Johnsonian Polytopes

[wendy]

For what it is worth, i do agree with Richard's use of *A to close loops, etc, since this is already in his hack of my notation.

In my particular notation, what is written is actually a 'trace' of the dynkin symbol, which matches in regular cases, but one needs other devices to complete the non-regular symboles. In any case, what is important is that the separate nodes be kept separate (although one can abreviate them). The thing is resolved as an oblique coordinate system, (which is why all of the arithmetic works), and thus while one can effectively drop 0 values or gather together equals under one head, it does no good to drop the coordinate system.

For example,

Code: Select all

         o 5  o  3  o  5  o  3  z      o 3 o 3 o 3 o 3 z B z  B zz
         1    2     3     4     5      1   2   3   4   5   6    7

       5 1  o--5--o  2                1 5 6   o-----o  2  7
            |     |                           | \ / |
            |     |                           | / \ |
         4  o--5--o  3                    4   o-----o  3

                                   Nodes 5,6 are z, are at the same place as 1
                                   Nodes 7 is zz, is at the same place as 2.
                                     The branch between 5 and 6 is an S-2, so
                                   goes from 5-2 to 6.
                                     The branch between 6 and 7 is also S-2,
                                   so goes 6-2 to 7, ie 5 to 7



One can, for example, write something like _3ox4_&xt is flat, for all values of _, but the essential graph must be kept.

Currently, the nodals are like this. a(A) constructs a short-chord of a polygon (A), which is expanded in the text.

The notation describes a good deal of what's been discussed here.

• a = atom node (these do not add in the way that x-nodes do: the vertex of the rhombo-dodecahedron lie at aq3o4a. Used to show stations in lattices.
• b = bevel node (these create a face perpendicular to this axis: eg the rhombododecahedron = o3b4o
• d = density
• e = reserved open.
• f = a(F) = shortchord of the pentagon the number 1.618033 = sqrt(2.618033)
• g = gyrate node, is the dual of 's', eg the dodecahedron could be written as o4g3g.
• h = a(H) = shortchord of the hexagon, the number 1.7320508 = sqrt(3)
• i = supplement modifier on the branch, so P/D becomes P/(P-D).
• j = overload symbol. It has no specific meaning, one must look up the overload table. currently: j5j2j5j = grand antiprism.
• k = a(K) = shortchord of the octagon: the number 1.847 759 065 = sqrt(3.41421356) [this used to represent sqrt(4/3)]
• m = mirror-node, used to construct duals over x. One could subset m to eg mq to produce the dual of a q-node.
• n = (reserved for generic arrays, eg na - nz, which might be expanded in an accompanying table).
• o = unmarked branch, notionally the vertex is bisected by this mirror.
• p(n) = shortchord of {n}, eg p(10) = 1.90211303259 = sqrt(3.73205080757)
• r = a(R) = shortchord of the digon. This is an infintesimally short edge, used to show a proto-figure as r4o3o = point-like cube
• s = snub node, here alternation and adjustment of a corresponding construction in 'x'
• t = lace-tower, the sections are stated in the proceeding list.
• tz = lace - ring. The lacing proceeds it.
• u = a(U) = shortchord of the horogon (U = W4). 2 = sqrt(4). This is the polygon inscribed in a euclidean straight arc.
• v = a(V) = shortchord of the pentagram (5/2) = 0.618033 = sqrt(0.381966)
• w(n) = sqrt(n), used to directly specify the shortchord of W(n) - mainly used of hyperbolics.
• x = a(S) = shortchord of the triangle, taken as 1 = sqrt(1)
• z = construct-node, representing a repetition of the first node
• zz = construct-node, repetition of the second node.
• : and :: = older varieties of z and zz
• $ (dollar-rune) = vertex-node. - mainly used of vertex-figures of corresponding x-form. eg vf o3x5o = o3$5o.
• *(letter), = construction-node, reperesenting the repetition of the node of that letter (counting A, B, C, ...) [Klitzing]
• +, *, ++ = asterix node, creates a 4/2, 6/3, or 8/4 branch from the previous to next branch. eg /3+3 = stella octangula.
• / = older form of x-node, /q = 'q' etc.
• % = edge needed to join two faces, eg in decorated orbiform notation.
• \ = older form of m node, \q = 'mq', etc.
• # = lace prism or tegum, being arranged according to a simplex arrangement.

The following letters might be used to construct branches. All kinds of branch are used, if one wants to set numbers directly against the symbols, eg 1S2S4. A branch connects S to O (which differ by 1), unless specified in the subject and object nodes.

The original use of this set was to replace Coxeter's use of subscripts at 2_21 etc, to 4B. It was then extended to include non-three branches, such as Q, F, H, V, eg 2F = {3,3,5}. Slashes were implemented to allow node marks, eg /1/1F = x3x3o5o. Extensions made to accomidate the 'second extension', that is hyperbolic groups of finite content, and some open forms (P, W, D). One could write PnS stellates to PnD2D1 and then PnD1D2, means {p,3} stellates to {p/2,p} and then {p,p/2}.

S was added to allow the system to be used as a coordinate system, with the branches as column-separators, eg "1.99 S 2.15 F0".

R and i were allowed to accomidate the socalled 'wythoff' (ie decorated schwarz-triangles) notation. So | 5 3 2 might become /FSR. The lower-case i is used to do supplement angles, so the triangle VSS 5/2 3 3 gives supplements ViSiS 5/3, 3/2, 3 and VSiSi. The slash and its backform allows one to create figures which give mirror-edge and the dual mirror-margin, eg F/SR is the icosahedron, and F\SR is the dodecahedron as a catalan dual. It is actually S/FR also. Note that R\SF is the rhombic tricontahedron o3m5o.

O allows circles and spheres to participate in products. The 'truncates' are read as increasing / or decreasing \ radii, allow ellipses. So /OO sphere, /O/O is a ellipsoid x < y = z. /OO/ is x = y < z. /O&x is a cylinder, \O&m is the corresponding circular tegum. In four dimensions, /O&/O is a duocylinder [(xx)(xx)], \O&\O is that figure's dual (bicurcular tegum) <(xx)(xx)>, /OO&x is a spherinder (spheric prism) = [(xxx)x]. Note that there is no notation for the crind product, eg ([xx]x) has no representation: figures follow notation: you got to watch that!

• A = second-subject node, ie a '3' branch connecting S-1 to O - usually at the tail: E is the second-object node
• B = third-subject node, a '3' branch connectiong S-2 to O - usually at the tail: G is the third-object node
• C = fourth-subject node, a '3' branch connecting S-3 to O
• D = density, a referrent to a previous P or W node. eg P12D5 is {12/5}
• E = second-object node, a '3' branch connecting S to O+1
• F = branch marked '5', representing an angle of 12p
• G = third-object node, a '3' branch connecting S to O+2
• H = branch marked '6', representing an angle of 10p
• I = creates a supplement angle, ie F = 5/1, FI = 5/(5-1): angle is 60p - proceding
• K = branch marked '8', representing an angle of 7p60s
• O = sphere-branch - O = circle, OO = sphere, etc.
• P = polygon, eg P12 is the dodecagon.
• Q = branch marked '4' represent an angle of 15p
• R = branch marked '2' representing an angle of 30p
• S = branch marked '3' representing an angle of 20p
• T = (occasionally used of the hexagram)
• U = branch marked (oo), representing an angle of 0p. This is the horogon W4 only. It is not used of bollogons.
• V = branch marked (5/2)
• W = branch to specify infinitogons one bollogons, giving the shortchord, rather than the number of sides, eg W(4/3) never closes, but is finite.
• & = orthogonal product.

The following do not create a solid per se, but ought be noted.

• ".." = quotes, used to delinearate a figure, eg "o3o5o".
• || = joins two parallel figures by x-lacing, after Klitzing. as in A atop B atop C gives A || B || C . This can be written by multiple node-values too.
• *A etc = return to the first, etc figure of a list, in Klitzing's 'atop' notation eg "o3x || x3x || x3o || o3o || *A" creates a loop (square), in four dimensions. This replicate his *a nodal.
• , = separates values of a general list, eg o3o5o, o3o5x, &c.

[Klitzing]

Just stumbled yesterday on a finding of A. Weimholt from way back in 2004. I don't know if you would be aware of that structure, so I'll mention it here now:

It is a CRF-honeycomb (haha: "C" - at least what could come as close to that as possible: its limes onto flat space) using cubes, dodecahedra and bilunabirotundae!

In fact it is built with pyrithohedral symmetry: each cube gets attached by "bilbiro"es in such a way that each adjacent pair of the latter becomes completely orthogonal. This would further attach the 2 different types of triangles of bilbiro to be adjoined with each other. And also the inner symmetry of the dodecahedron gets reduced to that of an inscribed brick x2x2x, i.e. there are vertices (exactly those of that inscribed brick), which opposite-adjoin it to a cube, while at the set of mutually completely orthogonal 6 edges it opposite-adjoines to the next dodecahedron. That is, both, the cubes and the dodecahedra each are placed at cubical lattice positions each. Either type would cage the other ones in.

