Truncation (geometry)

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Truncated hexahedron.png
A truncated cube - faces double in edges, and vertices replaced by new faces.
Truncated cubic honeycomb.jpg
A truncated cubic honeycomb - faces doubled in edges, and vertices replaced by new, octahedral, 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[edit]

Types of truncations shown on an edge isolated from a larger polygon or polyhedron with with red and blue vertices. The edge reverses direction after complete truncation.

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, while new facets are formed, centered at the old vertices.
  • Uniform truncations are a special case of this with equal edge lengths. The truncated cube, t{4,3}, with square faces becoming octagons, with new triangular faces are the vertices.
  • Antitruncation A reverse shallow truncation, truncated outwards off the original edges, rather than inward. This results in a polytope which looks like the original, but has parts of the dual dangling off its corners, instead of the dual cutting into its own corners.
  • Complete truncation or rectification - The limit of a shallow truncation, where edges are reduced to points. The cuboctahedron, r{4,3}, is an example.
  • Hypertruncation A form of truncation that goes past the rectification, inverting the original edges, and causing self-intersections to appear.
  • Quasitruncation A form of truncation that goes even farther than hypertruncation where the inverted edge becomes longer than the original edge. It can be generated from the original polytope by treating all the faces as retrograde, i.e. going backwards round the vertex. For example, quasitruncating the square gives a regular octagram (t{4,3}={8/3}), and quasitruncating the cube gives the uniform stellated truncated hexahedron, t{4/3,3}.
Truncations on a square
Types of truncation on square4.png
Types of truncation on a square, {4}, showing red original edges, and new truncated edges in cyan. A uniform truncated square is a regular octagon, t{4}={8}. A complete truncated square becomes a new square, with a diagonal orientation. Vertices are sequenced around counterclockwise, 1-4, with truncated pairs of vertices as a and b.
Truncations of the cube
Cube truncation 3.75.png
Cube truncation 0.00.png
Cube
{4,3}
Cube truncation 0.25.png
Cube truncation 0.50.png
Truncation
t{4,3}
Cube truncation 0.75.png
Cube truncation 1.00.png
Complete truncation
r{4,3}
Cube truncation 1.25.png
Cube truncation 3.50.png
Antitruncation
Cube truncation 1.50.png
Hypertruncation
Cube truncation 3.25.png
Cube truncation 3.00.png
Complete quasitruncation
Cube truncation 2.75.png
Cube truncation 2.50.png
Quasitruncation
t{4/3,3}
Cube truncation 2.25.png
Cube truncation 2.00.png
Complete hypertruncation
Cube truncation 1.75.png

Truncation operations on uniform polytopes:

  • Bitruncation A deeper truncation, like the hypertruncation, but leaves smaller faces, and new faces are created. 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, creating all new faces. 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 polytopes 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.

Others:

  • Edge-truncation - beveling or Chamfer for polyhedra, where edges are replaced by hexagons. In comparison, edge-truncation in 4-polytopes replaces edges with elongated bipyramid cells.

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

Uniform truncation[edit]

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[edit]

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[edit]

Truncations of the cube beyond rectification

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.

Cube truncation sequence.svg

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[edit]

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[edit]

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] Uniform polyhedron-33-t0.png
Tetrahedron
Uniform polyhedron-33-t01.png
Truncated tetrahedron
Uniform polyhedron-33-t1.png
Octahedron
Uniform polyhedron-33-t12.png
Truncated tetrahedron
Uniform polyhedron-33-t2.png
Tetrahedron
[4,3] Uniform polyhedron-43-t0.png
Cube
Uniform polyhedron-43-t01.png
Truncated cube
Uniform polyhedron-43-t1.png
Cuboctahedron
Uniform polyhedron-43-t12.png
Truncated octahedron
Uniform polyhedron-43-t2.png
Octahedron
[5,3] Uniform polyhedron-53-t0.png
Dodecahedron
Uniform polyhedron-53-t01.png
Truncated dodecahedron
Uniform polyhedron-53-t1.png
Icosidodecahedron
Uniform polyhedron-53-t12.png
Truncated icosahedron
Uniform polyhedron-53-t2.png
Icosahedron
[6,3] Uniform tiling 63-t0.png
Hexagonal
Uniform tiling 63-t01.png
Truncated hexagonal
Uniform tiling 63-t1.png
Trihexagonal
Uniform tiling 63-t12.png
Hexagonal
Uniform tiling 63-t2.png
Triangular
[7,3] Uniform tiling 73-t0.png
Heptagonal
Uniform tiling 73-t01.png
Truncated heptagonal
Uniform tiling 73-t1.png
Triheptagonal
Uniform tiling 73-t12.png
Truncated order-7 triangular
Uniform tiling 73-t2.png
Order-7 triangular
[8,3] Uniform tiling 83-t0.png
Octagonal
Uniform tiling 83-t01.png
Truncated octagonal
Uniform tiling 83-t1.png
Trioctagonal
Uniform tiling 83-t12.png
Truncated order-8 triangular
Uniform tiling 83-t2.png
Order-8 triangular
[4,4] Uniform tiling 44-t0.png
Square
Uniform tiling 44-t01.png
Truncated square
Uniform tiling 44-t1.png
Square
Uniform tiling 44-t12.png
Truncated square
Uniform tiling 44-t2.png
Square
[5,4] Uniform tiling 54-t0.png
Order-4 pentagonal
Uniform tiling 54-t01.png
Truncated order-4 pentagonal
Uniform tiling 54-t1.png
Tetrapentagonal
Uniform tiling 54-t12.png
Truncated order-5 square
Uniform tiling 54-t2.png
Order-5 square
[5,5] Uniform tiling 552-t0.png
Order-5 pentagonal
Uniform tiling 552-t01.png
Truncated order-5 pentagonal
Uniform tiling 552-t1.png
Order-4 pentagonal
Uniform tiling 552-t12.png
Truncated order-5 pentagonal
Uniform tiling 552-t2.png
Order-5 pentagonal

