## A curious coincidence

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

### A curious coincidence

Since you people are so quiet, I thought I should stir the pot a little.

Here's a challenge:

1) Name a uniform polyhedron with 8 faces, 18 edges, and 12 vertices.

2) Can you name a different polyhedron with 8 faces, 18 edges, and 12 vertices? Interesting coincidence, isn't it?

If you found the above too trivial, how about:

3) How many 4D uniform polychora can you name that contains the above two polyhedra as cells?

quickfur
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### Re: A curious coincidence

quickfur wrote:1) Name a uniform polyhedron with 8 faces, 18 edges, and 12 vertices.

a hexagonal prism?

Hugh
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### Re: A curious coincidence

quickfur wrote:2) Can you name a different polyhedron with 8 faces, 18 edges, and 12 vertices? Interesting coincidence, isn't it?

a truncated tetrahedron?

Hugh
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### Re: A curious coincidence

Excellent! Now can you answer question (3)?
quickfur
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### Re: A curious coincidence

Hexagonal duoprism
Secret
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### Re: A curious coincidence

Well, a 6,6-duoprism has hexagonal prisms for cells, but it doesn't have truncated tetrahedra. Which polychora have both?
quickfur
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### Re: A curious coincidence

I see four of them - there are runcitruncated simplex, runcitruncated 16-cell, runcitruncated 600-cell and a prism over truncated tetrahedron.
Mrrl
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### Re: A curious coincidence

Cool!! You caught a 4th one that I didn't catch: the truncated tetrahedron prism. The others are right, too. Congrats! Молодец!
quickfur
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### Re: A curious coincidence

Well, again a bit quiet lately.
So I thought I'd revisit this tiny thread from 2011, reusing it now, nearly 5 years later, for a completely different figure, a 6D lace prism.

In fact, I stumbled upon ox3oo3xo3oo3ox&#x, i.e. upon "dottascad" (dot || scad). Except that the used Dynkin symbols of either layer are mirror symmetric, this fellow does not look too exciting. But when knowing that the circumradius of dot is sqrt(3)/2 (- again: so what? -), that of scad is 1 (- okay, nice -), and then calculating the height of that lace prism to be 1/2 (- beginning to become intersting -), one can derive the 6D circumradius of the whole lace prism too. That one then evaluates to be 1 as well.

Therefore scad happens to be placed equatorially. Moreover we could mirror the lace prism at scad, i.e. derive the tower oxo3ooo3xox3ooo3oxo&#xt (= dot || scad || dot). That one then will be orbiform by construction (unit sized edges throughout and having an unique circumradius).

But that's still not all. I just had said that the height of the lace prism was 1/2. Therefore the 2 bases of that tower (the 2 "dot"s in parrallel, mirrored arrangement) instead could be connected directly as a mere prism (oo3oo3xx3oo3oo&#x = dotip). Therefore that just described tower also ought be some augmentation of this mere prism!

As scad features 30 vertices, and that number occurs in the incidence matrix of dotip also as the count of the oct prisms (oo3xx3oo&#x, ope), I'd suppose those vertices happen to be aligned most probably atop the centers of these opes - that is, atop the centers of some of the ridges!

Nice orbiform tower find, ain't it?
S.o. ever before considered that specific tower?
Does s.o. recognize that fellow, so that all these properties won't be so surprizing, after all?
E.g. can it be related to some uniform polypeton perhaps? (- Other than dotip, for sure.)

--- rk
Klitzing
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### Re: A curious coincidence

In three dimensions, one gets x5x2x and x5o3o have 12 faces, 30 edges and 20 verticies.

The runcitruncate x3x3o3x has as chora, both x3x3o tT and x3x2x hexagon-prism.

One of the reasons i opted out of using face-count as a name scheme, is when you get the likes of 27 in six dimensions (as eg

x3o2x3o2x3o tri-triangular prism, and xoo3oxo2oxo3oox2oox3xoo&#xz (student91's zero-height lacecompound rune). both involved in the same symmetry and having 27 vertices. The second feature resembles the three golden rectangles in an icosahedron xfo2oxf2fox&#xz, but the triangles are inverted, in the next product. It is in fact, /4B or 2_21.

The number of 216 and 240 gets involved in a number of interesting ways in eight dimensions too. The difference here is 24, which corresponds to the right tetra-hexagonal tegum, so we get 4_21 = /6B - hexagon^4 and the rather bisarre o3o3o4%hs having this 216 vertices, the latter has hexagonal symmetry. %h designates a complex node (%), with six verticies (h) on each edge.
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wendy
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### Re: A curious coincidence

Wow, xoo3oxo oxo3oox oox3xoo&#xz, i.e. the tegum sum of 3 mutually gyro-orthogonal triddips, looks to be the most interesting representation of jak!