For sure it would not be uniform, as its constituents, the bilbiroes, are not uniform. But it is not scaliform either. Sure it is CRF, but not all vertices are equivalent. Just consider those different vertex types mentioned above for the dodecahedra.

And here follows its incidence matrix as well (N -> oo):

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8N  * |   3   3  0 |  3   3   3 | 1  3 1
 * 6N |   0   4  1 |  0   2   6 | 0  3 2
------+------------+------------+-------
 2  0 | 12N   *  * |  2   1   0 | 1  2 0
 1  1 |   * 24N  * |  0   1   2 | 0  2 1
 0  2 |   *   * 3N |  0   0   4 | 0  2 2
------+------------+------------+-------
 4  0 |   4   0  0 | 6N   *   * | 1  1 0
 2  1 |   1   2  0 |  * 12N   * | 0  2 0
 2  3 |   0   4  1 |  *   * 12N | 0  1 1
------+------------+------------+-------
 8  0 |  12   0  0 |  6   0   0 | N  * *  cube
 8  6 |   8  16  2 |  2   8   4 | * 3N *  bilbiro
 8 12 |   0  24  6 |  0   0  12 | *  * N  doe


--- rk

[quickfur]

Just stumbled yesterday on a finding of A. Weimholt from way back in 2004. I don't know if you would be aware of that structure, so I'll mention it here now:

It is a CRF-honeycomb (haha: "C" - at least what could come as close to that as possible: its limes onto flat space) using cubes, dodecahedra and bilunabirotundae!

In fact it is built with pyrithohedral symmetry: each cube gets attached by "bilbiro"es in such a way that each adjacent pair of the latter becomes completely orthogonal. This would further attach the 2 different types of triangles of bilbiro to be adjoined with each other. And also the inner symmetry of the dodecahedron gets reduced to that of an inscribed brick x2x2x, i.e. there are vertices (exactly those of that inscribed brick), which opposite-adjoin it to a cube, while at the set of mutually completely orthogonal 6 edges it opposite-adjoines to the next dodecahedron. That is, both, the cubes and the dodecahedra each are placed at cubical lattice positions each. Either type would cage the other ones in.
[...]

Whoa. This just blew my mind. I've realized before that, based on the standard coordinates of the dodecahedron (that includes an inscribed cube) that it should be possible to build a network of dodecahedra (i.e. dodecahedra sharing vertices) with cubic symmetry. But never did it occur to me that it's possible to incorporate cubes into the mix too, to make a tiling.

This also means that the bilunabirotunda isn't as strange as it might seem at first glance: in a sense you can say it's the CRF resulting from cutting adjoining dodecahedra and cubes from 3-space!

This makes me wonder if other Johnson solids can be similarly derived, esp. those that don't have an obvious derivation from uniform polyhedra. And even more so, are there (incomplete) tilings of 4-space with uniforms (or even regular polychora) that would give rise to crown jewel 4D CRFs? Since tilings impose quite a lot of restrictions on the possibilities, this may be one area where searching for CRFs may be much easier than the fully-general case. All we have to do is to find a repeating structure assembled from gluing uniform (or maybe even currently-known CRF) polychora together, and see if we can fill in enough gaps to close up a CRF "hole" (which then becomes equivalent to a 4D tiling involving a new CRF in the shape of that gap).

[Klitzing]

Great to have you back in this thread, quickfur!
(Might be it could become April now! - I think you know what I mean?)

Most probably I've to disappoint you, so, wrt your just stated 4D hope. The thing, why this cube-doe-bilbiro honeycomb works, is a speciality of 3D. Doe has non-cubical symmetry, so produces interesting gaps. OTOH it still has a radius small enough, so that the gaps can easily tiled by convex figures (here cubes and bilbiroes). In 4D both regular 335-symmetrical figures are already much to large for that, i.e. are themselves multistratic with a stack piling of much more strata than 3D. And I fear the other non-tesseractic higher symmetry, 343, would be too close to tesseractal symmetry in that according gaps probably would come out to be not too surprising (if at all CRF). ... Nonetheless we might consider to look into that more deeply.

--- rk

[Klitzing]

... And I fear the other non-tesseractic higher symmetry, 343, would be too close to tesseractal symmetry in that according gaps probably would come out to be not too surprising (if at all CRF). ... Nonetheless we might consider to look into that more deeply.

--- rk

Just was roaming within according possibilities. But then soon remembered that using icoes at tessic lattice positions would just make up further ico gaps at the tessic body-centers: the tesselation of euclidean 4-space by icoes is not only a regular one, but moreover uses those icoes at the vertex positions of the body-centered tessic lattice. Or in other words: those icoes are just the Voronoi cells of that lattice.

In fact this can be seen from icot = x3o4o3o3o = o4o3x3o4o
x3o4o3o3o having cells = x3o4o3o . and verf = . x4o3o3o
o4o3x3o4o having cells = o4o3x3o . (tessic lattice point centered) + . o3x3o4o (centered at tessic body-centeres) and verf = o4x . x4o

You then might like to apply expansions at the hyperplanes of body-centeres (but not at those of the tessic lattice points). But this just results in srittit = o4o3x3o4x, with additional octahedral prisms, spacing those lattice point icoes apart, and those body-center icoes would become srits = o3x3o4x.

--- rk

[Klitzing]

Rethought once more.

The last consideration in fact was the idea described in my page on partial Stott expansion in the realm of tetracombs, in axial subsymmetry:

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o4o3x3o4o = icot °     ↔  ...      ↔  ...       ↔  ...      ↔  o4o3x3o4x = srittit °


(Details: icot, srittit.)
Sure, those partial ones here would be not so interesting, when Expansion takes place in axial directions individually, i.e. the outcome (those so far not elaborated dotted positions) would look more brick like.

But then thought about hexadecachoric tetracombal subsymmetry. Here we have a full 4 step partial Stott expansion series, each usind a further hex subsymmetry of ico as to be added expansion directions. So I confered my tables and found the entry

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o3o3o4o3x = icot °     ↔  thext °  ↔  batitit °  ↔  bithit °  ↔  o3o3o4x3x = ticot °


(Details: thext, batitit, bithit, ticot.)
which clearly does what is asked for, but still is somehow dissappointing in that all members of this series are known Wythoffian tetracombs. In fact those are nothing but (according to a lower symmetry) true Stott expansions:

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o3x3o *b3o *b3o  ↔  x3x3o *b3o *b3o  ↔  x3x3x *b3o *b3o  ↔  x3x3x *b3x *b3o  ↔  x3x3x *b3x *b3x


But then I thought: what about using those icoes in dual positioning? - This clearly works as well, and was also already contained in the table as line:

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o3x3o4o3o = icot °     ↔  ...      ↔  ...        ↔  ...       ↔  o3x3o4x3o = sibricot °


Even so those dots mark not yet fully elaborated tetracombs, the net content (i.e. cell types) of those was already shematically tabulated there in the provided table below (here extracted accordingly):

• icot consists of ico cells only
• phextex(icot) would have cells of type ico, poxic, triddip, and pexic (in the ratio 1 : 3 : 32 : 12)
• pabhextex(icot) would have cells of type poxic, pocsric, triddip, and bicyte ausodip (in the ratio 2 : 2 : 64 : 12)
• phextco(sibricot) would have cells of type pocsric, srico, triddip, and pacsrit (in the ratio 3 : 1 : 96 : 12)
• and sibricot is known to have cells of type srico, triddip, and srit (in the ratio 1 : 32 : 3)

Thus finally arrived with what you where after!
So, none of those is totally new: I elaborated those tables mostly in the spring of this year 2013 - i.e. exactly 100 years after Stotts original paper. - And all those mentioned individual CRFs were already discussed somewhere in this very thread.

--- rk

[quickfur]

Great to have you back in this thread, quickfur!
(Might be it could become April now! - I think you know what I mean?)

You mean my website hasn't been updated for a while? Yeah... I got very busy with other things, and haven't had the time to do any new renders. Plus, the only remaining uniform polychora are the last three most complex 120-cell family polychora, which due to a limitation in my current rendering program takes very long rendering times. I've been working on and off on a rewrite of the program that uses better algorithms for the rendering, but haven't made very much progress. So right now things are at a standstill.

Most probably I've to disappoint you, so, wrt your just stated 4D hope. The thing, why this cube-doe-bilbiro honeycomb works, is a speciality of 3D. Doe has non-cubical symmetry, so produces interesting gaps. OTOH it still has a radius small enough, so that the gaps can easily tiled by convex figures (here cubes and bilbiroes). In 4D both regular 335-symmetrical figures are already much to large for that, i.e. are themselves multistratic with a stack piling of much more strata than 3D. And I fear the other non-tesseractic higher symmetry, 343, would be too close to tesseractal symmetry in that according gaps probably would come out to be not too surprising (if at all CRF). ... Nonetheless we might consider to look into that more deeply.
[...]

