Truncation (geometry)

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Truncation in regular polyhedra and tilings

This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube

The middle image is the uniform truncated cube

Other truncations

Uniform polyhedron and tiling examples

Family	Original	Truncation	Rectification	Bitruncation (truncated dual)	Birecification (dual)
[3,3]	Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle 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....	Truncated tetrahedron Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....	Octahedron Octahedron 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 wikt:vertex....	Truncated tetrahedron Truncated tetrahedron The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....	Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle 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....
[4,3]	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	Truncated cube Truncated cube The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....	Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	Truncated octahedron Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....	Octahedron Octahedron 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 wikt:vertex....
[5,3]	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....	Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	Truncated icosahedron Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
[6,3]	Hexagonal tiling Hexagonal tiling In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....	Truncated hexagonal tiling Truncated hexagonal tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....	Trihexagonal tiling Trihexagonal tiling In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....	Hexagonal tiling Hexagonal tiling In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....	Triangular tiling Triangular tiling In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....
[7,3]	Order-3 heptagonal tiling Order-3 heptagonal tiling In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....	Order-3 truncated heptagonal tiling Order-3 truncated heptagonal tiling In geometry, the Truncated order-3 heptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two tetrakaidecagons on each vertex ....	Triheptagonal tiling Triheptagonal tiling In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two triangles and two heptagons alternating on each vertex ....	Order-7 truncated triangular tiling Order-7 truncated triangular tiling In geometry, the Order 7 truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex ....	Order-7 triangular tiling Order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
[8,3]	Order-3 Octagonal tiling	Order-3 truncated Octagonal tiling	Trioctagonal tiling	Order-8 truncated triangular tiling	Order-8 triangular tiling
[4,4]	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	Truncated square tiling Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	Truncated square tiling Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
[5,4]	pentagonal	truncated pentagonal	Rectified pentagonal	Truncated square	Square
[5,5]	Pentagonal	Truncated pentagonal	Rectified pentagonal	Truncated pentagonal	Pentagonal

Prismatic polyhedron examples

Family	Original	Truncation	Rectification (And dual)
[2,p]	Hexagonal hosohedron Hosohedron An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol .... (As spherical tiling)	Hexagonal prism Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce... t	Hexagonal dihedron Dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat.... (As spherical tiling)

rhombitruncated examples

Original	Rectification	Rhombitruncation
		Truncated octahedron Truncated octahedron The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
	Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	Truncated cuboctahedron Truncated cuboctahedron The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges.... or rhombitruncated cuboctahedron
	Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	Truncated icosidodecahedron Truncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges.... or rhombitruncated icosidodecahedron
	Trihexagonal tiling Trihexagonal tiling In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....	Truncated trihexagonal tiling Great rhombitrihexagonal tiling In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex .... or great rhombitrihexagonal tiling
	Triheptagonal tiling Triheptagonal tiling In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two triangles and two heptagons alternating on each vertex ....	Truncated triheptagonal tiling Great rhombitriheptagonal tiling In geometry, the great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one tetrakaidecagon on each vertex .... or great rhombitriheptagonal tiling
	Trioctagonal tiling	Truncated trioctagonal tiling or great rhombitriheptagonal tiling
	Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	Truncated square tiling Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....
	Order-5 square tiling Order-5 square tiling In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....	Order-5 truncated square tiling
	Order-5 pentagonal tiling	Order-5 truncated pentagonal tiling