Just for fun - haven't worked that out so far - we could proceed that idea of yours, and build instead of a ring of 3 trigonal-antiprismatically attached triddips - as a next step - a corresponding ring of 4:  xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx.
Hmm, would that figure then be an otherwise already well-known one, too?
Obviously that one ought to be 8D, and as each layer is a triddip, i.e. has 9 vertices each, that figure then has 36 vertices...

--- rk

Edit: could that one then possibly be rene = o3x3o3o3o3o3o3o ? - At least that one is 8D and it has 36 vertices ...
Klitzing
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### Re: A curious coincidence

The real fun comes from this.

o3o x3o2x3o2x3o + o3x × xoo3oxo2oxo3oox2oox3xoo&#xz + x3o * xoo3oox2oxo3xoo2oox3oxo&#xz + o3o. o3x2o3x2o3x.

This has 216 verticies, and comprises of eight tri-triangular prisms, aranged abstractly as a cube (ie 1,3,3,1). The middle two sets of three are the three bi-triangular prisms times the lead triangle.

You add the tetra-hexagonal tegum xooo3xooo2oxoo3oxoo2ooxo3ooxo2ooox3oooox&#(qu)z to this lot, and you get the 4_21.

That 1,6,15,20,15,6,1 on a fold around , gives 12,30,20, the icosahedral numbers, is not simply there as a numerical co-incidence. The hexateron has some truly interesting features. 20 = 1+9+9+1 = 1²+3²+3²+1² means that o3o3x3o3o contains cube cross-sections 6 at a vertex, over 20 vertices, or 15 in total.

In fact, 4_21 has a presentation of a lace-compound of E_n A_(8-n) for n = 2 \to 7. So there's a six figures of symmetry o3o3o 2 o3o3o. as E4 A4.

Your lace-tower is not rene. Rene has a circumdiameter2 of 28/9, where the lace tower makes only 24/9. I know you can write the 8-simplex in terms of a tetra-triangular pyramid, the diameter2 of a simplex is 16/9, that of a triangle is 12/9. This makes the altitude triangle as 4/9. This means that if in x0,x1 space you put a triangle edge 1/2, and put a edge-1 triangle at x2,x3 space, x4,x5 space, x6,x7 space, the centres of these centred on the initial triangle, you get an 8d simplex. Kind of cute, really.
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wendy
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### Re: A curious coincidence

Klitzing wrote:... Just for fun - haven't worked that out so far - we could proceed that idea of yours, and build instead of a ring of 3 trigonal-antiprismatically attached triddips - as a next step - a corresponding ring of 4:  xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx.
Hmm, would that figure then be an otherwise already well-known one, too?
Obviously that one ought to be 8D, and as each layer is a triddip, i.e. has 9 vertices each, that figure then has 36 vertices...

... could that one then possibly be rene = o3x3o3o3o3o3o3o ? - At least that one is 8D and it has 36 vertices ...

Wendy, you're right, the circumradius argument, which for rene is immediate, and for that new fellow can be obtained from the lace sum construction advice as that of either component (i.e. layer of notation) - all being just triddips - is evident: those cannot be the same thingy.

Thus I rethought that fellow a bit deeper.

Within  xoo3oxo oxo3oox oox3xoo&#zx (jak)  one uses  ooo3ooo ooo3ooo ooo3ooo&#x  for subelements (in fact there are 216 of those).
Does then  xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx  ought use some  ooo.3ooo. ooo.3ooo. ooo.3ooo. ooo.3ooo.&#x  (etc.) too?
Or rather some  oooo3oooo oooo3oooo oooo3oooo oooo3oooo&#xr ?

The first assumption then would imply that we will have  oo..3oo.. oo..3oo.. oo..3oo.. oo..3oo..&#x  (etc.) in all(!) 6 pairings. But then we could decide whether it's true by looking at the respective distancies between according vertices... (the "x" part in the trailing "&#x" - to be done).

The second, i.e. alternative hypothesis would rely on asking these "squares" to be flat... (the "r" (lace ring) part of the trailing "&#xr" - to be checked).

Those, after all, make the existance of that new fellow kind a bit more doubtable...

On the other hand, the mere found jak structure looks not being restricted to links being all "3". We also could consider  xoo-P-oxo-2-oxo-Q-oox-2-oox-R-xoo&#zx  or, more specially,  xoo-P-oxo-2-oxo-P-oox-2-oox-P-xoo&#zx  for any P>=3. Those (with all "P"s) even ought be at least scaliform then!

