## A curious coincidence

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

### Re: A curious coincidence

Coming back to my idea of looking out for some  xooo3oxoo oxoo3ooxo ooxo3ooox ooox3xooo&#zx .

Wendys spreadsheet for calculation of lace prism heights from given bases and lacing length, resp. conversely, for calculation of lacing length from given bases and height, immediately provides a lacing "edge" size of 2/sqrt(3) = 1.1547... for its "diametral" distances, i.e. the hypothetical lacings between layer 1 and 3 resp. between layer 2 and 4.
As that number is neither 1 nor sqrt(2) the subelemental "tetrahedra"  oooo3oooo oooo3oooo oooo3oooo oooo3oooo&#x  neither are regular nor are flat and degenerate (i.e. becoming mere squares).

That is, the assumption of that "&#zx" ending seems to be erroneous here, after all.
(The mere compound (without that suffix) surely exists, and so does its hull a.k.a. the tegum sum of these layers. That one simply seems to refuse to be an 8D CRF.)

--- rk
Last edited by Klitzing on Sat Aug 06, 2016 5:43 am, edited 1 time in total.
Klitzing
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### Re: A curious coincidence

I really have not been following the CRF material. Every now and then, I download a symbol, and see if i can read it out of context. This is the magic of a robust system.

Of course, the fact that i diasgree with this notation or that, does not mean that i discourage their use. Instead, if it works, and people are happy with it, then I am not standing in the way. The magic of notation is like a telescope, it sees less but further. This is why a good deal of thought has gone in the notions behind the notation.

Really, though, i have not seen the reasons advanced for prefix-z and suffix-z, but i am not denying they might exist. So i am not obstructing it. This effects to people telling other scientists what they can or can't study based on the objector's ideologies and sponsors. It has no place here and i don't plan to encourage it.

What Richard calls 'tegum sum' is a similar path i followed, but i use 'thatch' for this now. It's a kind of general process you use to make a convex hull, but convexity is not an issue. For example, thatching provides the skin for Jonathan Bower's companies of figures, even though the hedroframe (surtopes to 2d) suggest a convex polytope.

What Student91 was talking about making lines thicker, like the atomium.be in Brussel, is 'spheration'. It equates to running a sphere-tipped marker over the surtopes, prehaps changing the radius (usually to make it less as the dimension of the surtope goes up). So the afore mentioned atomium, is the latroframe (vertices and edges) of a cube, with the centre, and rays to the vertices. This latroframe is then spherated with a thick marker at the vertices, and a thinner one for the lines.
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wendy
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### Re: A curious coincidence

Klitzing wrote: 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.)

Ha, found the searched one!

In fact we have 1_2,2 = mo = pt||dot||scad||dot||pt = ooxoo3ooooo3oxoxo3ooooo3ooxoo&#xt.
Therefore that dot||scad||dot after all could be called "pabdimo", i.e. a parabidiminished mo.

Just as the medial 2 layers have height 1/2, the polar ones then have the same height.
These then are just dotpies (dot pyramids).
Thus orbiformity (here with radius 1) is not surprising anymore.

But the relation to dottip still is.
Moreover we see, that mo even hides a compound of 36 dottips!
In fact that one then is: "20  o3o3o3o3o *c3x [36  o3o3x3o3o x]".

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

You would get these directly out of the 'station-diagrams', ie 'standing-point tables' of the lamina in A_n. It's pretty clear.

A_n is the singly marked ring x(n+1 3's)z. eg x3o3o3z, x3o3o3o3o3z, u.s.w. Each of the nodes in the ring corresponds to a kind of hole in the tiling, the cell is given by removing the inntial x, and placing a new x there. So, relative to a given x3o3o3o3...z the o3x3o3o3..z is 'upwards simplex, o3o3x3o3..z is 'rectified upwards simplex, etc. The points in x3o3o3...z are the point itself, and then the runcinate, viz x3o3o..o3x.

So we get now a layer of spheres (or mootly half-edges), all in A_n, and we simply stack these up, layer after layer. If we one station (ie to the tetrahedron), we get then A_{n+1}, with two stations, we get B_{1+n}, with three E_{1+n} and four is possible too. These are primitives of k_01, k_11, k_21, and k_31 resp.

Since the produced lattices are the complete eutactic stars of the groups, (ie from a point, it is all possible points where one might move in unit steps perpendicular to the mirrors: ie a mirror-edge-star), we find all of the polytope vertices in here, given a sphere large enough.

So let's look at the example by Klitzing, which is based on A_5, or a six-node ring.

The nodal distances are 2n, 1.(n-1), 2.(n-2), 3.(n-3) &c, eg 12, 5, 8, 9, 8, 5.

If we now place the second layer on top of the first, the height is then 12-v, where v is the value in the row. But for the first case, the hole is not empty, so it sits at 12. So the thickness of the strata are then 12, 7, 4, 3, 4, 7... for the various lattices. The subsequent layers are then at the squares of these, eg h, 4h, 9h, 16h, &c, because we are using squares of distances.

Gosset-Elke figures, on these axies, are characterised by each layer being moved three to the right of the previous.

So when one plots the vertices of some simplex-figure on the base, the figure falls in the same column as where the central kernel is. For example, the group o3o3o has points (1)3(2)3(3), modulo 4. A figure x3x3o adds 1+2=3, and the central is o3o3x (3). When you put a double-edge length etc, then you simply multiply the node-value by the length, eg a tetrahedron of edge 2, has a central figure of 2,0,0 = 2 gives octahedron. A tetrahedron of edge 3 (which gives the tT too), gives (3,0,0) = 3 inverted figures.

So when you are placing the figures of .3.3.3.3.3. over a symmetry, the central cell is found entirely by the modulo of the nodes, ie o3o3x3o3o is in column 3, and x3o3o3o3x is 1+5, is in column 0. The progression of the three groups, for any shape, is that each subsequent layer must proceed by a modulo-sum of 1, 2, 3.

So we get eg oxo3xoo3[ooo3]oox3oxo&#xt gives the runcinated simplex in the next dimension.

Likewise oxo3ooo3xoo3[ooo3]oox3oxo&#xt gives the truncated orthotope of the next dimension,

The gosset figure can proceed to four layers, but the middle layer is here x3o..3x runcinated simplex, and the upper and lower are runcinates shifted by three.

So it's a simple matter, given the circumradius, to simply add height with radius. In the example here, we have

height = 0 ; x3o3o3o3o3x 12 together 12
height = 3 ; o3o3x3o3o3o 9 together 12
height = 12; o3o3xo3o3o3o 0 together 12

This makes the 1_22.

Gosset's polytope of diam 8, is satisfied in the same layers as

height 0 = o3x3o3o3o3o 8 together 8
height 3 = o3o3o3o3o3x 5 together 8
height 12 = (already too big).

Note that the figures have a column-modulus here of 2 and 5, This is one of the two orientations of the 2_21 in the off-positions (rather like the up and down triangles in the A2.)
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the dream we dream together is reality.

wendy
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