I see.

Well, I was thinking not so much about a direct analogue using regular polytopes to make tilings, but about 4D tesselations in general. By that I mean, let's say we start with our currently-known uniforms (or CRFs if you're daring), and find combinations for which the angle defect around, say, an edge, is zero. That would mean it's a potential seed for a 4D tesselation (this should be easier to find than arbitrary combinations of cells to make CRFs, since that requires handling of arbitrary unknown angle defects which complicates the search for possibilities). Then we check if it's possible to make some kind of repeating structure with this seed, maybe by adding a few more cells to make it possible to assemble multiple copies of the seed in a repeating pattern.

Once we have a repeating pattern, then in a sense we already have a tesselation, if we fill in the gaps between the repeating units, except that it may or may not be CRF. The challenge then is to look at how these gaps may be further filled with more CRFs such that at least some subset of it will be enclosed by CRF cells. The enclosed gap then, if it is convex, will be CRF by definition, since it will be enclosed by CRFs whose cells are themselves CRF. The hope is that even if the tessellation itself may contain some non-CRF tiles, among its other tiles there may be new CRFs.

[quickfur]

Along with your discovery of a tiling involving ico's (24-cells), I wonder if something interesting may be obtained if we substitute snub 24-cells for some of the positions in an ico tesselation, and see if it's possible to make a CRF tesselation out of it?

[Klitzing]

Hi snubophiles!

Quickfur was asking about Johnsonian tetracombs in general. here now would follow one, which was NOT derived by direct research, but rather did freely occur in an completely unrelated private discussion thread. In fact Tom Ruen just asked me privately about s4o3x3o4s'. (Note that snubbing here has to be applied at both ends separately (represented by the prime at the one snub node symbol), so in fact this becomes not a hemiation, rather it would be a quatersnub). But then it stuck to me to consider s4o3x3o4s as well. This one is much easier to be done, as only a single alternated faceting would be asked for. In this case one which alternates the contained octahedra (. o3x3o .) of scartit (x4o3x3o4x, small cellirhombated tesseractic tetracomb = stericantellated tesseractic tetracomb), cf. http://bendwavy.org/klitzing/incmats/scartit.htm.

Any guess so far what that fellow would look like? - Okay, here it comes.

The 4D cells of that tetracomb s4o3x3o4s then clearly would be:

• s4o3x3o . (and . o3x3o4s), i.e. in the first run being variations of thex, cf. http://bendwavy.org/klitzing/incmats/thex.htm.
• Then s4o3x 2 s (and s 2 x3o4s). Those would be derived from x4o3x x, i.e. sircope, cf. http://bendwavy.org/klitzing/incmats/sircope.htm by alternation of the . o3x . (triangles). Here the top sirco then becomes a tut (trincated tetrahedron), and likewise the bottom sirco. Alternation moreover asks that those tuts have to be anti-aligned, so we have as s4o3x2s what is now known a tutcup, i.e. the scaliform segmentochoron "tut || inv. tut", cf. http://bendwavy.org/klitzing/incmats/tut=invtut.htm, which in fact itself is nothing but the equatorial stratos of the rectified tesseract (rit, o4x3o3o) when considerered with axial tetrahedral symmetry.
• Further there are s4o 2 o4s, derived from x4o . o4x, the tesseract by alternating the vertices. Here the 8 cubes become tetrahedra each, and as sectioning facets (sefa) underneath further 8 tetrahedra will occur. Thus we get 16 tetrahedra in total, or, in other words, an hexadecachoron.
• Finally we have to consider sefa( s4o3x3o4s ), the sectioning facetes underneath the alternated elements, i.e. underneath the octahedra. Those octahedra (of x4o3x3o4x) are adjacent to x4o3x3o . and to . o3x3o4x, i.e. srits, cf. http://bendwavy.org/klitzing/incmats/srit.htm. So we have to consider those srits with oct first. The next sectioning underneath will be q3x3o, a variant of an truncated tetrahedron (tut). Thus it follows (after some more investigations) that those searched for cells likewise come out to be tutcups.

The mere alternated faceting process surely does exist. Then the corresponding incidence matrix is derived as

Code: Select all

s4o3x3o4s

demi( . . . . . ) | 48N |   4   1   2   1 |   2   2   4   4   6   6   4 |  1  2   2   2  2  2   6   4   6  2 | 1  2  1  2 1  4
------------------+-----+-----------------+-----------------------------+------------------------------------+----------------
demi( . . x . . ) |   2 | 96N   *   *   * |   1   1   1   1   0   0   1 |  1  1   0   0  1  1   2   0   2  1 | 1  1  0  1 1  2
      s4o . . .   |   2 |   * 24N   *   * |   0   0   0   4   4   0   0 |  0  2   2   0  0  2   4   2   0  0 | 1  2  1  0 0  2
      s . 2 . s   |   2 |   *   * 48N   * |   0   0   2   0   4   4   0 |  0  0   2   2  0  0   4   4   4  0 | 0  2  2  2 0  4
      . . . o4s   |   2 |   *   *   * 24N |   0   0   0   0   0   4   4 |  0  0   0   2  2  0   0   2   4  2 | 0  0  1  2 1  2
------------------+-----+-----------------+-----------------------------+------------------------------------+----------------
demi( . o3x . . ) |   3 |   3   0   0   0 | 32N   *   *   *   *   *   * |  1  1   0   0  0  0   1   0   0  1 | 1  1  0  0 1  1
demi( . . x3o . ) |   3 |   3   0   0   0 |   * 32N   *   *   *   *   * |  1  0   0   0  1  1   0   0   1  0 | 1  0  0  1 1  1
      s 2 x 2 s   |   4 |   2   0   2   0 |   *   * 48N   *   *   *   * |  0  0   0   0  0  0   2   0   2  0 | 0  1  0  1 0  2
sefa( s4o3x . . ) |   6 |   3   3   0   0 |   *   *   * 32N   *   *   * |  0  1   0   0  0  1   1   0   0  0 | 1  1  0  0 0  1
sefa( s4o . 2 s ) |   3 |   0   1   2   0 |   *   *   *   * 96N   *   * |  0  0   1   0  0  0   1   1   0  0 | 0  1  1  0 0  1
sefa( s 2 . o4s ) |   3 |   0   0   2   1 |   *   *   *   *   * 96N   * |  0  0   0   1  0  0   0   1   1  0 | 0  0  1  1 0  1
sefa( . . x3o4s ) |   6 |   3   0   0   3 |   *   *   *   *   *   * 32N |  0  0   0   0  1  0   0   0   1  1 | 0  0  0  1 1  1
------------------+-----+-----------------+-----------------------------+------------------------------------+----------------
demi( . o3x3o . ) |   6 |  12   0   0   0 |   4   4   0   0   0   0   0 | 8N  *   *   *  *  *   *   *   *  * | 1  0  0  0 1  0  oct
      s4o3x . .   |  12 |  12   6   0   0 |   4   0   0   4   0   0   0 |  * 8N   *   *  *  *   *   *   *  * | 1  1  0  0 0  0  tut
      s4o . 2 s   |   4 |   0   2   4   0 |   0   0   0   0   4   0   0 |  *  * 24N   *  *  *   *   *   *  * | 0  1  1  0 0  0  tet
      s 2 . o4s   |   4 |   0   0   4   2 |   0   0   0   0   0   4   0 |  *  *   * 24N  *  *   *   *   *  * | 0  0  1  1 0  0  tet
      . . x3o4s   |  12 |  12   0   0   6 |   0   4   0   0   0   0   4 |  *  *   *   * 8N  *   *   *   *  * | 0  0  0  1 1  0  tut
sefa( s4o3x3o . ) |  12 |  12   6   0   0 |   0   4   0   4   0   0   0 |  *  *   *   *  * 8N   *   *   *  * | 1  0  0  0 0  1  tut
sefa( s4o3x 2 s ) |   9 |   6   3   6   0 |   1   0   3   1   3   0   0 |  *  *   *   *  *  * 32N   *   *  * | 0  1  0  0 0  1  tricu
sefa( s4o 2 o4s ) |   4 |   0   1   4   1 |   0   0   0   0   2   2   0 |  *  *   *   *  *  *   * 48N   *  * | 0  0  1  0 0  1  tet
sefa( s 2 x3o4s ) |   9 |   6   0   6   3 |   0   1   3   0   0   3   1 |  *  *   *   *  *  *   *   * 32N  * | 0  0  0  1 0  1  tricu
sefa( . o3x3o4s ) |  12 |  12   0   0   6 |   4   0   0   0   0   0   4 |  *  *   *   *  *  *   *   *   * 8N | 0  0  0  0 1  1  tut
------------------+-----+-----------------+-----------------------------+------------------------------------+----------------
      s4o3x3o .   |  48 |  96  24   0   0 |  32  32   0  32   0   0   0 |  8  8   0   0  0  8   0   0   0  0 | N  *  *  * *  *  thex
      s4o3x 2 s   |  24 |  24  12  24   0 |   8   0  12   8  24   0   0 |  0  2   6   0  0  0   8   0   0  0 | * 4N  *  * *  *  tutcup
      s4o 2 o4s   |   8 |   0   4  16   4 |   0   0   0   0  16  16   0 |  0  0   4   4  0  0   0   8   0  0 | *  * 6N  * *  *  hex
      s 2 x3o4s   |  24 |  24   0  24  12 |   0   8  12   0   0  24   8 |  0  0   0   6  2  0   0   0   8  0 | *  *  * 4N *  *  tutcup
      . o3x3o4s   |  48 |  96   0   0  24 |  32  32   0   0   0   0  32 |  8  0   0   0  8  0   0   0   0  8 | *  *  *  * N  *  thex
sefa( s4o3x3o4s ) |  24 |  24   6  24   6 |   4   4  12   4  12  12   4 |  0  0   0   0  0  1   4   6   4  1 | *  *  *  * * 8N  tutcup

The remaining question now would be, whether this figure also has a variant, where all edges would have the same size (and faces remain planar for sure).