Prismatic polyhedron examples[edit]

As spherical tilings
Family Original Truncation Rectification
(And dual)
[2,6] Spherical hexagonal hosohedron.png
Hexagonal hosohedron
{2,6}
Spherical hexagonal prism.png
Hexagonal prism
t{2,6}
Hexagonal dihedron.png
Hexagonal dihedron
{6,2}

Cantitruncated examples[edit]

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} or t1{p,q}
Cantitruncation
(Truncated rectification)
tr{p,q} or t0,1,2{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
Uniform polyhedron-33-t0.png
Uniform polyhedron-33-t1.png
Uniform polyhedron-33-t012.png
Truncated octahedron
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t1.png
Cuboctahedron
Uniform polyhedron-43-t012.png
Truncated cuboctahedron
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t1.png
Icosidodecahedron
Uniform polyhedron-53-t012.png
Truncated icosidodecahedron
Uniform tiling 63-t0.png Uniform tiling 63-t1.png
Trihexagonal
Uniform tiling 63-t012.png
Truncated trihexagonal
Uniform tiling 73-t0.png Uniform tiling 73-t1.png
Triheptagonal
Uniform tiling 73-t012.png
Truncated triheptagonal
Uniform tiling 83-t0.png Uniform tiling 83-t1.png
Trioctagonal
Uniform tiling 83-t012.png
Truncated trioctagonal
Uniform tiling 44-t0.png Uniform tiling 44-t1.png
Square
Uniform tiling 44-t012.png
Truncated square
Uniform tiling 54-t0.png Uniform tiling 54-t1.png
Tetrapentagonal
Uniform tiling 54-t012.png
Truncated tetrapentagonal
Uniform tiling 552-t0.png Uniform tiling 552-t1.png
Order-4 pentagonal
Uniform tiling 552-t012.png
Truncated order-4 pentagonal

Truncation in 4-polytopes and honeycomb tessellation[edit]

A regular 4-polytope or tessellation {p,q,r}, truncated becomes a uniform 4-polytope or tessellation with 2 cells: truncated {p,q}, and {q,r} cells are created on the truncated section.

See: uniform 4-polytope and convex uniform honeycomb.

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] Schlegel wireframe 5-cell.png
5-cell (self-dual)
Schlegel half-solid truncated pentachoron.png
truncated 5-cell
Schlegel half-solid rectified 5-cell.png
rectified 5-cell
Schlegel half-solid bitruncated 5-cell.png
bitruncated 5-cell
[3,3,4] Schlegel wireframe 16-cell.png
16-cell
Schlegel half-solid truncated 16-cell.png
truncated 16-cell
Schlegel half-solid rectified 16-cell.png
rectified 16-cell
(Same as 24-cell)
Schlegel half-solid bitruncated 16-cell.png
bitruncated 16-cell
(bitruncated tesseract)
[4,3,3] Schlegel wireframe 8-cell.png
Tesseract
Schlegel half-solid truncated tesseract.png
truncated tesseract
Schlegel half-solid rectified 8-cell.png
rectified tesseract
Schlegel half-solid bitruncated 8-cell.png
bitruncated tesseract
(bitruncated 16-cell)
[3,4,3] Schlegel wireframe 24-cell.png
24-cell (self-dual)
Schlegel half-solid truncated 24-cell.png
truncated 24-cell
Schlegel half-solid cantellated 16-cell.png
rectified 24-cell
Bitruncated 24-cell Schlegel halfsolid.png
bitruncated 24-cell
[3,3,5] Schlegel wireframe 600-cell vertex-centered.png
600-cell
Schlegel half-solid truncated 600-cell.png
truncated 600-cell
Rectified 600-cell schlegel halfsolid.png
rectified 600-cell
Bitruncated 120-cell schlegel halfsolid.png
bitruncated 600-cell
(bitruncated 120-cell)
[5,3,3] Schlegel wireframe 120-cell.png
120-cell
Schlegel half-solid truncated 120-cell.png
truncated 120-cell
Rectified 120-cell schlegel halfsolid.png
rectified 120-cell
[4,3,4] Partial cubic honeycomb.png
cubic (self-dual)
Truncated cubic honeycomb.jpg
truncated cubic
Rectified cubic honeycomb.jpg
rectified cubic
Bitruncated cubic honeycomb.png
bitruncated cubic
[3,5,3] H3 353 CC center.png
icosahedral (self-dual)
H3 353-0011 center ultrawide.png
truncated icosahedral
H3 353 CC center 0100.png
rectified icosahedral
H3 353-0110 center ultrawide.png
bitruncated icosahedral
[4,3,5] H3 435 CC center.png
cubic
H3 435-0011 center ultrawide.png
truncated cubic
H3 435 CC center 0100.png
rectified cubic
H3 534-0110 center ultrawide.png
bitruncated cubic
(bitruncated dodecahedral)
[5,3,4] H3 534 CC center.png
dodecahedral
H3 534-0011 center ultrawide.png
truncated dodecahedral
H3 534 CC center 0100.png
rectified dodecahedral
[5,3,5] H3 535 CC center.png
dodecahedral (self-dual)
H3 535-0011 center ultrawide.png
truncated dodecahedral
H3 535 CC center 0100.png
rectified dodecahedral
H3 535-0110 center ultrawide.png
bitruncated dodecahedral

See also[edit]

References[edit]

External links[edit]

Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png Uniform polyhedron-43-s012.png
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}