E8

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

E8

Postby papernuke » Sat Feb 02, 2008 4:26 am

What is E8? not mathematically.
is it a geometric shape or what? and what is it used for.
how did scientists first discover it?

Also, whats a Lie Group?
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Postby wendy » Sat Feb 02, 2008 9:12 am

E8 is a Lie Group, a particular set of mathematical operations, that one largely misses even in University. None the same, there are those that dabble in it. Use Wiki or Google to seek Lie Groups. The symmetry group is represented in a great number of dimensions, such as 248D and 8D, and some _much_ larger.

Thorald Gosset discovered a series of polytopes, beginnining in E3 with the triangle-prism, and each subsequent figure having the previous as a vertex figure. This leads through E4, the rectiified pentachoron, E5 as the 'diminished penteract', E6, E7 and E8 as polytopes of 27, 56 and 200 (dec 240) vertices. The corresponding E9 is an infinite tiling.

It is associated with the 8d "Gosset Polytope", having 200 v (dec 240), and for faces, 1 2400 (dec 17280) simplexes, and 1800 (dec 2160) cross polytopes. The symmetry order here is implemented by reflections, has an order of 3.4372.0000 (dec 696729600).

E8 is one of the 26 sparodic Lie groups. This means that it's not a kind of "regular substitution for n=x", rather like the uniform polyhedra = regulars + prisms + archimedean This is a proposed symmetry that is being used as a model for the fundemental physical particles, in one of the grand unification theories.

It is getting into the realm of black mathematics.
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Re: E8

Postby papernuke » Fri Feb 08, 2008 3:43 am

But then how many faces verticies and edges does E8 have?
Whats black mathematics?
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Re: E8

Postby Keiji » Fri Feb 08, 2008 3:54 am

I'd like to know the answer to that [edit: what black mathematics is] too. Wikipedia and Google give nothing.
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Re: E8

Postby wendy » Sat Feb 09, 2008 8:09 am

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, 17280+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.
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Re: E8

Postby pat » Tue Feb 12, 2008 7:11 am

Wendy, refresh me on what number system you are using? Because 200 = 240dec looks like you're using base 2 sqrt(30), which seems slightly unnatural. I'm assuming your numbers aren't exactly in a classical place-value system.
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Re: E8

Postby wendy » Tue Feb 12, 2008 9:18 am

The classical number systems are alternating, like 20 or 60. I use twelfty as alternating columns of 12 and 10, largely to simplify calculations.
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Re: E8

Postby papernuke » Sun Feb 17, 2008 9:24 pm

Thorald Gosset discovered a series of polytopes, beginnining in E3 with the triangle-prism, and each subsequent figure having the previous as a vertex figure
-wendy
How did he discover it? Through mathematical equations and functions etc.
or did he actually create a model of the figure?
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Re: E8

Postby wendy » Mon Feb 18, 2008 7:22 am

It was discovered in the process of finding all semi-regular figures: that is, all polytopes whose faces are regular. Apart from this set, there are a few others in 4D (ie the snub 24choron, the rectified 335), and two semiregular tilings (in 3dt, 2 octahedra + 2 tetrahedra, in eight dimensions, the tiling of 2 cross-polytopes + 1 simplex at a margin).
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Re: E8

Postby papernuke » Mon Feb 18, 2008 10:38 pm

Ok, but from the same earlier post,
what is black mathematics?
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Re: E8

Postby wendy » Tue Feb 19, 2008 8:27 am

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.
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Re: E8

Postby zero » Wed Feb 20, 2008 4:28 am

Oh, but there are all kinds of logic. Careful with those recipes!
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Re: E8

Postby papernuke » Wed Feb 20, 2008 3:22 pm

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.
--wendy
Then what would be "white mathematics"? Would it be math you can act out?
If it was simply math you can act out, then wouldn't some of algebra be "white" mathematics? For example, motion. D=VT. Couldn't you act that out by simply going at the desired velocity within the time limit, and if its a vector, just go in the designated direction.