Right to that end applies the same reasoning, which was brought up by Wendy for prissi (which I did find as a similar alternated faceting, in fact s3s4o3x, way back in September 2005) in reply in about November 2005 (accordingly adopted): s4o3o3o4s clearly does exist as uniform tetracomb (in fact being the hexadecachoral tetrachomb, hext, cf. http://bendwavy.org/klitzing/incmats/hext.htm - sadly in the online version so far only s4o3o3o4o is outlined, but for sure s4o3o3o4s would run similar, and in my not yet uploaded copy already is contained). Knowing that, then s4o3x3o4s becomes nothing but a mere Stott expansion of that one. And that expansion clearly is a one parameter process: the ratio of the added edge size in respect to the edge sizes of s4o3o3o4s. Obviously that parameter then can be sized such that this ratio becomes unity. Therefore, it finally turns out that this new tetracomb s4o3x3o4s will have a scaliform scaling too.

(Uniformity clearly is not achievable, as the included elements (esp. tricu and tutcup) are not uniform in turn. But this would have been asked for by the hierarchical part of the according definition.)

--- rk

[Klitzing]

Along with your discovery of a tiling involving ico's (24-cells), I wonder if something interesting may be obtained if we substitute snub 24-cells for some of the positions in an ico tesselation, and see if it's possible to make a CRF tesselation out of it?

Here now follows, what has been searched for: a scaliform only tetracomb with incorporated snub icositetrachora (sadi, s3s4o3o).

It can be derived from the uniform snub icositetrachoron tetracomb (sadit, s3s4o3o3o) by means of a Stott addition with the hexadecachoral tetracomb (hext, o3o4o3o3x). In fact it results in shifting the snub icositetrachora of the former one edge length apart and filling in accordingly.

Here is its matrix:

Code: Select all

s3s4o3o3x   (N → ∞)

demi( . . . . . ) | 96N |    3    3    6 |   3   3   3    9   12   3 |   1   3   6   1   4    9   6   3 | 1   3   3  1   4
------------------+-----+----------------+---------------------------+----------------------------------+-----------------
demi( . . . . x ) |   2 | 144N    *    * |   2   0   1    0    4   0 |   1   0   2   0   0    3   4   2 | 0   1   2  1   3
      . s4o . .   |   2 |    * 144N    * |   0   0   1    2    0   2 |   0   1   0   1   2    2   0   2 | 1   1   0  1   2
sefa( s3s . . . ) |   2 |    *    * 288N |   0   1   0    2    2   0 |   0   2   2   0   1    2   1   0 | 1   2   1  0   1
------------------+-----+----------------+---------------------------+----------------------------------+-----------------
demi( . . . o3x ) |   3 |    3    0    0 | 96N   *   *    *    *   * |   1   0   0   0   0    0   2   1 | 0   0   1  1   2
      s3s . . .   |   3 |    0    0    3 |   * 96N   *    *    *   * |   0   2   2   0   0    0   0   0 | 1   2   1  0   0
      . s4o 2 x   |   4 |    2    2    0 |   *   * 72N    *    *   * |   0   0   0   0   0    2   0   2 | 0   1   0  1   2
sefa( s3s4o . . ) |   3 |    0    1    2 |   *   *   * 288N    *   * |   0   1   0   0   1    1   0   0 | 1   1   0  0   1
sefa( s3s . 2 x ) |   4 |    2    0    2 |   *   *   *    * 288N   * |   0   0   1   0   0    1   1   0 | 0   1   1  0   1
sefa( . s4o3o . ) |   3 |    0    3    0 |   *   *   *    *    * 96N |   0   0   0   1   1    0   0   1 | 1   0   0  1   1
------------------+-----+----------------+---------------------------+----------------------------------+-----------------
demi( . . o3o3x ) |   4 |    6    0    0 |   4   0   0    0    0   0 | 24N   *   *   *   *    *   *   * | 0   0   0  1   1  tet
      s3s4o . .   |  12 |    0    6   24 |   0   8   0   12    0   0 |   * 24N   *   *   *    *   *   * | 1   1   0  0   0  ike
      s3s . 2 x   |   6 |    3    0    6 |   0   2   0    0    3   0 |   *   * 96N   *   *    *   *   * | 0   1   1  0   0  trip
      . s4o3o .   |   4 |    0    6    0 |   0   0   0    0    0   4 |   *   *   * 24N   *    *   *   * | 1   0   0  1   0  tet
sefa( s3s4o3o . ) |   4 |    0    3    3 |   0   0   0    3    0   1 |   *   *   *   * 96N    *   *   * | 1   0   0  0   1  tet
sefa( s3s4o 2 x ) |   6 |    3    2    4 |   0   0   1    2    2   0 |   *   *   *   *   * 144N   *   * | 0   1   0  0   1  trip
sefa( s3s 2 o3x ) |   6 |    6    0    3 |   2   0   0    0    3   0 |   *   *   *   *   *    * 96N   * | 0   0   1  0   1  trip
sefa( . s4o3o3x ) |  12 |   12   12    0 |   4   0   6    0    0   4 |   *   *   *   *   *    *   * 24N | 0   0   0  1   1  co
------------------+-----+----------------+---------------------------+----------------------------------+-----------------
      s3s4o3o .   |  96 |    0  144  288 |   0  96   0  288    0  96 |   0  24   0  24  96    0   0   0 | N   *   *  *   *  sadi
      s3s4o 2 x   |  24 |   12   12   48 |   0  16   6   24   24   0 |   0   2   8   0   0   12   0   0 | * 12N   *  *   *  ipe
      s3s 2 o3x   |   9 |    9    0    9 |   3   3   0    0    9   0 |   0   0   3   0   0    0   3   0 | *   * 32N  *   *  triddip
      . s4o3o3x   |  32 |   48   48    0 |  32   0  24    0    0  32 |   8   0   0   8   0    0   0   8 | *   *   * 3N   *  rit
sefa( s3s4o3o3x ) |  16 |   18   12   12 |   8   0   6   12   12   4 |   1   0   0   0   4    6   4   1 | *   *   *  * 24N  tet||co


It is that very last cell, the segmentochoron tet || co, which requires this tetracomb to be scalliform (as that one on its own obviously would not be uniform). Btw., that very last figure can be seen also as a rotunda of spid (x3o3o3x).

--- rk

[Klitzing]

Knew already about the possibility for s3s3s4o3x to be relaxable from the mere alternated faceting of x3x3x4o3x (gicaricot = great cellirhombated icositetrachoral tetracomb) back to unit edges for long. Even so I did not come around to elaborate that one so far.

