TRUNCATION (GEOMETRY)

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

Contents

Truncation in regular polyhedra and tilings

Other truncations

Uniform polyhedron and tiling examples

Prismatic polyhedron examples

rhombitruncated examples

Truncation in polychora and honeycomb tessellation

See also

References

External links

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,...}.

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 truncated alternate vertices. (This operation can also apply to any zonohedron which have even-sided faces.)

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	Tetratetrahedron	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	Truncated triangular	Triangular
[7,3]	Heptagonal	Truncated heptagonal	Recified heptagonal	Truncated Triangular	Triangular
[8,3]	Octagonal	Truncated Octagonal	Recified Octagonal	Truncated Triangular	Triangular
[4,4]	Square	Truncated square	Square	Truncated square	Square
[5,4]	Pentagonal	Truncated pentagonal	Rectified pentagonal	Truncated square	Square
[5,5]	Pentagonal	Truncated pentagonal	Rectified pentagonal	Truncated pentagonal	Pentagonal

Prismatic polyhedron examples

{| class="wikitable"
!Family
!Original
!Truncation
!Rectification
(And dual)
|-
|[2,p]
|align=center|
Hexagonal hosohedron
(As spherical tiling)
{2,p}
|align=center|
Hexagonal prism
t{2,p}
|align=center|
Hexagonal dihedron
(As spherical tiling)
{p,2}
|}

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.
{| class="wikitable"
!Original
!Rhombitruncation
!Rectification
|-
|

|
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
|
|}

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.
{| class="wikitable"
!Family
[p,q,r]
!Parent
!Truncation
!Rectification
(birectified dual)
!Bitruncation
(bitruncated dual)
|-
![3,3,3]
|
5-cell
|
truncated 5-cell
|
rectified 5-cell
|
bitruncated 5-cell
|-
![3,3,4]
|
16-cell (self-dual)
|
'truncated 16-cell'
(Same as 24-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
|
truncated 24-cell
|
rectified 24-cell
|
bitruncated 24-cell
|-
![3,3,5]
|
600-cell
|
truncated 600-cell
|
rectified 600-cell
|rowspan=2|
bitruncated 600-cell
(bitruncated 120-cell)
|-
![5,3,3]
|
120-cell
|
truncated 120-cell
|
rectified 120-cell
|-
![4,3,4]
||
cubic
|
truncated cubic
|
rectified cubic
|
bitruncated cubic
|-
![3,5,3]
|
icosahedral
|(No image)
truncated icosahedral
|(No image)
rectified icosahedral
|(No image)
bitruncated icosahedral
|-
![4,3,5]
|
cubic
|(No image)
truncated cubic
|(No image)
rectified cubic
|rowspan=2|(No image)
bitruncated cubic
(bitruncated dodecahedral)
|-
![5,3,4]
|
dodecahedral
|(No image)
truncated dodecahedral
|(No image)
rectified dodecahedral
|-
![5,3,5]
|(No image)
dodecahedral
|(No image)
truncated dodecahedral
|(No image)
rectified dodecahedral
|(No image)
bitruncated dodecahedral
(bitruncated dodecahedral)
|}

See also

★ uniform polyhedron

★ uniform polychoron

★ Bitruncation (geometry)

★ Rectification (geometry)

★ Alternation (geometry)

★ Conway polyhedron notation

References

★ Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation)

External links

★ MathWorld: Truncation

★