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.
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Who is Norman Johnson?
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Re: E8

Postby wendy » Thu Feb 21, 2008 11:03 am

I can read algebra, if it's not too hard. I can even do some, if the answer is already found by some other means.

I understand what the runes mean, eg D=VT (or 11 = 10+1, in dimensional analysis-number), but i find it difficult to do extended algebra, and impossible to do all but the simplest calculus (save looking up the CRC tables). On the other hand, if i am familiar with the numbers, it all sort of makes sense.

Among the white arts are number theory, cyclotomic numbers, quantum geometry, division by zero, matrix-dot, and a few other things. These are the magic i can master in some way.

Of Norman Johnson, he is of the "Johnson Solids", a former student of HSM Coxeter, see http://en.wikipedia.org/wiki/Norman_Joh ... hematician)

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Re: E8

Postby quickfur » Fri Nov 14, 2008 12:58 am

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.

So if calculus is black magic, erm, mathematics, then what is set theory? (Esp. Cantorian set theory with its non-constructible sets, such as the ones featured in the Banach-Tarski "Paradox".) :lol:

I can do algebra in my head, but anything to do with actual numbers completely eludes me. I do most of geometry (esp. 4D geometry) via visualization. I couldn't compute a dot product if my life depended on it, but I dabble with rearranging algebraic expressions all the time. Symbols and Greek letters are my friends, and actual numbers are the bane of my existence. Perhaps that's why I'm a programmer: I write the symbols, and the computer does the real work.

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.

The way I usually go about it is to first derive things by intuition, and then prove them true or false after the fact. :) Sometimes, I prove things by intuition as well, the actual concrete proof being left as an exercise for the reader. :P
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Re: E8

Postby wendy » Fri Nov 14, 2008 9:20 am

Much of mathematics is more cultural than a representation of Mathematics in nature.

Cantor's theory of infinity assume that 'names' are always countable, and then shows that there must be larger sets. Then there is the proposition of how C ought connect to Aleph_0, Aleph_1, &c. It leads to too many paradoxes for it to be a reliable presentation of infinity.

The Banach-Tarski 'paradox' assumes the 'axiom of choice', which some people have been rather edgy about. In any case, it is based on the assumption that a set like B10 can represent all points on the number line (it relies on fractals, which are geometricly implemented added fractions). The fact that one can run something to "infinitely small" is elsewhere not counted as proof of equality with 'covering all points'.

Much of what I do with mathematics is to overlay 'discrete sets', to see if certian things are permissible. In essence, this is like overlaying the sets B7 and B10 (numbers expressed in base 7 and 10 respectively), to demonstrate something is an integer. For example, one can write in an {5,3,3} the vertices of a {3,3,5}. However, there is never a ring or glome constructable from the lattice {5,3,3,5/2}, that contains both {3,3,5} and {5,3,3}. For this to happen, we would need the number system Z5Z4, but the actual system that occurs is only Z5. One can never construct r5+r2 out of Xr5+Y, because these discrete sets do not overlap.
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Re: E8

Postby Klitzing » Sat Sep 15, 2012 4:34 pm

Here are some references to those numbers by means of complete incidence matrices:

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

http://bendwavy.org/klitzing/incmats/trip.htm

E4: 10; 30, 30, 5+5

http://bendwavy.org/klitzing/incmats/rap.htm

E5: 16; 80, 160, 120; 16+10

http://bendwavy.org/klitzing/incmats/hin.htm

E6: 27; 216, 720, 1080; 648, 72+27

http://bendwavy.org/klitzing/incmats/jak.htm

E7: 56; 756, 4032, 10080; 12096, 6048, 576+126

http://bendwavy.org/klitzing/incmats/naq.htm

E8: 240; 6720. 60480; 241920; 483840, 483840; 207360, 172800+2160

http://bendwavy.org/klitzing/incmats/fy.htm (17280, not 2 zeroes).


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.


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Re: E8

Postby wendy » Mon Sep 17, 2012 10:58 am

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.

