Truncation_(geometry)

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Truncation (geometry)

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

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 with two sets of overlapping vertices and edges.

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 truncates 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]
[4,3]
[5,3]
[6,3]
[7,3]
[8,3]
[4,4]
[5,4]
[5,5]

Prismatic polyhedron examples

Family	Original	Truncation	Rectification (And dual)
[2,p]

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

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]
[3,3,4]
[4,3,3]
[3,4,3]
[3,3,5]
[5,3,3]
[4,3,4]
[3,5,3]	(No image) truncated icosahedral	(No image) rectified icosahedral	(No image) bitruncated icosahedral
[4,3,5]	(No image) truncated cubic	(No image) rectified cubic	(No image) bitruncated cubic (bitruncated dodecahedral)
[5,3,4]	(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