"exotomes"

[Keiji]

I've recently been interested in the n-dimensional polytopes with CD xo*x (xx, xox, xoox, xooox, etc.).
Particularly because I've been trying to find an n-dimensional analog for the octagon for use as powertope bases - though the series of truncates (aka 2-tomes) with CD xxo* (xx, xxo, xxoo, xxooo, etc.) also forms a possibility for this.

Anyway, I wanted a good name for the first series mentioned. I noticed they involve precisely the first and last nodes being ringed (the "outside nodes", if you will), so I chose the Greek prefix "exo" for "outside". So:
*exotome = general family and dimension
*pyroxotome, stauroxotome, xyloxotome, rhodoxotome = specified family, general dimension
*exogon, exohedron, exochoron, exopeton, ... = general family, specified dimension
*rhodoxohedron, stauroxochoron, ... = specified family and dimension
Note I drop the 'e' in most situations because it would be a mouthful to go vowel-vowel, and I don't say e.g. stauroxotomochoron like I do with stauropantomochoron because exo- implies exotomo- (pantomo- is kept because panchoron etc. sound weird).

With these names we don't have to go:
truncated square (= octagon), cantellated cube/octahedron (= rhombicuboctahedron = cuboctahedral rectate), runcinated tesseract, stericated penteract, etc.
We can just go:
stauroxogon, stauroxohedron, stauroxochoron, stauroxopeton, etc.

Any comments on this naming scheme?

[wendy]

In the polygloss these are listed as 'runcinates'. This is the term lifted from Norman Johnson, who also uses it for xoox (only). It's the same group that leads to the 'prismato' branch by GO et al, because the bulk of faces are prisms. My ancient name for these are 'prism-circuits'. The runcinate class is also the generic /face expand/ by Mrs Stott.

The dual of the runcinates are the strombiates, these have antitegums as faces, the faces are formed by the intersection of margins of the figure and its dual.

[Keiji]

I had briefly considered calling them "runcinates", but I can't for the same annoying reason that I couldn't call them "cantellates" when I previously considered this series: in 5D and up, runcination is used to mean something completely different. That's why I wanted to come up with a new word.

Interesting info about the duals though, I will have to look into that.

[quickfur]

So these are essentially the facet expansions of the polytopes, e.g., square -> octagon, cube -> rhombicuboctahedron, tetracube -> runcinated tetracube, etc.. They all project into octagons, and have nice "octagonal"-like properties. :-) Such as being able to form a uniform n-space tiling with their corresponding cubes.

[wendy]

There is some interesting story here.

When I found these in 1977, i called them 'prism circuits'. This is because the bulk of the faces are prisms of an element of a polytope, and the orthogonal element of the dual. For example, in the twelftychoron 5,3,3, the pentagons are crossed by edges of 3,3,5, and give rise to a pentagonal prism in the PC.

The name by Jonathan Bowers for xoox is also 'prismato-', but he uses fixed positions, so uses different names for xooox (5D etc).

The names cantellate and runcinate are various words from Norman Johnson, of 'johnson polyhedra' fame. In his terminology, cantelate and runcinate refer to the second and third node from the primary, ie [o]xox[o] and [o]xoox[o]. The first node is the truncate.

Although NJ and JB uses the same scheme, the order of count is different. NJ counts l-r, while JB runs r-l. Something like xoxx, for example, gives either '11' or '13', depending on which end one starts at. In my notation, it might be Dc13 or Do11. NJ prefers the name Do11 (runcitruncated 'Do'), while Bowers uses Dc13, as prismatorhombo tesseract.

The m-cantellate and m-cantetrate in the PG, in the sense of rectified m-rectate truncated m-rectate correctly aligns with Norman Johnson's usage, even though his reading is different.

The sense of rectified correctly corresponds to 4d, but was classified as a synonym for 'prism circuit', or /face expand/. The original cartoon for this shows a cube attached to a bicycle pump, and in the next frame, the cube becomes the rhombo-cuboctahedron, with the same size face.