The symbol /5B refers to a marked node, then 5 '3' branches, and then a B branch. This notation replaces the Coxeter 3_21 style, but it is more precise because it allows one to write all sorts of uniforms based on it. A 'B' branch is actually a '3' branch (when seen as a verb), the subject is three nodes back (count S, A, B, C from the end to find the subject of the A, B, and C nodes.

The notation Gb1 is a reference to its location in the uniforms. G is the seventh letter, so it's seven dimensional. 'b' is a reference to a gosset-style polytope in the icosahedral (or leading 3's) form, and 1 refers to a sum of marked nodes in the form 1,2,4,8,16,...

The icoashadral form is with leading 3's, ie x3o5o = /1F = Ci1, and x3o3o5o = /2F = Di1. The 'dodecahedral form' is with the '3's trailing, so one can write things at the other end of the pattern, so x5o3o = /F1 = Cd1, and x5o3o3o = /F2 = Dd1. When one reverses A and B nodes, one gets E and G nodes. C nodes are not reversed. So

1_22 = 4B/ or /G4 or Fg1 while 2_21 = 4/B or G/4 = Fg2
1_32 = 5B/ or /G5 or Gg1, while 2_31 = 5/B = G/4 = Gg2
1_42 = 6B/ or /G6 or Hg1, while 2_41 = 6/B = G/6 = Hg2

So, in 6, 7, and 8 dimensions, the gosset figures are Nb1, Ng2, and Ng1.
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Re: E8

Postby Klitzing » Mon Sep 17, 2012 2:33 pm

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.
[...]

Well, Wendy, this all is a mere recoding of the Dynkin symbol. Whether Coxeter gave the Gosset figures as I_JK (where I, J, and K are just the numbers of nodes at either branch of the graph, and the one before the "_" just would be the one the end of which is being ringed) or you would write Nb1 (N being variously E (=5th letter, so representing a 5D thingy), ..., H (=8th letter, so representing a 8D polytope), and "b" represents a bifurcation at the b-th (i.e. 2nd) but last node of the linearized sequence with an off-spring of length "1"). Those are not truely names.

By history the Kepler naming scheme for instance uses facet counts. Such as the icosidodecahedron has 20 (icosa) triangles and 12 (d(u)o-deca) pentagons. This is what hedrondude was extending in his namings of polytopes. Sure, those soon get unwildy. This was the reason for his idea of using acronyms of those only.

"jak" (2_12) for instance stands for icosihepta(=27)-heptacontidi(=72)-peton(=having 5D facets). I suppose that he used here "j" as reference for 27, "k" as one for 72.
BTW, even all other quasiregular members of that symmetry group (i.e. having exactly 1 node ringed), those with 2 ringed nodes, and even some beyond are already calculated in detail, i.e. incidence matrices are provided at http://bendwavy.org/klitzing/dimensions/polypeta.htm. Esp. their element counts for sure.
Note, "mo" (1_22) does not provide a clue for that supposed correspondence for sure, as Jonathan uses facet count additions, if there are congruent facets (just as the rectified tetrahedron clearly is not known as tetra(=4)-tetra(=4)-hedron, but as octa(=8)-hedron). So we would have m = 54 = 27+27 in here.

"naq" (3_12) stands for hecatonicosihexa(=126)-pentacosiheptacontihexa(=576)-exon(=having 6D facets). As he calls 2_13 by "laq" - for pentacontihexa(=56)-pentacosiheptacontihexa(=576)-exon - the acronyms get easily accessible here: l = 56, n = 126, q = 576. Thus, 1_23 clearly would be abreviated as "lin", or would have the fullname pentacontihexa-hecatonicosihexa-exon.
Again, all incidence matrices of any quasiregular member of that symmetry group, and even some few beyond, are provided at http://bendwavy.org/klitzing/dimensions/polyexa.htm.

