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 -d ...

each of whose vertices are adjacent to exactly edges (also

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

). The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

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 to

simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...

s. The family of polytopes which are both simple and simplicial are

simplices 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. ...

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 toge ...

s. A ''simple polyhedron'' is a three-dimensional

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

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 only r ...

), 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 o ...

and

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

, and, among the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s, the

truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...

truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges. Geometric relations This polyhedron can be formed from a regular dodecahedron by truncat ...

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 squares. ...

, 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 Wooda ...

. They also include the

Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three pro ...

and

Fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...

s, 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 operati ...

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 operatio ...

, 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 s ...

. 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 antiprism. T ...

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 In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

. Simple

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...

include the

truncated 5-cell In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell. There are two degrees of truncations, including a bitruncation. Truncated 5-cell The truncated 5-cell, tr ...

truncated tesseract In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''. Truncated tesserac ...

truncated 24-cell In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. Truncated 24-cell The truncated 24- ...

truncated 120-cell In geometry, a truncated 120-cell is a uniform 4-polytope formed as the truncation of the regular 120-cell. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 600-cell''. Truncated 120-c ...

, and

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

s. 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 of ...

permutohedron In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...

, and all omnitruncated 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 and Peter Mani-Levitska.

Gil Kalai Gil Kalai (born 1955) is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics ...

shortly after provided a simpler proof of this result based on the theory of unique sink orientations.

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 ''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topic ...

, 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 The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applicat ...

, 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