# Truncation (geometry)

 A truncated cube - faces double in edges, and vertices replaced by new faces. A Truncated cubic honeycomb - faces doubled in edges, and vertices replaced by new cells.

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

## Types of truncations

Coxeter uses truncation to mean a number of related but topologically different operations:

• Shallow truncation - Edges are reduced in length, faces are truncated to have twice as many sides. Uniform truncations are a special case of this with equal edge lengths. The truncated cube t0,1{4,3} or t{4,3}, with square faces becoming octagons, with new triangular faces are the vertices.
• Uniform truncations can be applied to quasiregular polyhedra, by adjustment of the geometry. The truncated cuboctahedron, t0,1,2{4,3} or tr{4,3} is an example. This is also called a quasi-truncation or a cantitruncation.
• Complete truncation or rectification - Edges are reduced to points. The cuboctahedron t1{4,3} or r{4,3} is an example.
• Bitruncation A deeper truncation that leaves smaller faces. The truncated octahedron is a bitruncated cube: t1,2{4,3} or 2t{4,3} is an example.
• Birectification The limit of bitruncation that reduces faces to points. For polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube: {3,4} = 2r{4,3} is an example.
• Further deep truncations are defined by specific terms: Cantellation t0,2{p,q} or rr{p,q} cuts edges of polyhedra creaing new faces, runcination t0,3{p,q,r} cuts faces creating new cells to 4-polytopes in 4D, and sterication t0,4{p,q,r,s} or 2r2r{p,q,r,s} cuts cells and creates new 4-faces to 5-polytopes.
• Hypertruncation A form of truncation that goes past the rectification, but allows the dual faces to continue growing and intersect the original faces.
• Quasitruncation A form of truncation that goes even farther than hypertruncation. It can be generated from the original polyhedron by treating all the faces as retrograde, i.e. going backwards round the vertex. For example, quasitruncating the square gives a regular octagram, and quasitruncating the cube gives the uniform stellated truncated hexahedron.
• Antitruncation A form of truncation that goes even farther than quasitruncation. This results in a polyhedron which looks like the original, but has parts of the dual dangling off its corners, instead of the dual cutting into its own corners. Its resemblance to a "negative truncation" gives it this name.

Other usages express a truncation of only some of the vertices.

## Uniform truncation

In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation.

A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra.

More abstractly any uniform polytope defined by a Coxeter-Dynkin diagram with a single ring, can be also uniformly truncated, although it is not a geometric operation, but requires adjusted proportions to reach uniformity. For example Kepler's truncated icosidodecahedron represents a uniform truncation of the icosidodecahedron. It isn't a geometric truncation, which would produce rectangular faces, but a topological truncation that has been adjusted to fit the uniformity requirement.

## Truncation of polygons

A truncated n-sided polygon will have 2n sides (edges). A regular polygon uniformly truncated will become another regular polygon: t{n} is {2n}.

Star polygons can also be truncated. A truncated pentagram {5/2} will look like a pentagon, but is actually a double-covered (degenerate) decagon ({10/2}) with two sets of overlapping vertices and edges. A truncated great heptagram {7/3} gives a tetradecagram {14/3}.

## 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 t0,1{p,q,...} or t{p,q,...}.

## 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.)

The dual operation to truncation is the construction of a Kleetope.

## 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
{p,q}
Truncation
t{p,q}
t0,1{p,q}
Rectification
r{p,q}=r{q,p}
t1{p,q}
Bitruncation
(truncated dual)
2t{p,q}=t{q,p}
t1,2{p,q}
Birectification
(dual)
2r{p,q}={q,p}
t2{p,q}
[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

Truncated hexagonal

Trihexagonal

Hexagonal

Triangular
[7,3]
Heptagonal

Truncated heptagonal

Triheptagonal

Truncated order-7 triangular

Order-7 triangular
[8,3]
Octagonal

Truncated octagonal

Trioctagonal

Truncated order-8 triangular

Order-8 triangular
[4,4]
Square

Truncated square

Square

Truncated square

Square
[5,4]
Order-4 pentagonal

Truncated order-4 pentagonal

Tetrapentagonal

Truncated order-5 square

Order-5 square
[5,5]
Order-5 pentagonal

Truncated order-5 pentagonal

Order-4 pentagonal

Truncated order-5 pentagonal

Order-5 pentagonal

## Prismatic polyhedron examples

Family Original Truncation Rectification
(And dual)
[2,6]
Hexagonal hosohedron
(As spherical tiling)
{2,6}

Hexagonal prism
t{2,6}

Hexagonal dihedron
(As spherical tiling)
{6,2}

## Cantitruncated 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 cantitruncation requires adjustment to create square faces except when p = q (as then the rectified form is regular rather than simply quasiregular).

Original
(Regular)
{p,q}
Rectification
r{p,q}
t1{p,q}
Truncated rectification
Cantitruncation
Omnitruncation
tr{p,q}
t0,1,2{p,q}

Truncated octahedron

Cuboctahedron

Truncated cuboctahedron

Icosidodecahedron

Truncated icosidodecahedron

Trihexagonal

Truncated trihexagonal

Triheptagonal

Truncated triheptagonal

Trioctagonal

Truncated trioctagonal

Square

Truncated square

Tetrapentagonal

Truncated tetrapentagonal

Order-4 pentagonal

Truncated order-4 pentagonal

## 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.

Family
[p,q,r]
Parent
{p,q,r}
Truncation
t{p,q,r}
t0,1{p,q}
Rectification
(birectified dual)
r{p,q,r}
t1{p,q}
Bitruncation
(bitruncated dual)
2t{p,q,r}
t1,2{p,q}
[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
[4,3,4]
cubic (self-dual)

truncated cubic

rectified cubic

bitruncated cubic
[3,5,3]
icosahedral (self-dual)

truncated icosahedral

rectified icosahedral

bitruncated icosahedral
[4,3,5]
cubic

truncated cubic

rectified cubic

bitruncated cubic
(bitruncated dodecahedral)
[5,3,4]
dodecahedral

truncated dodecahedral

rectified dodecahedral
[5,3,5]
dodecahedral (self-dual)

truncated dodecahedral

rectified dodecahedral

bitruncated dodecahedral