Yesterday now I've done that one. And it turns out to be a rather pretty scaliform!
Here follows its incidence matrix:

Code: Select all

s3s3s4o3x

demi( . . . . . ) | 288N |    2    2    1    2    4 |   1   1    2    2    6    4    3    3    4   2 |   2   2   1   1   2   1    4    6   2   3   3 |  1   2   1   1 1   4
------------------+------+--------------------------+------------------------------------------------+-----------------------------------------------+---------------------
demi( . . . . x ) |    2 | 288N    *    *    *    * |   1   0    1    0    0    2    0    0    2   1 |   0   1   0   0   1   1    0    3   2   2   2 |  0   1   1   1 1   3
      s 2 s . .   |    2 |    * 288N    *    *    * |   0   0    1    0    2    0    2    0    0   0 |   1   0   1   0   0   0    2    2   0   2   0 |  1   1   0   1 0   2
      . . s4o .   |    2 |    *    * 144N    *    * |   0   0    0    0    0    0    2    2    0   2 |   0   0   1   1   0   1    2    0   0   2   2 |  1   0   0   1 1   2
sefa( s3s . . . ) |    2 |    *    *    * 288N    * |   0   1    0    0    2    2    0    0    0   0 |   2   2   0   0   0   0    1    2   1   0   0 |  1   2   1   0 0   1
sefa( . s3s . . ) |    2 |    *    *    *    * 576N |   0   0    0    1    1    0    0    1    1   0 |   1   0   0   1   1   0    1    1   0   0   1 |  1   1   0   0 1   1
------------------+------+--------------------------+------------------------------------------------+-----------------------------------------------+---------------------
demi( . . . o3x ) |    3 |    3    0    0    0    0 | 96N   *    *    *    *    *    *    *    *   * |   0   0   0   0   0   1    0    0   2   1   1 |  0   0   1   1 1   2
      s3s . . .   |    3 |    0    0    0    3    0 |   * 96N    *    *    *    *    *    *    *   * |   2   2   0   0   0   0    0    0   0   0   0 |  1   2   1   0 0   0
      s 2 s 2 x   |    4 |    2    2    0    0    0 |   *   * 144N    *    *    *    *    *    *   * |   0   0   0   0   0   0    0    2   0   2   0 |  0   1   0   1 0   2
      . s3s . .   |    3 |    0    0    0    0    3 |   *   *    * 192N    *    *    *    *    *   * |   1   0   0   1   1   0    0    0   0   0   0 |  1   1   0   0 1   0
sefa( s3s3s . . ) |    3 |    0    1    0    1    1 |   *   *    *    * 576N    *    *    *    *   * |   1   0   0   0   0   0    1    1   0   0   0 |  1   1   0   0 0   1
sefa( s3s . 2 x ) |    4 |    2    0    0    2    0 |   *   *    *    *    * 288N    *    *    *   * |   0   1   0   0   0   0    0    1   1   0   0 |  0   1   1   0 0   1
sefa( s 2 s4o . ) |    3 |    0    2    1    0    0 |   *   *    *    *    *    * 288N    *    *   * |   0   0   1   0   0   0    1    0   0   1   0 |  1   0   0   1 0   1
sefa( . s3s4o . ) |    3 |    0    0    1    0    2 |   *   *    *    *    *    *    * 288N    *   * |   0   0   0   1   0   0    1    0   0   0   1 |  1   0   0   0 1   1
sefa( . s3s 2 x ) |    4 |    2    0    0    0    2 |   *   *    *    *    *    *    *    * 288N   * |   0   0   0   0   1   0    0    1   0   0   1 |  0   1   0   0 1   1
sefa( . . s4o3x ) |    6 |    3    0    3    0    0 |   *   *    *    *    *    *    *    *    * 96N |   0   0   0   0   0   1    0    0   0   1   1 |  0   0   0   1 1   1
------------------+------+--------------------------+------------------------------------------------+-----------------------------------------------+---------------------
      s3s3s . .   |   12 |    0    6    0   12   12 |   0   4    0    4   12    0    0    0    0   0 | 48N   *   *   *   *   *    *    *   *   *   * |  1   1   0   0 0   0   ike
      s3s . 2 x   |    6 |    3    0    0    6    0 |   0   2    0    0    0    3    0    0    0   0 |   * 96N   *   *   *   *    *    *   *   *   * |  0   1   1   0 0   0   trip
      s 2 s4o .   |    4 |    0    4    2    0    0 |   0   0    0    0    0    0    4    0    0   0 |   *   * 72N   *   *   *    *    *   *   *   * |  1   0   0   1 0   0   tet
      . s3s4o .   |   12 |    0    0    6    0   24 |   0   0    0    8    0    0    0   12    0   0 |   *   *   * 24N   *   *    *    *   *   *   * |  1   0   0   0 1   0   ike
      . s3s 2 x   |    6 |    3    0    0    0    6 |   0   0    0    2    0    0    0    0    3   0 |   *   *   *   * 96N   *    *    *   *   *   * |  0   1   0   0 1   0   trip
      . . s4o3x   |   12 |   12    0    6    0    0 |   4   0    0    0    0    0    0    0    0   4 |   *   *   *   *   * 24N    *    *   *   *   * |  0   0   0   1 1   0   tut
sefa( s3s3s4o . ) |    4 |    0    2    1    1    2 |   0   0    0    0    2    0    1    1    0   0 |   *   *   *   *   *   * 288N    *   *   *   * |  1   0   0   0 0   1   tet
sefa( s3s3s 2 x ) |    6 |    3    2    0    2    2 |   0   0    1    0    2    1    0    0    1   0 |   *   *   *   *   *   *    * 288N   *   *   * |  0   1   0   0 0   1   trip
sefa( s3s 2 o3x ) |    6 |    6    0    0    3    0 |   2   0    0    0    0    3    0    0    0   0 |   *   *   *   *   *   *    *    * 96N   *   * |  0   0   1   0 0   1   trip
sefa( s 2 s4o3x ) |    9 |    6    6    3    0    0 |   1   0    3    0    0    0    3    0    0   1 |   *   *   *   *   *   *    *    *   * 96N   * |  0   0   0   1 0   1   tricu
sefa( . s3s4o3x ) |    9 |    6    0    3    0    6 |   1   0    0    0    0    0    0    3    3   1 |   *   *   *   *   *   *    *    *   *   * 96N |  0   0   0   0 1   1   tricu
------------------+------+--------------------------+------------------------------------------------+-----------------------------------------------+---------------------
      s3s3s4o .   |   96 |    0   96   48   96  192 |   0  32    0   64  192    0   96   96    0   0 |  16   0  24   8   0   0   96    0   0   0   0 | 3N   *   *   * *   *   sadi
      s3s3s 2 x   |   24 |   12   12    0   24   24 |   0   8    6    8   24   12    0    0   12   0 |   2   4   0   0   4   0    0   12   0   0   0 |  * 24N   *   * *   *   ipe
      s3s 2 o3x   |    9 |    9    0    0    9    0 |   3   3    0    0    0    9    0    0    0   0 |   0   3   0   0   0   0    0    0   3   0   0 |  *   * 32N   * *   *   triddip
      s 2 s4o3x   |   24 |   24   24   12    0    0 |   8   0   12    0    0    0   24    0    0   8 |   0   0   6   0   0   2    0    0   0   8   0 |  *   *   * 12N *   *   tut || inv tut
      . s3s4o3x   |  288 |  288    0  144    0  576 |  96   0    0  192    0    0    0  288  288  96 |   0   0   0  24  96  24    0    0   0   0  96 |  *   *   *   * N   *   prissi
sefa( s3s3s4o3x ) |   12 |    9    6    3    3    6 |   2   0    3    0    6    3    3    3    3   1 |   0   0   0   0   0   0    3    3   1   1   1 |  *   *   *   * * 96N   {6} || trip


I.e. implementing both 3D and 4D segmentotopes (tricu = {3} || {6}, tutcup = tut || inv tut, and {6} || trip) as well as the snubs ike and sadi, and then moreover prissi as well!

s3s3s4o3x still needs to get both, a long name and an OBSA (--> @Hedrondude).

--- rk

[wendy]

I would had been supprised if s3s3s4o3x did not work. Remember that the 'x' edges are completely orthogonal to the space containing 's' edges. This happens as long as the 'x' edges are in the 'around-mirrors' relative to the 's' edges. This happens if there is no marked edge that has an s at one end and an x at the other.

[Klitzing]

It just occured to me an interesting point about some good old polyhedra!
In fact, we will come here back again to the original meaning of this subject line, to some Johnson solids (and to the non-convex generalizations of that subset): the cupolae.

I realized that we dare consider the n/d-gonal cupola (=  xx-n/d-ox&#x  =  x-n/d-o || x-n/d-x ) as an alternated faceting (a precisely defined localized snubbing procedure) applied to the 2n/d-gonal antiprism (=  xo-2n/d-ox&#x  =  x-2n/d-o || o-2n/d-x ), when we would consider as alternatedly to be replaced objects (by their sectioning facets underneath) only half of the top layer vertices! I.e. by writing those cupolae as  so-n/d-ox&#x .

Note that this latter symbol both can be read as  s10o || o10x , i.e. as applying the snubbing first and than stacking, but well also the other way round! (That latter view will be outlined below in some more detail.)

(For sure, an afterwards to be applied relaxation back to unit edges - if possible at all - might be understood, if wanted for, as usual.)

E.g. for n/d=5, i.e. the pentagonal cupola, an according incidence matrix here would read:

Code: Select all

so10ox&#x

demi( o.10o.    ) | 5  * | 2  2 0 0 | 1 1 0 2
      .o10.o      | * 10 | 0  1 1 1 | 0 1 1 1
------------------+------+----------+--------
sefa( s.10o.    ) | 2  0 | 5  * * * | 1 0 0 1
demi( oo10oo&#x ) | 1  1 | * 10 * * | 0 1 0 1
demi( ..  .x    ) | 0  2 | *  * 5 * | 0 0 1 1
demi( ..  .x    ) | 0  2 | *  * * 5 | 0 1 1 0
------------------+------+----------+--------
      s.10o.      | 5  0 | 5  0 0 0 | 1 * * *
demi( ..  ox&#x ) | 1  2 | 0  2 0 1 | * 5 * *
      .o10.x      | 0 10 | 0  0 5 5 | * * 1 *
sefa( so10ox&#x ) | 2  2 | 1  2 1 0 | * * * 5


(As usual: "sefa" means "sectioning facet underneath the omitted objects".)