Provided that "x" part in the trailing "&#zx" remains applicable - which might impose some additional restriction on P.

Any guesses/ideas to the addressed points so far?

--- rk
Klitzing
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### Re: A curious coincidence

The terminology of lacing here follows that of a pyramid and pyramid products, since they ultimately rest on the same background.

A square pyramid, is oo4ox&#x. If we divide by the symmetry, we ge a------b That is 'apex to base', separated by 'altitude'. That is, the line ab is not parallel to the space of a or b. It adds new dimensions. In the pyramid, the line ab appears as the four lines coming down from apex to corner. In lace-towers and lace-cities, we get the altitude figure, as a kind of structure of lines, etc, which exist to add extra dimensions of one or two spaces.

For altitude, one counts the number of bases (an apex is a point as base), as individual points (since they are mutually orthogonal to each other and the altitude). So a simple pyramid of this kind has two bases, and the altitude features in the volume calculation. (the volume is 'a! * altitude * \product (b! * base) / (a + \sum b)!), when prismatic units are used.

The thing represented by #xz etc, is the 'lacing'. It's always connected. In otherwords, in something like the 'jak', there are not just 216 triangles, but each triangle falls in the overall symmetry of 3,2,3,2,3. The fact that the lacing has no overall mirror-symmetry, means that there is only one per cell.

If you are looking at four points, as in your second example, the lacing has four vertices. This can be either a tetrahedron (which makes six edges, by #x), or a square (#xr). But in the 'altitude' these will be arranged exactly as described.

You would have to consider all six lacing-edges, bot only for one case. In &#x, we arrange the points at a simplex, which is how it comes from the vertex-figure of a wythoff-figure. Of course, this is not &#xt (lace tower), or &#xr (lace ring). For example, the xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zr would have 8+2=10 dimensions too, and the same circumdiam as the jak. On the other hand, something like xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx would give 11 dimensions, since there are four points in the altitude, and eight dimensions across the bases, makes 8+4-1=11.

The lacing here has no real effect on the overall structure. We have basically only defined a cycle-of-four in the name, and it can be as easily a flat square as it is a Petrie zigzag.

Thus zzzz&#x is a tetrahedron, and zzzz&#xt is a lace tower o----o----o----o, and zzzz&#xr is a square. Here z means as i use it, translates into your notation as "*a", ie a self-referent node, which removes even the bilateral symmetry that o or x implies. one node is one mirror, zero nodes is zero mirrors!

Since nothing to the left of &x# tells us exactly how to place each of the parts of the compound, we rely entirely on the brief code of what's to the right to do the deed.
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wendy
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### Re: A curious coincidence

Klitzing wrote:...
On the other hand, the mere found jak structure looks not being restricted to links being all "3". We also could consider  xoo-P-oxo-2-oxo-Q-oox-2-oox-R-xoo&#zx  or, more specially,  xoo-P-oxo-2-oxo-P-oox-2-oox-P-xoo&#zx  for any P>=3. Those (with all "P"s) even ought be at least scaliform then!

Provided that "x" part in the trailing "&#zx" remains applicable - which might impose some additional restriction on P.
...

Well, that  xoo-P-oxo-2-oxo-P-oox-2-oox-P-xoo&#zx  surely exists in general, but we then would have not only 3D subelements  ox-P-oo&#x  (i.e. P-gonal pyramids), which require for a strictly positive height that P < 6, but for 4D subelements of the form  xo-2-ox-P-oo&#x  (i.e. P-gonal scalenes or alternatively described as rosettes of P tetrahedra around a common edge, which is slided somewhere into the orthogonal direction). Already this restriction would imply a value of P < 5.104299, else the height of those 4D subelements would become degenerate.

Moreover we will have a restriction from the next dimensional elements too: we also would have 5D subelements of the form  xo-P-oo-2-ox-P-oo&#x  (i.e. P-gon || perp P-gon). For P=3 we have: 3-gon || perp 3-gon = hix. 3-gon || perp 4-gon (as used in the more general context with arbitrary P, Q, R) also is known already to be squete (= a wedge of tac = x3o3o3o4o). But the next one required here, i.e. 4-gon || perp 4-gon becomes degenerate already, featuring a height of zero and thus becomes  xo4oo ox4oo&#zx , which happens to be nothing but hex = x3o3o4o.

That is, we are thus down to P < 4. And therefore, when looking at convex solutions only, we cannot vary that value of P=3 anymore!

