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 is belatedly (and retroactively) being 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).
For the full list of possibilities, see the table of regular polytopes by elemental name.
Uniform polytopes
To be added
