Glossary

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

A

 * Abstract polytope
 * A simplified version of a polytope that disregards the constraints of solid geometry, only caring about which elements are connected to one another.


 * 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.


 * Antiprism
 * 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 alternattions of prisms.


 * Archimedean solid
 * A convex, uniform, finite polyhedron that is not a prism or antiprism and is not also regular.


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

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
 * A three-dimensional element of a polytope.


 * Central symmetry/Central inversion symmetry
 * 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.


 * 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.


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


 * Convex
 * 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
 * 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.


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


 * Cupola (plural Cupolae or Cupolas)
 * 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


 * Dimension
 * The number of dimensions of a space is the number of coordinates it takes to uniquely identify a point in that space.


 * Dual

E

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


 * Element
 * A component of a polytope. Also a polytope itself.

F

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


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


 * Flag
 * A series of elements of a polytope containing a nullitope, vertex, edge... all the way up to a facet, such that all of the elements contain or are contained by one another.

G

 * Grand
 * 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.
 * See also stellation.

H

 * Hypercube
 * 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

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

J

 * Johnson solid
 * A non-uniform, convex, regular-faced polyhedron. There are 92 Johnson solids.

K

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

N

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

O

 * OBSA
 * 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 where each vertex corresponds to one symmetry.


 * Orbiform
 * A polytope that can be inscribed within a hypersphere, that is, whose vertices all lie on the surface of the hypersphere.


 * Orthoplex
 * 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

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


 * Polychoron (plural polychora)
 * A four-dimensional polytope.


 * Polygon
 * A two-dimensional polytope.


 * Polyhedron (plural polyhedra)
 * A three-dimensional polytope.
 * Polytope
 * A type of geometrical figure that generalizes the idea of "flat" shapes to higher dimensions.
 * Peak
 * One of the elements of a polytope that has the third-highest dimension. For a polyhedron, the peaks are the vertices.


 * Prism


 * Pyramid

R

 * Rank
 * The rank of a polytope is the number of dimensions it is defined by?


 * 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
 * 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.


 * Regular
 * A polytope that is transitive upon its flags, or whose symmetries can move any of the polytope's elements to any other one of the elements of the same rank. Put simply, a polytope in which all the vertices are the same, all the edges are the same, and so on up to the facets.


 * Ridge
 * 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

 * Scaliform
 * 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.


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


 * Segmentotope
 * A polytope whose vertices all lie on two parallel hyperspheres. It is orbiform. Pyramids, prisms, antiprisms, and cupolae are segmentotopes.


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


 * Simplex (plural simplices or simplexes)
 * 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.


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


 * Stellation
 * 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.


 * Symmetry
 * A relation 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.

T

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


 * 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

 * Uniform
 * 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)
 * A zero-dimensional element of a polytope.


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