geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a -dimensional simple polytope is a -dimensional

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

each of whose vertices are adjacent to exactly

edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...

(also facets). The vertex figure of a simple -polytope is a -

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...

. Simple polytopes are topologically

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

s. A ''simple polyhedron'' is a three-dimensional polyhedron whose vertices are adjacent to three edges and three faces. The dual to a simple polyhedron is a ''simplicial polyhedron'', in which all faces are triangles.

Examples

Three-dimensional simple polyhedra include the prisms (including the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

), the regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

and dodecahedron, and, among the Archimedean solids, the truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron,

truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squa ...

, and

truncated icosidodecahedron In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Wood ...

. They also include the Goldberg polyhedron and Fullerenes, including the

chamfered tetrahedron In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this opera ...

chamfered cube In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this oper ...

, and

chamfered dodecahedron In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in ...

. In general, any polyhedron can be made into a simple one by truncating its vertices of valence four or higher. For instance,

truncated trapezohedron In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with pyramids truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an antipris ...

s are formed by truncating only the high-degree vertices of a trapezohedron; they are also simple. Four-dimensional simple polytopes include the regular

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...

and tesseract. Simple uniform 4-polytope include the truncated 5-cell, truncated tesseract, truncated 24-cell, truncated 120-cell, and duoprisms. All bitruncated, cantitruncated or omnitruncated four-polytopes are simple. Simple polytopes in higher dimensions include the ''d''-

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...

associahedron In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application ...

, permutohedron, and all

omnitruncated In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...

polytopes.

Unique reconstruction

Micha Perles Micah (; ) is a given name. Micah is the name of several people in the Hebrew Bible (Old Testament), and means "Who is like God?" The name is sometimes found with theophoric extensions. Suffix theophory in ''Yah'' and in '' Yahweh'' results in ...

conjectured that a simple polytope is completely determined by its 1-skeleton; his conjecture was proven in 1987 by

Roswitha Blind Roswitha Blind (also published as Roswitha Hammer) is a German mathematician, specializing in convex geometry, discrete geometry, and polyhedral combinatorics, and a politician and organizer for the Social Democratic Party of Germany in Stuttgart ...

and Peter Mani-Levitska. Gil Kalai shortly after provided a simpler proof of this result based on the theory of

unique sink orientation In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one vertex for which all adjoining edges are oriente ...

Notes

{{reflist, refs= {{citation , last1 = Blind , first1 = Roswitha , author1-link = Roswitha Blind , last2 = Mani-Levitska , first2 = Peter , doi = 10.1007/BF01830678 , issue = 2-3 , journal = Aequationes Mathematicae , mr = 921106 , pages = 287–297 , title = Puzzles and polytope isomorphisms , volume = 34 , year = 1987. {{citation , last = Cromwell , first = Peter R. , isbn = 0-521-66405-5 , page = 341 , publisher = Cambridge University Press , title = Polyhedra , url = https://books.google.com/books?id=OJowej1QWpoC&pg=PA341 , year = 1997 {{citation , last = Kalai , first = Gil , authorlink = Gil Kalai , doi = 10.1016/0097-3165(88)90064-7 , issue = 2 , journal = Journal of Combinatorial Theory , mr = 964396 , pages = 381–383 , series = Series A , title = A simple way to tell a simple polytope from its graph , volume = 49 , year = 1988, doi-access = free . {{citation , last = Ziegler , first = Günter M. , author-link = Günter M. Ziegler , isbn = 9780387943657 , page = 8 , publisher = Springer , series = Graduate Texts in Mathematics , title = Lectures on Polytopes , url = https://books.google.com/books?id=xd25TXSSUcgC& , volume = 152 , year = 2012 Euclidean geometry