In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a simplicial polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

is a d-polytope whose facets

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

 are all simplices

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

.

For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem

Steinitz's theorem

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs...

 to a maximal planar graph.

They are topologically dual to simple polytope

Simple polytope

In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges . The vertex figure of a simple d-polytope is a -simplex....

s. Polytopes which are both
simple and simplicial are either simplices or two-dimensional polygons.

Examples

Simplicial polyhedra include:

• Bipyramid
  Bipyramid
  An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
  
  s
• Gyroelongated dipyramids
• Deltahedra
  Deltahedron
  A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
  
   (Equilateral triangles)
  • Platonic
    Platonic solid
    In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
    
    
    • tetrahedron
      Tetrahedron
      In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
      
      , octahedron
      Octahedron
      In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
      
      , icosahedron
      Icosahedron
      In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
      
      
  • Johnson solid
    Johnson solid
    In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
    
    s:
    • triangular dipyramid
      Triangular dipyramid
      In geometry, the triangular bipyramid is the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces....
      
      , gyroelongated square dipyramid
      Gyroelongated square dipyramid
      In geometry, the gyroelongated square bipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves...
      
      , triaugmented triangular prism
      Triaugmented triangular prism
      In geometry, the triaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by attaching square pyramids to each of the three equatorial faces of the triangular prism...
      
      , snub disphenoid
      Snub disphenoid
      In geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
      
      
• Catalan solid
  Catalan solid
  In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
  
  s:
  • triakis tetrahedron
    Triakis tetrahedron
    In geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron...
    
    , triakis octahedron
    Triakis octahedron
    In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more...
    
    , tetrakis hexahedron
    Tetrakis hexahedron
    In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....
    
    , disdyakis dodecahedron
    Disdyakis dodecahedron
    In geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...
    
    , triakis icosahedron
    Triakis icosahedron
    In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron...
    
    , pentakis dodecahedron
    Pentakis dodecahedron
    In geometry, a pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron...
    
    , disdyakis triacontahedron
    Disdyakis triacontahedron
    In geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons...
    
    

Simplicial tilings:

• Regular:
  • triangular tiling
• Laves tilings:
  • tetrakis square tiling
    Tetrakis square tiling
    In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is square tiling with each square divided into four triangles from the center point, forming an infinite arrangement of lines....
    
    , triakis triangular tiling
    Triakis triangular tiling
    In geometry, the Triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles from the center point....
    
    , bisected hexagonal tiling

Simplicial 4-polytopes include:

• convex regular 4-polytope
  Convex regular 4-polytope
  In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
  
  
  • 4-simplex
    Pentachoron
    In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
    
    , 16-cell
    16-cell
    In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....
    
    , 600-cell
• Dual convex uniform honeycomb
  Convex uniform honeycomb
  In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
  
  s:
  1. Disphenoid tetrahedral honeycomb
     Disphenoid tetrahedral honeycomb
     The disphenoid tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of identical nonregular tetrahedral cells....
     
     
  2. Dual of cantitruncated cubic honeycomb
     Cantitruncated cubic honeycomb
     The cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.- Uniform colorings :...
     
     
  3. Dual of omnitruncated cubic honeycomb
     Omnitruncated cubic honeycomb
     The omnitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3.A wireframe rendering- References :...
     
     
  4. Dual of cantitruncated alternated cubic honeycomb
     Cantitruncated alternated cubic honeycomb
     The cantitruncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cuboctahedra, truncated cubes and truncated tetrahedra in a ratio of 1:1:2.- References :...
     
     

Simplicial higher polytope families:

• simplex
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
  
• cross-polytope
  Cross-polytope
  In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
  
  (Orthoplex)