-wendyThorald Gosset discovered a series of polytopes, beginnining in E3 with the triangle-prism, and each subsequent figure having the previous as a vertex figure
--wendyblack magic^H^H^H^H^Hmathematics is anything involving calculus, algebra, trignometry, or any of those write-only mathematical languages. One is said to "cast spells" in the field.
--wendyAside. Norman Johnson once compared me to a priestess who cooked up assorted things without any sign of mathematical rigour or any kind of logic. I suppose it's true.
wendy wrote:black magic^H^H^H^H^Hmathematics is anything involving calculus, algebra, trignometry, or any of those write-only mathematical languages. One is said to "cast spells" in the field.
Aside. Norman Johnson once compared me to a priestess who cooked up assorted things without any sign of mathematical rigour or any kind of logic. I suppose it's true.
wendy wrote:Here is a table of the gosset-figures, and their surtope consist. All of these are simplexes, except what follows the plus-sign, which is a cross-polytope.
The count is from S0 to S(n-1), eg 0d vertex; 1d edges, 2d hedra, 3d chora; 4d tera, 5d peta, 6d ecta; 7d zetta and 8d yotta.
Semicolons (;) follow every third dimension, to assist reading.
E2: 3; 1+2
E3 : 6; 9, 2+3
E4: 10; 30, 30, 5+5
E5: 16; 80, 160, 120; 16+10
E6: 27; 216, 720, 1080; 648, 72+27
E7: 56; 756, 4032, 10080; 12096, 6048, 576+126
E8: 240; 6720. 60480; 241920; 483840, 483840; 207360, 172800+2160
E9: 1; 120, 2240, 15120; 483840, 40320, 69120; 60480, 25920, 1920+135
These figures are done in the mind, so there might be errors. To calculate E6 from E5, we see
27; 216 = 27*16/2; 720 = 27*80/3; 1080 = 27*160/4; 648 = 27* 120/5; 72 = 27*16/6; 27 = 27*27/10 (the last is 2*n)
E9 is proportional figures of the tiling that has E8 as a vertex-figure: for each vertex, 120 edges, 2240 triangles, 15120 tetrahedra, &c.
All figures are in decimal. They don't look familiar to me.
wendy wrote:They have real names as well. In the notation meant to replace Coxeter's 2_21 etc,
E5 = 1_21 becomes /3B or Eb1
E6 = 2_21 becomes /4B or Fb1
E7 = 3_21 becomes /5B or Gb1
E8 = 4_21 becomes /6B or Hb1.
[...]
wendy wrote:One of the problems with the kepler series of names, is that they suppose base 10, and that a face-count would be unique. Much of the calculations were done in base 120, and one still sees from me the /2F = {3,3,5}, is variously the fifhundchoron (ie 5.00 ch, twe) or sixhundchoron (600ch dec). Likewise, one uses twelftychoron, since 'twelfty' is a word used to describe long count.
One should not be overly facny to assume that Bower's names are not related to the dynkin graph. It's more opaque, but there is a one to one correspondence between the stem and the location on the graph. A name like Cc7 is somewhat less opaque, and since the name is a path to the construction, Cc7 suffices for this purpose. Fb1 is as short as 'jak', and is much more directly connected to the CD diagram.
It should be noted that even the CD diagram in the 'ringed node' form is an abbreviation or allowance for script. The 'full' form is to have the vertex node separate. It is this full form that provides the necessary insight into lace prisms usw..
One trouble with face-counts, is that for the tegum- and prism- products, the faces are a sum of lesser faces, but for the tegum- product, it's the product of faces. So in 24d, one has face-counts of 25, 48, and 985.1016 (dec 16 777 216). The individual polytopes are the 25, 24 and 24th powers of a point, line and line.
Still, the gosset-elte series extends from 2 to 9 dimensions, and one even writes 0_21, X_21 and Y_21 for these in 4, 3 and 2 dimensions. X and Y are perfectly regular forms, even something like X_33 and Y_33 has meaning.
Even 'dodecahedron' is not a good name for the {5,3}, since the decagonal prism is a dodecahedron, with 30 edges and 20 vertices to boot. There was a more ancient scheme here to use the elements (zb 'comsochorid'), which would extend to cosmoterid {5,3,3}, makes some more sense. At the moment, the only polytopes i have given distinct names to are o3x4x3o = Dq6 'octagonny', and the d73 o3x4/3x3o 'octagrammy' , both in 4d. These tile space in a discrete way, related to the compound symmetry 'bb' (it's like qq, but the q's are the same size).
In any case, something like 'tegmayottid' (tegum-8-solid) is a more suffice name for the regular 256-zetton, to match prismayottid (prism-8d-solid), and pyroyottid (pyramid 8-solid), works by me. The gosset-series has three tails, and one would have to work on three separate leaders. The dodecahedral series has two, ditto.
Still, it is important that there is a coordinate system to place things on, and a quick way to find these things, then to tedious grecko-lædinate names of little benefit either to users or the polytopes.
Well, wendy, the main problem here is, what is a name at all? We would even call some figure "Anton", that should be valid too. But it obviously does not tell much about it. Keplers face-count names at least tell something about face counts. The Dynkin symbol on the other hand is more kind a construction device, than a name - at least in the classical sense of it. None the less, it clearly specifies the construction and therefore exactly that very single polytope. (In fact, it is rather the other way round, that several Dynkin symbols can construct the same polytope!)
Eh, this is nopt fair! You well know that non congruent shapes are counted separately. Like the icosi(=20)-dodeca(=12)-hedron. So the regular dodecahedron has 12 congruent faces, while the decagonal prism would have 2 decagons and 10 squares. But, you are right, you would get in this greekish way of naming: dodeca(=12)-hedron respectively do(=2)-deca(=10)-hedron, which without those hyphens would become exactly the same.
Still, you most often would need names as inline strings. So we need to linearize those diagrams. This has been done both by you and by me. So that is no problem any longer. Even so in general we not only have different Dynkin graphs for the same polytope, but also different linearizations for the same Dynkin diagram! (E.g. starting at different nodes thereof.)
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