It would have been nice to switch the two[cantellate and runcinate], so that rr (runcinated) = both r (rectified) r (rectified), but convention is established.

W

[quickfur]

A thought just occurred to me, that we could call these polytopes "rhombi-(base polytope)s", because they correspond with the rhombi-polyhedra in 3D, and the rhombi- prefix hasn't been allocated in higher dimensions yet AFAIK. However, "rhombification" may already have a conflicting definition, so I'm not sure if this terminology is viable.

[Secret]

I'm not familiar with this topic
Can you recommend any external source for me to learn more about it?

Poison

[wendy]

I do not know many external sources. Much of the terminology is new here. The problem is that words are changing meanings, to the extent that a dog is a cat, by way that cats have four legs, and thus are quadrupeds, and dogs, being quadrupeds, must also be cats.

Rhombus is a geometric quadrilateral, being of equal sides. The rhombi-ID and rhombi-CO result from the intersection of an ID, CO with a figure with rhombic faces. Norman Johnson introduced the term 'rhombirtruncate-' in the sense that the truncate- of the ID or CO can never be uniform: the vertex figure is a rectangle, not a square.

Taken in terms of the dynkin graph, the truncate is x--x--o, the rhombi is x--o--x, and the rhombi-truncate is x--x--x, ie x--t--r, as marked. Norman Johnson did not carry rhombi- into 4d, but the Olshevsky-Bowers notation does. In four dimensions, Norman uses x--t--c--r, where c = cantelated, r = runcinated. The O/B notation extends x--t--r--p with 'p', prismato.

In four dimensions, the faces not at the end of the dynkin chain are indeed prismatic. It's the same when the first and last of any chain in any dimension are the only marked nodes, which is why i designated these as Prism-Circuits.

Much of the work in higher dimensions is done with those figures that have dynkin graphs, but as we see in three dimensions, this counts for just the platonics, the prisms, the antiprisms, and some 13 extra solids. Most of the crystrographic things (which are catalans), simply don't occur in the list. Nor does anything else.

Still, the O/B, notation, like NJ's extend one dimension at a time, and little progress is taken to the view from six dimensions etc. This is why i have a lot of notations that are based on solid figures in N dimensions. Much of this proves unpopular with the folk who think a word has a specific dimension, but the natural language says otherwise (tilings, of any dimensions, have 'cells' and 'walls').

exotomes are an example of marking the first and last node of the dynkin graph. The original term for this was 'expand', the faces of a solid being expanded away from the centre, while retaining the same size. However, expand equally applies to any part of the surface, eg edge, vertex, and one can construct the polytope by expands of the v--e--h--c--t etc where v=vertex, e=edge, h=hedron, c=choron, etc. The t_ notation, in the form of t_0,1 is precisely this.

Of course, the terms in both Johnson and in O/B are pretty much using words with gay abandon, with little addition to insight: notation without notion. There really isn't any real reason why all of the 128 uniform figures derived from 3_21 need specific names. It does just as well, eg in the short notation, to describe the graph of say 5B, as eg 5/B (for 2_31).

This frees terms up for real notions. Bevel, for example, is used in the PG in the sense of to create a face perpendicular to a direction: an edge-beveled cube gives rise to something with four squares and six long hexagons. The dual by 'descent of faces' is a vertex-bevel, to the point of completely removing the whole former surface. The truncates, bitruncates, etc are the bevels before the edge, hedron, etc disappear. The rectates are the point where the edge, hedron, etc are reduced to a single point at their respective centres.

The exotemes are just another sense of 'first and last'. I use Norman's c-node word 'runcinate' to match his 4d use, but to also to express the notion of pumping out the faces of a polytope with no size change of face. New faces connect the bits of the old face together, by a series of bevels. Specifically, they form prisms of the X element of the face and the F-X element of the dual's face. This is where the prismato and prism circuit come from, but because of different assignments of meaning, only the latter carries to all dimensions with the correct meaning.