Truncation (geometry)

From Wikipedia, the free encyclopedia

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

Contents

[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.

Image:Cube truncation sequence.png

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,...}.

[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 truncated 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

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

[edit] Prismatic polyhedron examples

Family Original Truncation Dual
[p,2]
Hexagonal dihedron
(As spherical tiling)

Hexagonal prism

Hexagonal hosohedron
(As spherical tiling)

[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 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

[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

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
[3,4,3]
24-cell

truncated 24-cell

rectified 24-cell

bitruncated 24-cell
[3,3,5]
600-cell

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

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
(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)

[edit] See also

[edit] References

[edit] External links