"fy" (4_12) stands for dischiliahectohexaconta(=2160)-myriaheptachiliadiacosioctaconta(=17280)-zetton(=having 7D facets). 2_14 he calls "bay", for diacositetracont(=240)-myriaheptachiliadiacosioctaconta-zetton. 1_24 accordingly is called "bif" (diacositetracont-dischiliahectohexaconta-zetton). That is, he just uses here b = 240, f = 2160, y = 17280.
Here too all the incidence matrices of all quasiregular members of the are linked at http://bendwavy.org/klitzing/dimensions/polyzetta.htm!

(Suppose that these counts are not even all being found in wikipedia, hehe. :lol: )

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Re: E8

Postby wendy » Tue Sep 18, 2012 7:22 am

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.
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Re: E8

Postby Klitzing » Tue Sep 18, 2012 12:42 pm

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..


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!)

Your names like Cc7 etc. could be considered both, either a true name, more kind of "aerochoron" etc., even so it is not so good pronouncable in general, - or as a coded re-description of a Dynkin symbol.

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.


Admitted, there are quite different, and even huge numbers to be used. So a good counting system would be useful here. Twelfty makes the numbers shorter (and even sometimes better to remember). Bowers usage in replacing specific numbers by some letters, clearly would run out of those rather soon.

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).


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.

BTW, this is why dinogeorge most probably would like to introduce therebetween the small greek word "kai" (=plus), i.e. dodeca versus dokaideca.

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.


Yep, this is true. The more thoughts about names would be come into the orbit, the more different naming shemes would arise. So you finally would need huge dictionaries which do nothing else but translate one name into an other. What we truely need instead is an easy advice to get to the geometric properties of the to be considered object.

Here, you are right we have as a useful device the Dynkin symbol for the Wythoffian polytopes. We even could include the alternated facetings by additionally allow for snub nodes.

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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Re: E8

Postby wendy » Thu Sep 20, 2012 9:44 am

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!)


The names by Norman Johnson and by Jonathan Bowers are thinly descised stott-constructions. And why not? Stott discovered most of the {3,3,5} figures by her method. Face-counts are pretty good when there's a little number of them, but when one gets many more, something different needs to be fancy.

Different ring-constructions suppose different constructions, with different degrees of freedom (except, say o3x3o4o, vs o3m3o4o vs x3o4o3o). In some cases, one has cube vs square prism vs rectanguloid vs triangular antitegum, all lead to the same thing, and s3s3s vs s3s4o vs x3o5o, ditto. But these have varying degrees of freedom, and so while any x3o5o is an s3s4o, it's not true the other way around. Coxeter, in introducing the gosset-construction for the {3,3,5}, varies the x3x4o3o from x3o4o3o to o3x4o3o, before showing that a variety of x3x4o gives s3s4o.

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.


It's not fair when you use multiple bases either. Still, it's pretty bizare to count faces, when the focus and natural order is by a vertex-count. Much of modern mathematics is based on vertex and (equal) edges, so 'dodecahedron' is really a 20-vertex chorid. But no, i don't rely on face-counts, because there are polytopes without face-counts, but none the less come under study (the ones like the laminate x4x3o8/3o comes to mind).

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.)


Seeing that this is the main construction of nearly every polytope, it's not a bad thing. It is pretty useful to have the two alternate forms (icosahedral vs dodecahedral), since sometimes the action happens at one end or the other. This is specifically accomidated for in the system (ie E vs A, and G vs B). It's pretty much the same way as saying 'truncated dodecahedron' (ie Cd3) vs 'bitruncated icosahedron' (ie Ci6). These have a regular reverse, and is no great deal.

The platonic elements do have something useful to say here. What i did with the tegmic, pyric- and prismic, is that these form regular names in all dimensions, akin to coxeter's alpha, beta, and gamma, but the tegmic, prismic, and crind- also generate coherent units. So there is more than just the polytope that is at hand: there is a unit of "tegmoteric foot", 24 of which make the prismiteric foot or tesseractic foot or biquadrate foot. The vertex angle of a simplex is known to lie between 1 and sqrt(n/4) of units of the "tegmic radian". The simplex of side sqrt(2), has a volume of sqrt(n+1) tegmisolid edges.
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