--- rk

[wendy]

Isn't oo10oo&#x just a line?

[Klitzing]

Sure, oo10oo&#x clearly is a (set of) line(s), in fact those lacing ones.
This is why it is given in the same dimensional block as sefa( s.10o. ), also a line (set).

But, I've to admit that you might have been irritated by the 2 entries below in this block.
Those should have been given rather as demi( .. .x ) each.

Sorry for that typo.
(I also just fixed it in the previous mail.)

--- rk

[student91]

Hello, I'm new to this forum.
Before I registered to this forum I already did some research on CRF-polytopes. For some reason I never encountered this forum, but I did encounter quickfur's website, which was extremely helpful in understanding the 4th dimention, and Klitzing's article about Sechmentotopes, which gave me the idea of stacking these (the #&xt way). I am currently enumerating stacks of sechmentotopes for a school project. During my research about the 4th dimention I came up with a limited way of algebraically calculating dihedral and dichoral angles. (it is quite tedious, and uses induction. I guess you already have good ways to calculate these. However, if not, I might post this if you're interesed) This, of course, is the only tool needed for checking whether a stack is convex or not.
Furthermore, I had a lot of loose thoughts about what CRF-polytopes might be possible. This way I thought of the diminishings of the 600-cell and other uniform polytopes, and other modifications of the uniform polytopes. I also thought that in a euclidean space with dimention 4 or up, a CRF-polytope must be much more closely related to a symmetry-group than in dimention less than 4. This thought was based on the fact that very few snub 4-polytopes are regular, which means that slight rearrangements of vertices, e.g. the rearrangement to make a s4s3s regular, are almost impossible. I think investigating this limitation of rearrangements may give clues of what a crown juel might look like. However, I hope that my interference won't stop your discusion about those lattices, as I'm interested in the outcome as well.

student91

[student91]

This also means that the bilunabirotunda isn't as strange as it might seem at first glance: in a sense you can say it's the CRF resulting from cutting adjoining dodecahedra and cubes from 3-space!

This makes me wonder if other Johnson solids can be similarly derived, esp. those that don't have an obvious derivation from uniform polyhedra. And even more so, are there (incomplete) tilings of 4-space with uniforms (or even regular polychora) that would give rise to crown jewel 4D CRFs? Since tilings impose quite a lot of restrictions on the possibilities, this may be one area where searching for CRFs may be much easier than the fully-general case. All we have to do is to find a repeating structure assembled from gluing uniform (or maybe even currently-known CRF) polychora together, and see if we can fill in enough gaps to close up a CRF "hole" (which then becomes equivalent to a 4D tiling involving a new CRF in the shape of that gap).

While I was thinking of deriving "non-obvious" johnson-solids in the evening, I found out that J84, J85 and J90 can be derived from other CRF-polyhedra by this weird "snubbing" operation that is not really snubbing.
e.g. one can derive a snub disphenoid (J84) from a disphenoid (tetahedron (digonal antiprism)) by cutting it in two parts and placing triangles between them.
J85 (snub square antiprism) can be derived of a square antiprism by cutting it over a zig-zag line along all triangles, and placing triangles in between.
If one tries this way of snubbing on a trigonal antiprism (octahedron), one gets a icosahedron.
J90 can be derived from a gyrobifastigium by this snubbing operation.
note that this way of snubbing isn't easily extrapolated to the 4th dimention, at least it will not easily give CRF's

I can't find a way of constructing J86 up to J89 from other CRF's

J91 and J92 are however very interesting, as they are intimately connected to other CRF's.
The bilunabirotunda (J91) has three verfs: 3.5.3.4; 3.5.3.5 and 5.3.5 . Of these, 3.5.3.5 is transitive to the vertex of an icosidodecahedron (not only by polyhedron count, but also by dihedral angles), and 5.3.5 is transitive to the vertex of a metabidiminished icosahedron.
What is interesting, is that the (3,5)-dihedral angle of a metabidiminished icosahedron( arccos(-sqrt((5-2sqrt(5))/15)) ) and the (3,5)-dihedral angle of a icosidodecahedron ( arccos(-sqrt((5+2sqrt(5))/15)) ) add up to the (5,5)-dihedral angle of a dodecahedron( arccos(-sqrt(5)/5).
The triangular hebesphenorotunda has verfs 3.5.3.4; 3.5.3.5; 3.3.3.5 and 3.3.4.6 . Of these, 3.5.3.4 is transitive to the 3.5.3.4 of J91, 3.5.3.5 is transitive to the vertex of a icosidodecahedon (and tus to the 3.5.3.5 of J91), and 3.3.3.5 is transitive to a pentagonal antiprism (and thus the (3,5)-dihedral angle of this vertex equals the (3,5)-dihedral angle of the 3.5.5 vertex of J91)

student91

[Klitzing]

A lot on axial polyhedra as well as reflections upon some of the Johnson solids can be found on the web site of Jim McNeill:
http://www.orchidpalms.com/polyhedra/index.html

--- rk

[Klitzing]

...
I realized that we dare consider the n/d-gonal cupola (=  xx-n/d-ox&#x  =  x-n/d-o || x-n/d-x ) as an alternated faceting (a precisely defined localized snubbing procedure) applied to the 2n/d-gonal antiprism (=  xo-2n/d-ox&#x  =  x-2n/d-o || o-2n/d-x ), when we would consider as alternatedly to be replaced objects (by their sectioning facets underneath) only half of the top layer vertices! I.e. by writing those cupolae as  so-n/d-ox&#x .

Note that this latter symbol both can be read as  s10o || o10x , i.e. as applying the snubbing first and than stacking, but well also the other way round! (That latter view will be outlined below in some more detail.)

(For sure, an afterwards to be applied relaxation back to unit edges - if possible at all - might be understood, if wanted for, as usual.)
...

Today it occured to me similarily that we also could rewrite that crown jewel segmentochoron  cube || ike  in a similar fashion too!

It already was known that  cube = o3o4x  and that  ike = s3s3s = s3s4o  and thus we could have, in the just stacking sense, clearly something like  o3o4x || s3s4o .

But then, would that also be true, if stacking would have been taken first, and alternation only thereafter?

Here now follows the proof thereof:

Code: Select all

os3os4xo&#x

      o.3o.4o.      | 8  * |  3  3 0  0 | 3  3  3  3  0 0 | 1 3  3 1 0
demi( .o3.o4.o    ) | * 12 |  0  2 1  4 | 0  1  2  4  3 2 | 0 1  3 2 1
--------------------+------+------------+-----------------+-----------
      .. .. x.      | 2  0 | 12  * *  * | 2  1  1  0  0 0 | 1 2  1 0 0
demi( oo3oo4oo&#x ) | 1  1 |  * 24 *  * | 0  1  1  2  0 0 | 0 1  2 1 0
      .. .s4.o      | 0  2 |  *  * 6  * | 0  0  2  0  2 0 | 0 1  2 0 1
sefa( .s3.s ..    ) | 0  2 |  *  * * 24 | 0  0  0  1  1 1 | 0 0  1 1 1
--------------------+------+------------+-----------------+-----------
      .. o.4x.      | 4  0 |  4  0 0  0 | 6  *  *  *  * * | 1 1  0 0 0
demi( .. .. xo    ) | 2  1 |  1  2 0  0 | * 12  *  *  * * | 0 1  1 0 0
sefa( .. os4xo&#x ) | 2  2 |  1  2 1  0 | *  * 12  *  * * | 0 1  1 0 0
sefa( os3os ..&#x ) | 1  2 |  0  2 0  1 | *  *  * 24  * * | 0 0  1 1 0
sefa( .s3.s4.o    ) | 0  3 |  0  0 1  2 | *  *  *  * 12 * | 0 0  1 0 1
      .s3.s ..      | 0  3 |  0  0 0  3 | *  *  *  *  * 8 | 0 0  0 1 1
--------------------+------+------------+-----------------+-----------
      o.3o.4x.      | 8  0 | 12  0 0  0 | 6  0  0  0  0 0 | 1 *  * * * cube
      .. os4xo&#x   | 4  2 |  4  4 1  0 | 1  2  2  0  0 0 | * 6  * * * trip
sefa( os3os4xo&#x ) | 2  3 |  1  4 1  2 | 0  1  1  2  1 0 | * * 12 * * squippy
      os3os ..&#x   | 1  3 |  0  3 0  3 | 0  0  0  3  0 1 | * *  * 8 * tet
      .s3.s4.o      | 0 12 |  0  0 6 24 | 0  0  0  0 12 8 | * *  * * 1 ike


I.e. those lacing trips occur right in the sense of the mentioned cupolae, here as a faceting of squap. The lacing tets likewise occur as corresponding snubs of hexagonal pyramids, and the lacing squippies come out as the added sectioning facets underneath those omitted vertices!

