Elemental naming scheme (ConceptTopic, 3)
From Hi.gher. Space
The elemental naming scheme is a method of naming many symmetry groups and uniform polytopes devised mostly by Keiji, though the original idea and root words were suggested by Tamfang. It used Tamfang's name for a long time, but has belatedly (and retroactively) been renamed due to his wish to be disassociated from the modifications made by Keiji.
Classically, polytopes are named after their facet counts, or the facet counts of other polytopes from which they are derived. This, however, can lead to confusion, since the facet counts do not match up between one dimension and another. Furthermore, some such polytopes have unwieldly long names as a result. Under the elemental naming scheme, most polytopes have shorter, simpler names, and immediately convey some important properties about the polytope, such as their symmetry.
Regular polytopes
The prefixes pyro-, geo-, aero-, xylo-, cosmo- and hydro- are used to refer to the basic symmetries of the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell and 600-cell respectively, as well as the analogous symmetries in higher and lower dimensions. The suffixes -gon, -hedron, -choron, -teron, -peton and so on are applied to refer to the dimension, just as they are in classic facet-count naming.
The above prefixes clearly come from the classical elements of fire, earth, air and water. However, since there was one extra regular polyhedron, the cosmo- prefix (meaning universe) was adopted for the dodecahedron and its analogues. Similarly, the xylo- prefix (meaning wood) was adopted for the 24-cell.
Finally, one can refer to a regular polytope and its dual collectively by using the prefixes stauro- (meaning cross, for geotopes and aerotopes together) and rhodo- (referring to plants with 5-fold symmetry, for cosmotopes and hydrotopes together). Note that because the pyrotopes and xylotopes are self-dual, those prefixes are their own collective terms.
For the full list of possibilities, see the table of regular polytopes by elemental name.
Uniform polytopes
To extend the naming scheme to the (convex, non-prismatic, Wythoffian) uniform polytopes, one must first notice that they divide into binary asymmetric and binary symmetric polytopes. The latter are those for which the binary representation of their Dx number is palindromic; the former are those for which it is not. The binary asymmetric polytopes are said to have one parent, which is the regular polytope of the same symmetry it is "closest" to, and the binary symmetric polytopes are said to have two parents, as they are equally close to one parent and its dual (though if the parent is self-dual, both parents are the same). We then name the binary asymmetric polytopes with the same elemental prefix as their parent polytope, and the binary symmetric polytopes with the collective elemental prefix containing their two parents (pyro-, stauro-, xylo- or rhodo-).
To distinguish two polytopes of the same symmetry and different Dx numbers, we then use one or more of the following infixes:
- -tomo-, rings the first two nodes, from 3D up (from tome, which was previously being used to refer to the various truncates collectively, and now refers to only the Dx 3 truncate)
- -canti-, rings the first and third nodes, from 4D up
- -runci-, rings the first and fourth nodes, from 4D up
- -recti-, rings the second node only, from 4D up
- -meso-, rings the middle node (in odd dimensions) or the middle two nodes (in even dimensions), from 3D up
- -peri-, rings the outside nodes, from 3D up (from peripheral or perimeter, meaning outside, or boundary)
- -panto-, rings all nodes, from 3D up (though the 2D case also exists, it simply becomes a regular n-gon with twice the number of sides) (from pan, meaning all)
If multiple infixes are used, they are ordered with the last ringed node first. For example, runci comes before canti, which comes before tomo.
For the full list of possibilities in 4D, see the list of uniform polychora.
Extension beyond 4D
Mecejide pointed out in 2021 that some convex uniform polytopes were not included in the elemental naming scheme, specifically those that have only branching CD diagrams. These are the demihypercubes (an infinite family of unique uniform polytopes from 5D and up; the equivalents in lower dimensions are reduced symmetry constructions of the aerochoron in 4D, the pyrochoron in 3D, and the degenerate digon in 2D) and the En family of polytopes, which are only convex in 6, 7 and 8 dimensions.
The demihypercube family is named osteo-, after bone, related to both wood (a reduced symmetry form of the xylochoron shares the same star arrangement CD symbol as the osteochoron, or demitesseract) and earth (the osteotopes are formed by alternating the geotopes).
The En family in 7 and 8 dimensions have three unique branches. By analogy with the stauromorphs and rhodomorphs, which are divided into geo-/aero- and cosmo-/hydro- respectively, the En family is collectively named metallo-, divided into sidero- for 321, argyro- for 231 and chryso- for 132 respectively, after iron, silver and gold. (chalco-, after copper, was originally suggested for 321, but unfortunately shares a prefix with chryso-, so sidero- is less confusing.) These are named with the most valuable metal being the most complex figure, similar to how the cosmotopes could be considered more complex than the hydrotopes due to their pentagonal (instead of trigonal) faces. The 8D equivalents 421, 241 and 142 use the same prefixes, while in 6D, chryso- is used for 122 and sidero- is used for 221 as the symmetry in the CD diagram means there is no unique third branch in 6D.
