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 nth 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:

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.
"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.
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.