It should be noted here that the starting figure itself, i.e. ox3ox4xo&#x, cannot be realized with unit edges throughout as a tiling of spherical space, i.e. is not embeddable into euclidean 4-space. It rather would require hyperbolic space embedding. (There would occur e.g. hexagonal pyramids for cells.) It would require for taller lacing edges in order to be valid for an euclidean embedding.

To solve this trouble, it should be noted then however, that snubs usually are considered as their relaxations down to all unit edges. Thus when relaxing those longer edges, derived by the alternated faceting procedure, we could aim for, in addition, to relax that asked for longer lacing edges simultanuously.

(Sure esp. in 4D and above, the here imposed relaxation requirements often surpass by far the available degrees of freedom. So the hope to impose here such a further requirement would not be too promising. But, in fact, here we did come from the opposite direction: we already know that this jewel indeed exists as a figure with all unit edges.)

Therefore in conclusion: we are well allowed to write that crown jewel as a true snubbed lace prism, as  os3os4xo&#x .

--- rk

[Klitzing]

Having now provided (in my recent mail of this thread) that segmentochoral gem, the cube || ike, in a (partially) snubbed lace prism notation, i.e. as os3os4xo&#x, it gets quite immediate where to find higher dimensional CRFs, which would contain right that fellow for subelements.


In fact, today I've evaluated the convex segmentoteron rico || sadi. It turns out, that it not only belongs to the right geometry (all edges can be chosen to have the same unit length, without drifting then into euclidean space (height becoming zero) or even an hyperbolic one (height becoming imaginary)), by construction it additionally solves that mentioned problem as well.

I also managed to calculate its height to
  sqrt[3 sqrt(10)-2 sqrt(5)+9 sqrt(2)-14]/2 = 0.967292
and its circumradius to
  sqrt[(2986+1443 sqrt(2)+950 sqrt(5)+717 sqrt(10))/3056] = 1.755538
(Here those huge coefficients only got derived from removing all roots from the divisor.)
Edit: cf. http://teamikaria.com/hddb/forum/viewtopic.php?p=19547#p19547

How did I get to consider rico || sadi? - Just by extension of the Dynkin symbol of that mentioned segmentochoron: Accordingly its incidence matrix too can be derived from os3os4xo3oo&#x as

Code: Select all

os3os4xo3oo&#x

      o.3o.4o.3o.      | 96  * |   6   3   0   0 |   6  3   6   3   6  0   0  0 |  2  3  3  1   6   6  3  0  0  0 | 1  2  3  3 0
demi( .o3.o4.o4.o    ) |  * 96 |   0   3   3   6 |   0  0   3   6   6  3   9  3 |  0  0  1  3   3   9  3  3  1  4 | 0  3  1  4 1
-----------------------+-------+-----------------+------------------------------+---------------------------------+-------------
      .. .. x. ..      |  2  0 | 288   *   *   * |   2  1   1   0   1  0   0  0 |  1  2  1  0   2   1  1  0  0  0 | 1  1  2  1 0
demi( oo3oo4oo3oo&#x ) |  1  1 |   * 288   *   * |   0  0   2   2   2  0   0  0 |  0  0  1  1   2   4  1  0  0  0 | 0  2  1  2 0
      .. .s4.o ..      |  0  2 |   *   * 144   * |   0  0   0   0   2  0   2  2 |  0  0  0  0   1   2  2  1  1  2 | 0  1  1  2 1
sefa( .s3.s .. ..    ) |  0  2 |   *   *   * 288 |   0  0   0   1   0  1   2  0 |  0  0  0  1   0   2  0  2  0  1 | 0  2  0  1 1
-----------------------+-------+-----------------+------------------------------+---------------------------------+-------------
      .. o.4x. ..      |  4  0 |   4   0   0   0 | 144  *   *   *   *  *   *  * |  1  1  0  0   1   0  0  0  0  0 | 1  1  1  0 0
      .. .. x.3o.      |  3  0 |   3   0   0   0 |   * 96   *   *   *  *   *  * |  0  2  1  0   0   0  1  0  0  0 | 1  0  2  1 0
demi( .. .. xo ..&#x ) |  2  1 |   1   2   0   0 |   *  * 288   *   *  *   *  * |  0  0  1  0   1   1  0  0  0  0 | 0  1  1  1 0
sefa( os3os .. ..&#x ) |  1  2 |   0   2   0   1 |   *  *   * 288   *  *   *  * |  0  0  0  1   0   2  0  0  0  0 | 0  2  0  1 0
sefa( .. os4xo ..&#x ) |  2  2 |   1   2   1   0 |   *  *   *   * 288  *   *  * |  0  0  0  0   1   1  1  0  0  0 | 0  1  1  1 0
      .s3.s .. ..      |  0  3 |   0   0   0   3 |   *  *   *   *   * 96   *  * |  0  0  0  1   0   0  0  2  0  0 | 0  2  0  0 1
sefa( .s3.s4.o ..    ) |  0  3 |   0   0   1   2 |   *  *   *   *   *  * 288  * |  0  0  0  0   0   1  0  1  0  1 | 0  1  0  1 1
sefa( .. .s4.o3.o    ) |  0  3 |   0   0   3   0 |   *  *   *   *   *  *   * 96 |  0  0  0  0   0   0  1  0  1  1 | 0  0  1  1 1
-----------------------+-------+-----------------+------------------------------+---------------------------------+-------------
      o.3o.4x. ..      |  8  0 |  12   0   0   0 |   6  0   0   0   0  0   0  0 | 24  *  *  *   *   *  *  *  *  * | 1  1  0  0 0  cube
      .. o.4x.3o.      | 12  0 |  24   0   0   0 |   6  8   0   0   0  0   0  0 |  * 24  *  *   *   *  *  *  *  * | 1  0  1  0 0  co
demi( .. .. xo3oo&#x ) |  3  1 |   3   3   0   0 |   0  1   3   0   0  0   0  0 |  *  * 96  *   *   *  *  *  *  * | 0  0  1  1 0  tet
      os3os .. ..&#x   |  1  3 |   0   3   0   3 |   0  0   0   3   0  1   0  0 |  *  *  * 96   *   *  *  *  *  * | 0  2  0  0 0  tet
      .. os4xo ..&#x   |  4  2 |   4   4   1   0 |   1  0   2   0   2  0   0  0 |  *  *  *  * 144   *  *  *  *  * | 0  1  1  0 0  trip
sefa( os3os4xo ..&#x ) |  2  3 |   1   4   1   2 |   0  0   1   2   1  0   1  0 |  *  *  *  *   * 288  *  *  *  * | 0  1  0  1 0  squippy
sefa( .. os4xo3oo&#x ) |  3  3 |   3   3   3   0 |   0  1   0   0   3  0   0  1 |  *  *  *  *   *   * 96  *  *  * | 0  0  1  1 0  trip
      .s3.s4.o ..      |  0 12 |   0   0   6  24 |   0  0   0   0   0  8  12  0 |  *  *  *  *   *   *  * 24  *  * | 0  1  0  0 1  ike
      .. .s4.o3.o      |  0  4 |   0   0   6   0 |   0  0   0   0   0  0   0  4 |  *  *  *  *   *   *  *  * 24  * | 0  0  1  0 1  tet
sefa( .s3.s4.o3.o    ) |  0  4 |   0   0   3   3 |   0  0   0   0   0  0   3  1 |  *  *  *  *   *   *  *  *  * 96 | 0  0  0  1 1  tet
-----------------------+-------+-----------------+------------------------------+---------------------------------+-------------
      o.3o.4x.3o.      | 96  0 | 288   0   0   0 | 144 96   0   0   0  0   0  0 | 24 24  0  0   0   0  0  0  0  0 | 1  *  *  * *  rico
      os3os4xo ..&#x   |  8 12 |  12  24   6  24 |   6  0  12  24  12  8  12  0 |  1  0  0  8   6  12  0  1  0  0 | * 24  *  * *  cube || ike
      .. os4xo3oo&#x   | 12  4 |  24  12   6   0 |   6  8  12   0  12  0   0  4 |  0  1  4  0   6   0  4  0  1  0 | *  * 24  * *  co || tet
sefa( os3os4xo3oo&#x ) |  3  4 |   3   6   3   3 |   0  1   3   3   3  0   3  1 |  0  0  1  0   0   3  1  0  0  1 | *  *  * 96 *  trippy
      .s3.s4.o3.o      |  0 96 |   0   0 144 288 |   0  0   0   0   0 96 288 96 |  0  0  0  0   0   0  0 24 24 96 | *  *  *  * 1  sadi


Its facets thus are: 1x rico as top base, 24x cube || ike, 24x co || tet, 96x trippy (here coming in as {3} || tet, for sure), and 1x sadi as bottom base. - Its polyhedral elements, now given cummulative, are 24x cube, 24x co, 24x ike, 288x squippy, 312x tet, and 240x trip. - It uses 1152 triangles and 432 squares. - It has a total of 1008 edges and 192 vertices.