For the more general setup we thus just could consider  xoo3oxo oxo3oox oox4xoo&#zx  at most ... (which then is obviously no longer scaliform).

--- rk
Klitzing
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### Re: A curious coincidence

Whence the reference to j5j2j5j. This is a 'gap', in terms of the bipentagonal symmetry. Evidently j7j2j7j exists, but is not scalaform. I played around with j5j2j5j2j5j to see if it made sense, but it did not, as far as i know.

On &#zx. The notation by a single size-letter, like x or o, is a single mirror. It's not the digonal group x2o, but a single line group 'x'. This leaves no symbol for a zero-dimension group, ie a point-altitude for example, since both o and x are lines. In this case, the loop-operator z in the first node, creates a loop of zero nodes, and so 'z' is actually a mirrorless group, while o has a direction. In lace towers evidently #z sets the mirror-group to z, but it's basically a mirrorless group that z would occupy, like #t is a mirrorless group a tower occupies.

So writing something like xfo2oxf2fox&#zx actually demands that the figure has unit lacing and point-altitude. It either exists or it doesn't. For example xxo2xox2oxx&#x would exist in 5D as a separate figure. It would evidently be scaliform, among other things, consisting of loops of tetrahedral prisms.

Code: Select all

a     x   x  o          1--b
/    \
b     o   x  x        a      2
\    /
c     x   o  x          3--c

1  4  vertex       1ab  2 squares       123a  square pyra
2  4  vertice      2bc  2 squares       1abc  (nothing)
3  4  vertices     3ca  2 sqiares       1a2c  tetrahedron
1c2  4 triangles     1ab2  digon ap.
12  8 edges        1c3  4 triangles
13  8 edges        2a1  4 triangles     123ab  tetra prism.
23  8 edges        2a3  4 triangles     12abc  tetra || square
1a  4 edges        3b1  4  do
1b  4 edges        3b2  4  do
2a  4 edges        123  8 triangles     123abc  itself
2b  4 edges
3a  4 edges
3b  4 edges
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### Re: A curious coincidence

Klitzing wrote:For the more general setup we thus just could consider  xoo3oxo oxo3oox oox4xoo&#zx  at most ...

Outch! Not even that!

This is because of the subelement  ... oxo oxo ... oox4xoo&#x , which again contains that degenerate  ox ox oo4xo&#x = ox4oo oo4xo&#x , which in turn thus should better be described  ox4oo oo4xo&#zx  (= hex).

--- rk
Klitzing
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### Re: A curious coincidence

wendy wrote:For example xxo2xox2oxx&#x would exist in 5D as a separate figure. It would evidently be scaliform, among other things, consisting of loops of tetrahedral prisms.

Code: Select all

a     x   x  o          1--b
/    \
b     o   x  x        a      2
\    /
c     x   o  x          3--c

1  4  vertex       1ab  2 squares       123a  square pyra
2  4  vertice      2bc  2 squares       1abc  (nothing)
3  4  vertices     3ca  2 sqiares       1a2c  tetrahedron
1c2  4 triangles     1ab2  digon ap.
12  8 edges        1c3  4 triangles
13  8 edges        2a1  4 triangles     123ab  tetra prism.
23  8 edges        2a3  4 triangles     12abc  tetra || square
1a  4 edges        3b1  4  do
1b  4 edges        3b2  4  do
2a  4 edges        123  8 triangles     123abc  itself
2b  4 edges
3a  4 edges
3b  4 edges

Hehe, yes, this is known. In fact this is tedrix, i.e. the tri-diminished rix (where rix = o3x3o3o3o).

You best could understand its relationship to rix, when considering the lace city of the latter:
Code: Select all
o3o3o   x3o3o

x3o3o   o3x3o

The first diminishing chops off the single vertex at the upper left. The other 2 diminishings chop of one vertex each of the lower right octahedron (diametrally opposite ones, in fact). Thus the lace figure reduces just to s.th. like
Code: Select all
T
T=tet
T   4       4=square

Then consider pairs of opposite edges of these tets. Those span 3D-orthogonal squares. And each of those furthermore is 3D-orthogonal to the one marked "4" above. Thus we are back to what your symbol said.

--- rk
Klitzing
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### Re: A curious coincidence

wendy wrote:Whence the reference to j5j2j5j. This is a 'gap', in terms of the bipentagonal symmetry.

Even so I know that your j5j2j5j = gap, I never truely figured out how your "j" does work as an opaerator.
Evidently j7j2j7j exists, but is not scalaform.

