Can somone explain the naming system Bowers and Coxeter uses

Higher-dimensional geometry (previously "Polyshapes").

Can somone explain the naming system Bowers and Coxeter uses

Postby ubersketch » Wed Apr 04, 2018 7:39 pm

title.
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Coxeters is really marvellous, but this title box is too sma

Postby student91 » Wed Apr 04, 2018 8:42 pm

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Re: Can somone explain the naming system Bowers and Coxeter

Postby Klitzing » Fri Apr 06, 2018 4:35 pm

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Re: Can somone explain the naming system Bowers and Coxeter

Postby wendy » Sat Apr 07, 2018 6:09 am

The names represent the crosses and dots on the Coxeter-Dynkin diagrams.

In essence, the name consists of how many unmarked nodes appear in front, and the pattern of the marked noded.

If there is only one marked node, like o--o--x--o--o-4-o, you start at the end with the fewest o's, here the left. Count 1,2,x, gives bi- (twice) and the pattern 'x' is 'rectified. This is a bi-rectified hexaract (6-cube).

If there are several unmarked nodes, like o--o--x--o--o-4-x the pattern is still the same, but the count is different. Now you count 2,3,x from the left, and x from the right. You would still use the greek prefix as above, here (L= tri- ) and (R= none).

The next step is to convert the pattern xoox. The first x is always there, and the next positions go truncated, rhombi- and prismato- in Bowers system. The order is from right to left, so xoox is a prismato-.

A pattern of o's and x' lie oooxoxxoooo, would number 2,3,4,x from the left and 2,3,4,5,x from the right. So you use the left form, ie TETRA. The middle pattern is xoxx. which gives xorp, becomes RHOMBO-PRISMATIC. So this is a tetra-rhombo-prismatic (whatever the numbers spell).

Coxeter used Stott's system, which is to give a prefix t_a,b,c added to the base figure. The node count starts from 0, so 0o 1o 2o 3x 4o 5x 6x 7o 8o .. becomes a 'three-five-six-truncated ....
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Re: Can somone explain the naming system Bowers and Coxeter

Postby Klitzing » Sat Apr 07, 2018 11:48 am

Wendy's reply just amounts to the prefix.
But the base figure names usually amount into a facet count part plus a dimension (of those facets) part.

E.g. an octahedron has 8=octa hedra, i.e. 8 2-dimensional facets.
E.g. an icosidodecahedron has 20=icosa + 12=dodeca hedra, i.e. 20 polygons of one type plus 12 polygons of an other type.
E.g. the decachoron has 10=deca chora, i.e. 10 3-dimensional facets.
And Gosset's figure 4_21 would be called a "dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton", i.e. a 2160 + 17280 faceted polytope, where all facets are of dimension 7=zetta.

hedra = 2D facets
chora = 3D facets
tera = 4D facets
peta = 5D facets
exa = 6D facets
zetta = 7D facets
yotta = 8D facets
xenna = 9D facets

Accordingly a general polytope of that dimension, i.e. with uncounted "many" facets of this type, then is a poly-hedron, poly-choron, poly-teron, poly-peton. poly-exon, poly-zetton, etc. (poly = many).

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Re: Can somone explain the naming system Bowers and Coxeter

Postby ubersketch » Sat Apr 07, 2018 6:09 pm

This doesn't explain how hemi is used which is one of my concerns.
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Re: Can somone explain the naming system Bowers and Coxeter

Postby Klitzing » Sun Apr 08, 2018 9:50 am

hemi is just some sort of hemiation, i.e. taking every second vertex, or so.
Thus a hemicube is nothing but the tetrahedron.
The hemitesseract is nothing but the hexadecachoron.

In fact, this is what Norman Johnson once made up his extended application of the snubbing symbols for. E.g.
Code: Select all
s-4-o-3-o-3-o-3-o  =

o_
  3_
   _o-3-o-3-o
 _3
x

Code: Select all
s-4-o-3-x-3-x-3-o  =

o_
  3_
   _x-3-x-3-o
 _3
x

Note that he did that only on a mere symbolic level then. But this usage can be made more rigorous. Infact, every Dynkin symbol with any mixture of unringed nodes (i.e "o"), ringed nodes (i.e. "x"), and snub nodes (i.e. "s") truely makes sense, at least as a mere alternated faceting, not necessarily as rescaled, all unit-edged polytope. (The ability for the latter step depends on the available degree of freedom only.)
Cf. https://bendwavy.org/klitzing/explain/dynkin.htm#snub.

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Re: Can somone explain the naming system Bowers and Coxeter

Postby Klitzing » Sun Apr 08, 2018 10:00 am

The above explains how hemi is being used as an operant, i.e. as an adjective.
But when being used as integral part of the name, it usually means that the according element runs through the body center. Thus an according polytope will be non-orientable.

E.g. cho = cubohemioctahedron and oho = octahemioctahedron both are facetings of the cuboctahedron (co), both using the 3 diametral hexagons, the first one additionally the 6 squares of co, the second one the 8 triangles of co.

Thus here too hemi means some sort of hemiation. But here the number of according faces is divided by 2. (As those else would get doubled up, when runing through the center.)

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Re: Can somone explain the naming system Bowers and Coxeter

Postby ubersketch » Sat Apr 14, 2018 9:50 pm

What about Bower's system for 4D polytopoids? For example lets take the medial omnicircumfacetopental triakishecatonicosachoron or mom fapathi. What does its name mean?
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