Fascinating, ain't it?

--- rk

[Keiji]

That is a great find! I wonder, are we going to be searching for >4D CRFs too now?

[quickfur]

That is a great find! I wonder, are we going to be searching for >4D CRFs too now?

We haven't even finished enumerating 4D CRFs yet! At the very least, duoprism augments haven't been fully enumerated yet, and there may still be a number of interesting CRFs to be obtained from there (one example that comes to mind is the omni-5gon||10gon-augmented 10,20-duoprism, which should have the elongated pentagonal bicupola among its cells -- it also has a variation that sports elongated pentagonal gyrobicupolae).

As for >4D CRFs, we've already found a few interesting ones, like the 5D teddies (including the 24-cell 5-teddy, and the 600-cell 6-teddy, the latter of which sports a 600-cell, 600 tetrahedral teddies, 120 icosahedral pyramids, and a rectified 600-cell: quite an impressive CRF, I must say!). I think there should be quite a few interesting ones based on the self-duality of the 24-cell -- a rather unique property among the regular polytopes!

[Klitzing]

The easiest example here would be the icositetrachoral antiprism, cf. icoap = ico || dual ico!

--- rk

[quickfur]

The easiest example here would be the icositetrachoral antiprism, cf. icoap = ico || dual ico!

--- rk

Well, that would be the trivial example. I was thinking more along the lines of exploiting the self-duality of the 24-cell to produce polytera that would have been non-CRF had they been based on non-self-dual polychora.

Nevertheless, ico||dual ico is an interesting shape. It would be a true 5D antiprism that is directly analogous to the 3D antiprisms! So it should contain two 24-cells, 48 bisected 16-cells (==octahedral pyramids), and 192 5-cells. Did I miss anything?

[Klitzing]

Just click to the provided link - you'll see that you're totally right, indeed!
--- rk

[quickfur]

Yay! I wasn't sure I didn't miss anything, because I'm not as facile with 5D constructions than 4D constructions. The lacings from cells to vertices are easy -- just octahedral pyramids -- but I wasn't immediately sure about the lacings between edges and ridges. For a moment I thought there would be some manner of duoprism cells, but then I realized that the self-duality of the 24-cell means that edges are laced with ridges interchangeably, so there is only 1 type of cell. So then it's just a matter of figuring out what line||triangle is, and it's pretty easy to figure out that it must be a 5-cell.

I'll probably have a harder time with 5D segmentotera with non-self dual bases, though.

[quickfur]

[...]
J91 and J92 are however very interesting, as they are intimately connected to other CRF's.
The bilunabirotunda (J91) has three verfs: 3.5.3.4; 3.5.3.5 and 5.3.5 . Of these, 3.5.3.5 is transitive to the vertex of an icosidodecahedron (not only by polyhedron count, but also by dihedral angles), and 5.3.5 is transitive to the vertex of a metabidiminished icosahedron.
What is interesting, is that the (3,5)-dihedral angle of a metabidiminished icosahedron( arccos(-sqrt((5-2sqrt(5))/15)) ) and the (3,5)-dihedral angle of a icosidodecahedron ( arccos(-sqrt((5+2sqrt(5))/15)) ) add up to the (5,5)-dihedral angle of a dodecahedron( arccos(-sqrt(5)/5).
The triangular hebesphenorotunda has verfs 3.5.3.4; 3.5.3.5; 3.3.3.5 and 3.3.4.6 . Of these, 3.5.3.4 is transitive to the 3.5.3.4 of J91, 3.5.3.5 is transitive to the vertex of a icosidodecahedon (and tus to the 3.5.3.5 of J91), and 3.3.3.5 is transitive to a pentagonal antiprism (and thus the (3,5)-dihedral angle of this vertex equals the (3,5)-dihedral angle of the 3.5.5 vertex of J91)
[...]

Whoa. This is very interesting!! This means that J91 and J92 can be thought of as the result of cutting out pieces of other CRF polyhedra's surface, and gluing them together, possibly adding some additional pieces to close up the shape. From your observations, J91 can be obtained by cutting out two 3.5.3.5 strips from the icosidodecahedron's surface, and gluing them together, while closing up the rest of the shape with line prisms (aka squares) and line wedges (aka triangles). This makes me wonder if we can do the same with 4D CRFs. Can we cut out some swaths of polyhedra from the surface of an existing CRF (or maybe uniform), such that they can be glued together and closed up by adding a few more CRF polyhedra?

I think searching for these kinds of constructions might be easier than the brute-force try-all-combinations approach to find crown jewels. Combinatorial explosion just makes that kind of approach intractible in the general case. Whereas using a pre-assembled surface, so to speak, allows us to work at a higher level than just trying to fit individual Johnson solids together.

In fact, some of the CRF diminishings of uniform polychora may be understood in this way: for example, the 600-cell wedges that I discovered can be thought of as cutting out lune-shaped strips from the 600-cell's surface, and closing them up with pentagonal rotundae and pentagonal pyramids. What if we cut out other kinds of strips from the 600-cell's surface, and find ways of gluing the pieces together?

P.S. More importantly, working with preassembled strips allows us to find pieces of existing CRF surfaces that have some common sub-sections -- like J92 having bits that look both like pentagonal antiprisms and J91: these cell configurations are a kind of hint that we can merge the common sections of these surfaces together to produce a hybrid surface made from two different existing CRFs, with the common part bridging between them. These constructions are probably more likely to produce CRF results than assembling random CRF polychora together, it would seem, at least intuitively.

[quickfur]

[...] I can't find a way of constructing J86 up to J89 from other CRF's
[...]

This morning I was looking at the diminished icosahedron (or gyroelongated pentagonal pyramid, J11), and I suddenly realized that if you make a cut that removes 3 triangles adjacent to the pentagon and cuts the pentagon across one of its chords, you'll end up with a non-CRF solid that has 12 triangles and two quadrilaterals (basically truncated pentagons). Since vertices surrounded by >3 faces are flexible (i.e., dihedral angles between the faces can be varied by distorting the solid), you can "squeeze" the long edge that join the two quadrilaterals until it becomes the same length as the other edges. The quadrilaterals then become squares, and the result is none other than the sphenocorona, J86.

This suggests another avenue for discovering 4D CRFs: make cuts that are initially non-CRF, but that can be distorted in such a way that it becomes CRF. Perhaps we should be looking at non-CRF cuttings of polychora like the 600-cell, because all the 600-cell's edges has degree >3, which means there's some extra degrees of freedom for distortion that may give enough leeway to distort a non-CRF cutting into a CRF.

[Klitzing]

That is a great find!

Thanks, Keiji!

I wonder, are we going to be searching for >4D CRFs too now?

Why not? Sure, we did not finish 4D (and will not in near future), but that does not mean to close ones eyes from higher dimensions, ain't it? We already have several interesting finds (beyond the several more obvious ones).

E.g., even so being more or less "obvious", those 2 still are kind of very special in directly extending the antiprism idea to 5D:
pen || dual pen = xo3oo3oo3ox&#x - coming out to be nothing but tac (triacontaditeron, pentacross, x3o3o3o4o)
ico || dual ico = xo3oo4oo3ox&#x

Similarily, but even less obvious ones:
hex || gyro hex = xo3oo3ox *b3oo&#x - coming out to be nothing but hin (hemipenteract, x3o3o *b3o4o)
rap || inv rap = oo3xo3ox3oo&#x - coming out to be nothing but dot (dodecateron, o3o3x3o3o)

But then those recent findings of mine:
rico || sadi = os3os4xo3oo&#x
{4} || ortho tepe = x2x2o || xo2ox2xx&#x = xxo2xox2oxx&#x (as lace simplex, not as lace tower: i.e. not using "&#xt" here!)

--- rk

[quickfur]

[...]
I wonder, are we going to be searching for >4D CRFs too now?

Why not? Sure, we did not finish 4D (and will not in near future), but that does not mean to close ones eyes from higher dimensions, ain't it? We already have several interesting finds (beyond the several more obvious ones).

I did discover some interesting trends about >4D CRF segmentotopes with n-cube symmetry; did you have a chance to review it? Basically, I discovered that many families of such segmentotopes only go up to 19D, and from 20D onwards, only 3 classes exist (and only 2 distinct heights exist: 2 and sqrt(2)). I also found an interesting boundary at 8D for segmentotopes with n-simplex symmetry, though I was not able to rigorously prove my results for the latter case.

E.g., even so being more or less "obvious", those 2 still are kind of very special in directly extending the antiprism idea to 5D:
pen || dual pen = xo3oo3oo3ox&#x - coming out to be nothing but tac (triacontaditeron, pentacross, x3o3o3o4o)

Conjecture (almost certainly a theorem, but I don't have a proof): in any dimension n, (n-1)-simplex || (n-1)-simplex = n-cross.