--- rk
Klitzing
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### Re: A curious coincidence

wendy wrote:You would have to consider all six lacing-edges, bot only for one case. In &#x, we arrange the points at a simplex, which is how it comes from the vertex-figure of a wythoff-figure. Of course, this is not &#xt (lace tower), or &#xr (lace ring). For example, the xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zr would have 8+2=10 dimensions too, and the same circumdiam as the jak. On the other hand, something like xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx would give 11 dimensions, since there are four points in the altitude, and eight dimensions across the bases, makes 8+4-1=11.

Ehhh, no.
For stacked Dynkin symbols we have the following endings, e.g. cf. here:
• ...(none) = just the compound of the symbol layers. Dimension equates to the number of node positions.
• ...&#x = lace prism (for 2 layers) resp. lace simplex (in general). Dimension equates to the number of node positions + number of layers - 1.
• ...&#xt = lace tower (for >2 layers). Dimension equates to the number of node positions + 1.
• ...&#xr = lace ring (for >3 layers). Dimension equates to the number of node positions + 2.
• ...&#zx = tegum sum, i.e. hull of mere compound of layers with additional information that all existing lacing edges happen to be of size "x". Accordingly here the dimension again equates to the number of node positions. - In fact, it is nothing but a virtual lace simplex (&#.x), where all mutual displacements happen to be zero (..z.). (This is what this ending is describing; at least as student91 was defining it.)
Therefore,  xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx  would, provided it exists indeed, truely be 8D. In fact, what is described here, is nothing but the hull of the mere compound (a.k.a. tegum sum). The additional requirement of this "&#zx" is the (so far) assumption, that all existing lacing edges would be of size "x". This is what has to be checked. - And as I had designed that symbol like a circuit of layers (1234), my additional question here just was, whether lacing edges between layers 1-3 resp. 2-4 would have to be considered as existing, or not.

--- rk
Klitzing
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### Re: A curious coincidence

I suppose that if you don't understand what 'j' does in j5j2j5j does, you're keeping up with me. I don't either.

One pokes a lot of different things into the meanings of these symbols, to see if any stick. Specifically, at the moment j is an irregular operator which is defined at instance. That is, when you see the 'j' symbol, the thing is to dig out the book of irregular verbs out and see how ich, du, sie goes. Its meaning changes with each symmetry.

A similar case where this happens is when i write something like Ci8 or Di16. Of course, the quoted number is written in reverse binary, thus ooox or oooox, around Ci 3,5 and Di 3,3,5, but you see this overflows, ie o3o5ox and o3o3o5ox. So you have to look up the table of irregulars to find Ci8 is the snub icosahedron, and Di16 is the gap. It's a handy way of mopping up the ugly bits at the end.
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### Re: A curious coincidence

I use a few different symbols, too.

R is the rectangular group, or digonal group '2', and thus 'r' is its shortchord. r differs from o, in that an r edge still has two vertices, but no edge, so the vertices sit either side of the mirror, viz "o|o" rather than a mirror through a vertix Ø. Its main use is to allow one to shrink a cube to a pointkin, without loosing the cube, ie r4o3o rather than o4o3o.

$is a vertex node. Replacing a marked node by a$ equates to creating a row of a lace-prism. For example, the figure ye describe as tedrix, can be written as "$3o3o$3o3o$3o3z", The symmetry when the$ nodes are removed is o2o2o, and the $connects to an o, converts the o into an x, q, f, v, h as the branch is 3, 4, 5, 5/2 or 6. The difference here is that$p\$ does not equate to &#c(p) but &#c(2p) [ c for chord, c(x,y) makes the y'th chord of x. default y=2. The function is a real one, so you can evaluate c(2pi, 3).

"i" is used to create a negative value, or the supplement figure. It derives from the attempts to write things like the 'wythoff symbol' = 'decorated schwarz-triangles' in a way that makes calculation. So if 532 is a schwarz-triangle, then then the four co-lunes are 5i3i2, 5i32i, 53i2i, 532. You then add a slash to make marked-unmarked, vis icosahedron = 5/32, dodecahedron = 3/52, etc. The duals are formed regularly by replacing / with \, so the rhombic tricontahedron is 2\35. Student91 and a few others mentioned negative coordinates in the vector matrix, and that's how we write it.

&#m is a 'lace tegum', the dual of 'lace prism' The fugre is the intersection of lace cones. One starts with imagining B as a base of the compount, and b as a point further out from b. Then b-B becomes a pyramid-head, which extends infinitely out the B direction. If this does not occupy the dimensions of space, one multiplies it by the 'ortho-space'. A lace tegum is then the intersection of lace cones.

