A
**tessellation**, possibly specifically one of rank 3, depending on author.

- A

**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**.

- An operation done to a polytope. Can be thought of either as "cut slightly beneath the

## U[edit | edit source]

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

- A polytope that is transitive upon its

## V[edit | edit source]

**Vertex**(plural*vertices*)*n*- A zero-dimensional
**element**of a polytope.

- A zero-dimensional

**Vertex figure***n*- A special polytope that represents which
**facets**come together at the vertex of a given polytope.

- A special polytope that represents which

## W[edit | edit source]

**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.

- A

**Wythoffian***adj*- A polytope that is able to be represented with a
**Coxeter diagram**.

- A polytope that is able to be represented with a