# Truncation (InstanceTopic, 3)

### From Hi.gher. Space

**Truncation** is the process of cutting polytope facets to produce new polytopes. The kind of truncation can be specified by a Dx number.

## Classic truncations

The classic truncations have the first and *n*th nodes ringed, and have the same Dx number in each dimension. These are *truncation* (Dx 3) for *n*=2, *cantellation* (Dx 5) for *n*=3 and *runcination* (Dx 9) for *n*=4. The words can be concatenated (in reverse order of dimension) to count the various nodes that are ringed. For example, *cantitruncation* has the first, second and third nodes ringed and *runcicantellation* has the first, third and fourth nodes ringed.

## Special truncations

There are three types of truncations which have a different Dx number in each dimension, while appearing to perform analogous operations on the actual polytope:

- Mesotruncation
- Truncates the polytope to its "midpoint". In odd dimensions, this is represented as the middle Coxeter-Dynkin node ringed. In even dimensions, the middle two nodes are ringed. Keiji had previously erroneously called this "rectification", however they are only the same in 3D: true rectification is represented with only the second node ringed, regardless of dimension.
- Peritruncation
- "Expands" the polytope. This is represented with the first and last nodes ringed. Etymology: the Romanian for "outside" is "periferic" - you can imagine the ringed nodes as being on the "outside" (at the ends) of the diagram. Interestingly, peritruncating a polyhedron or polychoron is equivalent to mesotruncating it twice, but it is not yet known whether this extends to higher dimensions.
- Omnitruncation
- Gives the "largest", most "spherical" version of the root polytope, with the highest number of total elements. This is represented with all nodes ringed. Omnitruncating a polyhedron is equivalent to mesotruncating it and then 1-truncating the result, but it is not yet known whether this extends to higher dimensions.