An example of a lace cone, is to imagine the tetrahedron, as the lace product of a line segment 1,0,1 to -1,0,1 over the one 0,1,-1 to 0,-1,-1. Here the altitude is in the z-axis, we see then that the first produces an ever-expanding sector from 0,0,2 outwards in the x,z plane. Since this does not occupy all-space, it is multiplied by the orthospace (or orthogonal space), gives x not just as the point 0,0,2, but the whole line 0,y,2. This turns the sector into a what you get with an open book. The second lace-cone, gives one spreading from x,0,-2, through the line at 0,y,-1, is an open book, upside down and at right angles. You see these intersect to make a tetrahedron.

And this basically fits in with the notion of the dual of a figure is by converting x to m. (and the choice of using 'lace prism' over 'lace simplex'!)

The actual debate was rather thorny, but it basically comes down to that an 'anti-prism' xoPox&#x is dual to an 'anti-tegum' moPom&#m.

I use 'z' in the manner that i tried to figure out what student91 meant. He particularly seemed concerned when the lace tower has a zero height, as some of the lace-compounds do. Of course &# is a branch creating a new axis of symmetry, a new height, and z gets rid of it.

There is room for something that you are meaning by &#.z is something that is best described by using a ÷ sign to 'divide space against this symmetry. It can be used in any place then. So the kiddies making Zome-tool projections of the twelftychoro, are making "x5o3o3o÷" ie, a projection into a lesser space 5,3, by dividing it down that axis. So your &#.z is the same as &x.÷ No formal symbol is created for it. So we can remove a dimension with ÷, it basically allows you to correctly describe a lace city in line by removing axies, but it is not always the case that the lace city has a notation that is a subset of the whole, eg 5,3,3 does not conveniently contain 5,2,5 as a symmetry, but the symmetry can be described as 52÷5÷ or the like.
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the dream we dream together is reality.

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### Re: A curious coincidence

Things like Ci8 notation you already described elsewhere. As I remember from that:
• C is the dimension prefix. A = 1D, B = 2D, C = 3D, D = 4D.
• i is the group familly. i = [(3,)* 5]. I.e. Bi = [5], Ci = [3,5], Di = [3,3,5].
• The number is that reverse binary code. I.e. 0 = ...oo, 1 = ...oox, 2 = ...ooxo, ..., 7 = ...ooxxx, etc.
Thus Ci1 = o3o5x = doe, Ci4 = x3o5o = ike, Ci7 = x3x5x = grid. But "Ci8" then happens to be some (undefined?) overflow.

So you here are intending to use such overflow numbers as an individual exceptional relative, if existing.
Thus Ci8 probably will be s3s5s = snid, and Di16 you already mentioned to be j5j2j5j = gap.
Then Df16 might be considered s3s4o3o = sadi (if "f" would represent the icoic symmetry group - don't remember the right character here)?

Okay; but my question just was:
what would represent your  "j7j2j7j" ?

--- rk
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### Re: A curious coincidence

wendy wrote:c for chord, c(x,y) makes the y'th chord of x. default y=2. The function is a real one, so you can evaluate c(2pi, 3).

That function is exactly, what I represented here as
Code: Select all
x(n/d, m) = sin(π md/n)/sin(π d/n)
and thence
Code: Select all
x(n/d) = x(n/d, 2) = sin(π 2d/n)/sin(π d/n) = 2cos(π d/n)

--- rk
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### Re: A curious coincidence

wendy wrote:I use 'z' in the manner that i tried to figure out what student91 meant. He particularly seemed concerned when the lace tower has a zero height, as some of the lace-compounds do. Of course &# is a branch creating a new axis of symmetry, a new height, and z gets rid of it.

There is room for something that you are meaning by &#.z is something that is best described by using a ÷ sign to 'divide space against this symmetry. It can be used in any place then. So the kiddies making Zome-tool projections of the twelftychoro, are making "x5o3o3o÷" ie, a projection into a lesser space 5,3, by dividing it down that axis. So your &#.z is the same as &x.÷ No formal symbol is created for it. So we can remove a dimension with ÷, it basically allows you to correctly describe a lace city in line by removing axies, but it is not always the case that the lace city has a notation that is a subset of the whole, eg 5,3,3 does not conveniently contain 5,2,5 as a symmetry, but the symmetry can be described as 52÷5÷ or the like.