Truncation in polychora and honeycomb tessellation

Family [p,q,r]	Parent	Truncation	Rectification (birectified dual)	Bitruncation (bitruncated dual)
[3,3,3]	5-cell (self-dual)	truncated 5-cell Truncated 5-cell In Fourth dimension geometry, the truncated 5-cell or truncated pentatope is a uniform polychoron bounded by 10 cell : 5 tetrahedron, and 5 truncated tetrahedron....	rectified 5-cell Rectified 5-cell In Fourth dimension geometry, the Rectification 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cell ....	bitruncated 5-cell Bitruncated 5-cell In Fourth dimension geometry, the bitruncation 5-cell, or bitruncated pentachoron, is a 4-dimensional polytope, or polychoron, composed of 10 cell in the shape of truncated tetrahedron....
[3,3,4]	16-cell 16-cell In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....	truncated 16-cell Truncated 16-cell In geometry, the truncated 16-cell or cantic tesseract is a uniform polychoron which is bounded by 24 cell_: 8 regular octahedron, and 16 truncated tetrahedron....	rectified 16-cell (Same as 24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell .... )	bitruncated 16-cell (bitruncated tesseract Bitruncated tesseract In geometry, the Bitruncation tesseract is a uniform polychoron.... )
[4,3,3]	Tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....	truncated tesseract Truncated tesseract In geometry, a truncated tesseract is a uniform polychoron which is bounded by 24 cell_: 8 truncated cubes, and 16 tetrahedron....	rectified tesseract Rectified tesseract In geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron bounded by 24 cell_: 8 cuboctahedron, and 16 tetrahedron....	bitruncated tesseract (bitruncated 16-cell)
[3,4,3]	24-cell 24-cell In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....  (self-dual)	truncated 24-cell Truncated 24-cell In geometry, the truncated 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cell_: 24 cubes, and 24 truncated octahedron. Each vertex contains three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure....	rectified 24-cell Rectified 24-cell In geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cell_: 24 cubes, and 24 cuboctahedron.It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4], or even D4....	bitruncated 24-cell Bitruncated 24-cell In geometry, the bitruncated 24-cell is a 4-dimensional uniform polytope derived from the 24-cell. It is constructed by bitruncation the 24-cell ....
[3,3,5]	600-cell 600-cell In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....	truncated 600-cell Truncated 600-cell In geometry, the truncated 600-cell is a uniform polychoron. It is derived from the 600-cell by truncation_....	rectified 600-cell Rectified 600-cell In geometry, the Rectification 600-cell is a convex uniform polychoron composed of 600 regular octahedra and 120 icosahedra cell . Each edge has two octahedra and one icosahedron....	bitruncated 600-cell (bitruncated 120-cell Bitruncated 120-cell In geometry, the Bitruncation 120-cell is a uniform polychoron.... )
[5,3,3]	120-cell 120-cell In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....	truncated 120-cell Truncated 120-cell In geometry, the truncated 120-cell is a uniform polychoron, constructed by a uniform Truncation of the Convex regular 4-polytope 120-cell polychoron....	rectified 120-cell Rectified 120-cell In geometry, the Rectification 120-cell is a convex uniform polychoron composed of 600 regular tetrahedron and 120 icosidodecahedron cell . Its vertex figure is a triangular prism, with 3 icosidodecahedra and 2 tetrahedra meeting at each vertex....
[4,3,4]	cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....  (self-dual)	truncated cubic Truncated cubic honeycomb The truncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cubes and octahedron in a ratio of 1:1....	rectified cubic Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1....	bitruncated cubic Bitruncated cubic honeycomb The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb....
[3,5,3]	icosahedral (self-dual)	(No image) truncated icosahedral	(No image) rectified icosahedral	(No image) bitruncated icosahedral
[4,3,5]	cubic	(No image) truncated cubic	(No image) rectified cubic	(No image) bitruncated cubic (bitruncated dodecahedral)
[5,3,4]	dodecahedral	(No image) truncated dodecahedral	(No image) rectified dodecahedral
[5,3,5]	(No image) dodecahedral (self-dual)	(No image) truncated dodecahedral	(No image) rectified dodecahedral	(No image) bitruncated dodecahedral

See also

• uniform polyhedron
  Uniform polyhedron
  
  A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
  
• uniform polychoron
  Uniform polychoron
  
  In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
  
• Bitruncation (geometry)
• Rectification (geometry)
  Rectification (geometry)
  
  In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
  
• Alternation (geometry)
• Conway polyhedron notation
  Conway polyhedron notation
  
  Conway polyhedron notation is used to describe polyhedron based on a seed polyhedron modified by various operators.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prism s , antiprisms and pyramid s ....
  

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