Truncation (geometry)

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A truncated cube - faces double in sides, and vertices replaced by new faces.

A Truncated cubic honeycomb - faces doubled in sides, and vertices replaced by new cells.

In geometry, a truncation is an operation in any dimension that cuts a polytope vertices, creating a new facet in place of each vertex.

1 Truncation in regular polyhedra and tilings
2 Other truncations
3 Uniform polyhedron and tiling examples
4 Prismatic polyhedron examples
5 rhombitruncated examples
6 Truncation in polychora and honeycomb tessellation
7 See also
8 References
9 External links

[edit] Truncation in regular polyhedra and tilings

When the term applies to truncating platonic solids or regular tilings, usually "uniform truncation" is implied, which means to truncate until the original faces become regular polygons with double the sides.

This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron.

The middle image is the uniform truncated cube. It is represented by an extended Schläfli symbol t_0,1{p,q,...}.

[edit] Other truncations

In quasiregular polyhedra, a truncation is a more qualitative term where some other adjustments are made to adjust truncated faces to become regular. These are sometimes called rhombitruncations.

For example, the truncated cuboctahedron is not really a truncation since the cut vertices of the cuboctahedron would form rectangular faces rather than squares, so a wider operation is needed to adjust the polyhedron to fit desired squares.

In the quasiregular duals, an alternate truncation operation only truncates alternate vertices. (This operation can also apply to any zonohedron which have even-sided faces.)

[edit] Uniform polyhedron and tiling examples

This table shows the truncation progression between the regular forms, with the rectified forms (full truncation) in the center. Comparable faces are colored red and yellow to show the continuum in the sequences.

Family	Original	Truncation	Rectification	Bitruncation (truncated dual)	Birecification (dual)
[3,3]	Tetrahedron	Truncated tetrahedron	Octahedron	Truncated tetrahedron	Tetrahedron
[4,3]	Cube	Truncated cube	Cuboctahedron	Truncated octahedron	Octahedron
[5,3]	Dodecahedron	Truncated dodecahedron	Icosidodecahedron	Truncated icosahedron	Icosahedron
[6,3]	Hexagonal tiling	Truncated hexagonal tiling	Trihexagonal tiling	Hexagonal tiling	Triangular tiling
[7,3]	Order-3 heptagonal tiling	Order-3 truncated heptagonal tiling	Triheptagonal tiling	Order-7 truncated triangular tiling	Order-7 triangular tiling
[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	Truncated square tiling	Square tiling	Truncated square tiling	Square tiling
[5,4]	pentagonal	truncated pentagonal	Rectified pentagonal	Truncated square	Square
[5,5]	Pentagonal	Truncated pentagonal	Rectified pentagonal	Truncated pentagonal	Pentagonal

[edit] Prismatic polyhedron examples

Family	Original	Truncation	Rectification (And dual)
[2,p]	Hexagonal hosohedron (As spherical tiling) {2,p}	Hexagonal prism t{2,p}	Hexagonal dihedron (As spherical tiling) {p,2}

[edit] rhombitruncated examples

These forms start with a rectified regular form which is truncated. The vertices are order-4, and a true geometric truncation would create rectangular faces. The uniform rhombitruction requires adjustment to create square faces.

Original	Rectification	Rhombitruncation
		Truncated octahedron
	Cuboctahedron	Truncated cuboctahedron or rhombitruncated cuboctahedron
	Icosidodecahedron	Truncated icosidodecahedron or rhombitruncated icosidodecahedron
	Trihexagonal tiling	Truncated trihexagonal tiling or great rhombitrihexagonal tiling
	Triheptagonal tiling	Truncated triheptagonal tiling or great rhombitriheptagonal tiling
	Trioctagonal tiling	Truncated trioctagonal tiling or great rhombitriheptagonal tiling
	Square tiling	Truncated square tiling
	Order-5 square tiling	Order-5 truncated square tiling
	Order-5 pentagonal tiling	Order-5 truncated pentagonal tiling

[edit] Truncation in polychora and honeycomb tessellation

A regular polychoron or tessellation {p,q,r}, truncated becomes a uniform polychoron or tessellation with 2 cells: truncated {p,q}, and {q,r} cells are created on the truncated section.

See: uniform polychoron and convex uniform honeycomb.

Family [p,q,r]	Parent	Truncation	Rectification (birectified dual)	Bitruncation (bitruncated dual)
[3,3,3]	5-cell (self-dual)	truncated 5-cell	rectified 5-cell	bitruncated 5-cell
[3,3,4]	16-cell	truncated 16-cell	rectified 16-cell (Same as 24-cell)	bitruncated 16-cell (bitruncated tesseract)
[4,3,3]	Tesseract	truncated tesseract	rectified tesseract	bitruncated tesseract (bitruncated 16-cell)
[3,4,3]	24-cell (self-dual)	truncated 24-cell	rectified 24-cell	bitruncated 24-cell
[3,3,5]	600-cell	truncated 600-cell	rectified 600-cell	bitruncated 600-cell (bitruncated 120-cell)
[5,3,3]	120-cell	truncated 120-cell	rectified 120-cell	bitruncated 600-cell (bitruncated 120-cell)
[4,3,4]	cubic (self-dual)	truncated cubic	rectified cubic	bitruncated cubic
[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	(No image) bitruncated cubic (bitruncated dodecahedral)
[5,3,5]	(No image) dodecahedral (self-dual)	(No image) truncated dodecahedral	(No image) rectified dodecahedral	(No image) bitruncated dodecahedral