As you can see here, there is quite a difference between &#zx and &#xz ! (Position of z matters!)
• &#zx = tegum sum of layers (a.k.a. hull of compound of layers) where (additionally) the existing lacing edges happen to be of size "x".
• &#xz = advice to first build &#x and thereafter shrink the displacements to zero. (Thereby the lacing edges, formerly being of size "x", clearly would shrink in some fashion.)
E.g. "xx5oo&#xz" then is nothing but your "r x5o".
While "qo oq&#zx" is just a unit square (x4o) which is spanned by its diagonals.

--- rk
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### Re: A curious coincidence

http://bendwavy.org/klitzing/explain/dynkin.htm

In 2014 Mrs. Krieger extended her zoo of lace prisms, towers, and simplexes a bit. She introduced a further qualifier "z" (zero). If this qualifier precedes the lacing edge lengths ("...&#zx.."), then the respective segmental heights happen to become zero. (In that this qualifier adds just some optional further information to the reader.) But there is a bit more to that: For lace prisms the bases usually are part of the figure. And for lace towers the extremal layers similarily. But not the intermediate ones. Those happen to be just pseudo facets, sectioning the tower into segments. But for degenerate lace towers there might occur situations, where this splitting of the vertex set into subsets (those co-hyperrealmic "layers") would result in pseudo facets solely. This then is a case, where this preceding "z" qualifier surely would be required! – A simple example here would be the ike, when being represented in briquet symmetry: fxo ofx xof&#zx. (None of those golden rectangles belongs to the elements of ike.)

She then also allowed for a suffixing "z" too. This one would build up the structure as devised without that additional "z" first, and thereafter scales all lacing edges down, such that the segmental heights all become zero. – This probably would be rather seldomly used explicitly. But implicitly it provides a different pictoral representation for the segmentochora, in fact the "telescope view" from infinity. (The pictures provided within this website however usually provide views from nearby, thus relatively scaling the two bases additionally in a perspectivic way.)

A further concept closely related to the former zoo is that of lace cities. Here the name derives from the picturesque description of a city: being a set of towers. So re-consider the just described lace towers. If each of those layers on its own can be described as a tower, if further all those towers of one dimension less are describable within the same symmetry group, then there is an arrangement of the higher tower in the manner of a street, while the lower towers are given as sets of layers at the specific grounds. This yields a 2 dimensional graphical arrangement of dynkin symbols, each of 2 dimensions less. Take for example the octahedron (oct). It can be given as the lace tower (in fact only a lace prism) xo3ox&#x, i.e. x3o || o3x. Now x3o = x o&#x and o3x = o x&#x. Thus we have for lace city

z has only one meaning, the construction-node that joins to the first node. That's it. The flat lace-tower is a programming feature.

The whole notation that i use, is specifically designed to be 'fairly easy' for a computer to parse, and someone like quickfur to turn a polytope. But even though it's all written before lace-towers etc, the thing is reasonably robust.

So let's parse what's going on.

When the symbol is linearised, the 'penciled in' line that describes the symmetry is the 'trace'. So o---o---o-5-o is a picture which has no direction, but if i write it as o3o3o5o, it's a trace. Normally we follow the regular line, if one is there, to minimalise the trace. The trace-count is 1--2--3-F-4 (which is why i have alternate letters for the structural branches!). The main reason the trace reset exists, is that z points to the first node after a reset, and zz the second. So if you write, eg o3o3o3z&o3o3o3z, the second z points to trace-node 5, and would be *e in Klitzing-style, while the first is *a. If you write a3o3o3z2o3o3o3z, both z's refer to the same node a, the group consists of a triangle-loop, with a tail of o3o3o3o.

& resets the trace-count for the z and zz node. 2 (or R), does not.

# creates a mirrorless mirror node. That means, that if you put an x after a 2, it creates a prism in both cases, so x3o2x and x3o&x are trips or triangle-prisms. But if we want to kill the mirror, we need to put something like # (now i have your attention, gosub hashnode...) So putting x3o2# is actually valid, but it's defined as separate to the base-figure, so we use x3o&#.

A z, creates a loop. It's different in function to the * marker, since the * identifies the current trace-node with a node marked 'a', but this structure does not have a reset-trace scheme, so *a when it comes to be the second part of a product, could become *g. So, eg the z points to a in both a3o3o2o3o3o3z and o3o3o&a3o3o3z , you see here we used the & to reset the trace count, so we go 1,2,3;1,2,3,4=1. This would be eg o3o3o o3o3o3*a vs o3o3o o3o3o3*c resp.

Putting z at the beginning of an trace then creates one of two effects. By itself, it is self-referent, so z = *a (by itself), which loops and removes the z node, and hence the whole symmetry. It's cunning really, but the CD diagram has no way of removing symmetry, or describing a point, since a single o by itself, is a line, not a point.

So putting z after &# makes &#z, which does in order

& new symmetry, with reset trace counters (ie no references back to the previous symmetries)
# create an axis with no mirror
z subsume this node into the first node of the current trace (ie it copies node 1)

The effect is to create an axis, and then to immediately destroy it. But it allows lacings that are in the same node to operate lace operations.

The order of precidence is that 'structural devices' build the kaleidoscope, are always interpreted before 'decorative devices' which create the thing the kaleidoscope shows.

You see, I keep a close eye on all this stuff, because the key thing is that one needs to program it with minimum effort.

The j node is a decorative node, which tells you to look for irregular(symmetry), eg j5j2j5j = j(5,2,5). This is a simple lookup table that returns the irregular figure as requested. But since j5j2j5j has a symmetry of 5,2,5, we still allow the base structure to shine through.

The idea then with the mirror-less mirror, is that you could create a compound before the &#, which extends only as far as the previous &, and then 'lace the bits together, in the order given'. A lace-tower, is like those chinese-lantens or the flat-pack christmas decorations that create a solid device when you pull it out. Something like oxoo5ooxo is simply four figures written in the same symmetry. When you add &#, it adds the third dimension. When you put 'x', it connects the vertices over every symmetry, with a length x (decorative). When you put 't', (structural), it means pick the mess up from the first row and let it dangle. So the order of precidence is tht # and t are structural, and x is decorative, so the structural takes precidence over the decorative.

The role of z then is to zap the current node to the first trace-count. It does not reset the trace-count itself. It's a structural node, which means that 333z is different to 333 by symmetry (the first is the triangle-tilings, the second is the fivechoron, so z affects how the kaleidoscope is made, and is thus structural). When you impose decorations on it, it's like moving one or more points in say 333, and then creating a depth perpendicular to it &#, and then pulling the stuff around so the lacing is tight.

Something like qoo3oqo3oooAooq&#xz uses the group 3,3,A of order 192, and creates inside it, three different 16chora, of edge q=sqrt 2. The vertical into 5d is created by &# and destroyed by z, and the x node is then some kind of lacing to these figures. The z for t just highlights the fact it is actually flat relative to the &# axis.
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wendy
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### Re: A curious coincidence

Dear Wendy,

the point here is a bit more intricate. One thing is what you have coded when parsing these extended Dynkin symbols. This well has be done with careful purpose, everything self contained; okay.

But then (probably around that year 2014) student91 came up, and he suggested / asked for a symbolification of these lace simplices and/or lace towers, which have a zero displacement, and where the components, as far as lacing edges exist, such would be restricted to some specific length (usually x). This then led - after some discussions right in this forum (I'd have to digg out references), where you also where involved, - to that very notion of "&#zx" in right that sense, which I just had described above.

Actually, the whole stuff of those EKF (expanded kaleido facetings) researches is heavily based on that very notion. - Whether this is conform to your parsing routines, or not. - And, at the very end, a couple of hundreds of incmats files meanwhile already are using that very notation.

Moreover, in the run of those discussions then it was you as well, who came up additionally with the notion of what I just described by "&#xz" (reverse order!). And it was from a post of yours, that I deduced that further notation and copied it into my cited pageref. - I'd suppose that for this notation not too many incmats files would exist. (So I probably rather easily could drop the related reference in future updates.)

Therefore it turns out to be just one thing, to insist on the coding of your program, and quite an other to face the meanwhile emerged further usages. - How could we solve that? Esp. how would you like to add the since arosen notion (i.e. that, what I described as the meanwhile heavily used meaning of "&#zx") into your coding? And what would then be the "cost" of changing all these incmat files as well as all the worldwide spread usages of that notation?

Clearly, usages are not only heavily within this forum, they are also in the polyhedron list archive, used within facebook posts and discussions, in lots of private mails around the world, etc.... And, last but not least that notion as well as its notation would be stuck already in several brains!

--- rk
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### Re: A curious coincidence

Just digged out the corresponding first email of student91 on that "&zx"-stuff.
The first idea occured here.
Subsequently it became an own thread.

Student91 originally was suggesting "&U" for "(lace) union".
Wendy in this mail of this thread was suggesting rather to use "z" instead.

--- rk
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### Re: A curious coincidence

The notation is robust enough to support what you are reading into it. We just have to add a context point here.

Thanks.
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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### Re: A curious coincidence

that is?
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