student91 wrote:There is a challenge for this generalization to work:
when you replace all octahedra of o4o3x3o with icosahedra to get o4s3s3s, you do not change the symmetry very much ([4,3]->[4,3][sup]+[\sup]+some extra symmetry for free).
Notice that here, [4,3] is the actual symmetry of the octahedra in the o4o3x3o, i.e. all symmetries that leave the o3x3o invariant, also leave the complete o4o3x3o invariant.
What this means is, when you change o3x3o into s3s3s, you don't need to care about how the ox3xo&#xt is oriented when you change it into a ofox3xofo&#xt, because the symmetry of ofox3xofo&#xt is almost the same as before.
However, when you change e.g. ox2xo&#xt into oBox2xoBo&#xt (for some weird number B), you do get a much smaller symmetry, and thus you do need to think about the orientation of the ox2xo&#xt's before expansion.
Of course, this is very well possible. All we need to do is think of an orientation for the tets in either x3o3o3o or o4o3o3x (I guess s4o3o3o will give such an orientation quite easilly, I will look into this)
quickfur wrote:[...]I just realized a fatal oversight in my original guess, though. In the 24-cell -> snub 24-cell construction, the edges of the octahedron are partitioned into 1 vertex each, so that 12 edges -> 12 vertices of the icosahedron. However, there is no analogous construction from the tetrahedron to snadow, because the tetrahedron has only 6 edges, yet snadow has 8 vertices. So there is no actual analogy here; the snub line antiprism construction is a different analysis of the octahedron -> icosahedron transformation, and doesn't apply directly to the snub 24-cell construction. So it's unlikely that a parallel construction with snadow will yield CRF holes in the result.
Still, the snub (X) antiprism construction seems very compelling, and I can't help wondering if there is some construction in which such an analogy would produce a CRF. If we can find a way to make it work, it could give us a handle on building a CRF from the snub square antiprism too. Thinking along these lines, apart from the snub 24-cell construction which now seems unlikely to be fruitful, I'm wondering about existing CRFs in which the octahedron functions as a triangular antiprism lacing cell in some kind of sandwich-like construction. The bisected 24-cell, for example, can be thought of as o4o3x || o4x3o with some lacing cells being octahedra serving as triangular antiprisms (o3x || x3o). Is there an analogous CRF in which there are icosahedral cells which function as snub triangular antiprisms in the same positions? I.e., replace the o3x || x3o with icosahedra, possibly modifying the top/bottom cells to remain CRF? Such a CRF would have 8 icosahedra as lacing cells. If it exists, we may be able to extract some kind of analogy or hint as to what kind of modifications are necessary in transforming o4o3x || o4x3o to this CRF, and perhaps it could give us a pathway to a similar construction involving tetrahedra -> snadow? Or, indeed, square antiprisms -> snub square antiprisms (e.g., in x4o3o || o4x3o).
quickfur wrote:Your "swirly-whirly" function is just the symmetry subgroup of the Hopf fibration of the 3-sphere. It features prominently in 4D geometry, especially in the uniform polytopes, because they can be thought of as discrete approximations of the surface of the 3-sphere, and their high level of symmetry makes them isomorphic to discrete subgroups of the Hopf fibration.
I think your idea is worth investigating, though I'm a bit doubtful about whether it will produce CRFs containing snadow, because the high level of symmetry generated by these subgroups impose a lot of restrictions on what dichoral angles are possible on the surface elements. So unless there is some hidden connection I'm not aware of between snadow and the Hopf fibration, I'd be quite surprised if such a construction could close up in a CRF way. My idea about using modified bistratic CRFs to find snadow-containing CRFs was based on the idea that a lower-symmetry construction would (hopefully!) grant more leeway in what cell shapes can close up in a CRF way.
But then again, you never know... before CRFebruary we all thought that J91 and J92 were too strange to close up in a CRF way, yet look at how many CRFs we have found since then that contain them! While I don't expect a similar large number of snadow-based CRFs, I'm hopeful that there should be at least a small number other than the snadow-prism. Among the "crown jewel" Johnson solids, snadow is one of the more symmetrical ones, so it would seem to suggest that it's more likely to have some way to close up in a CRF way than stranger cells like sphenocorona or sphenomegacorona.
Actually, we have been discussing this before. This then led to the conclusion that one needs to consider the biggest prime number that any exponent is divisible by.quickfur wrote:[…]But if we allow nested square roots, then we'd have to allow sphenocorona, because it involves degree 4 polynomials (or was it degree 8? I don't remember now -- but it's an even number). But I'm highly doubtful that sphenocorona is "constructible" in the sense you're describing.
Weird, the Wikipedia-page on the snub disphenoid has a similar construction, and that one does include cube roots. This means something has to be wrong somewhere. Best way to check this I think is by looking if all distances work out to be 1, when calculated the normal way.Marek14 wrote:Well, looking through here (viewtopic.php?f=32&t=1927&p=20187), it looks like snub disphenoid (and sphenocorona) are actually constructible. Though they yield weird numbers.
Marek14 wrote:Well, x^6 - 1 = 0 is a trivial example of a sixth-degree equation solvable with square roots. Still, I might have made some mistake in my original calculation...
Marek14 wrote:Well, x^6 - 1 = 0 is a trivial example of a sixth-degree equation solvable with square roots. Still, I might have made some mistake in my original calculation...
student91 wrote:Marek14 wrote:Well, x^6 - 1 = 0 is a trivial example of a sixth-degree equation solvable with square roots. Still, I might have made some mistake in my original calculation...
However, your result concerning the sphenocorona surely is correct! I asked Wolfram for coordinates of the sphenocorona, and he gave me exact, constructible coordinates .
So according to wolfram, the sphenocorona can be given as oaox2xxbo&#xt, with a=(6+sqrt(6)+y)/15≈1.70545388569283371, b=(9-sqrt(6)+y)/15≈1.57885525332174329772195, and the newly introduced y=2sqrt(213-57sqrt(6))≈17.1323185426093275640265628819748750.
This means that it is a little bit more likely that a polytope with sphenocoronae exists than a polytope with snub disphenoids.
Still working on a coordinate-agnostic approach though. I am experiencing problems exactly at the point where I try to impose local constraints. If you e.g. want to impose local angle defects, you have to find a way to calculate the ditopal angles of the surtopes, solely from the representation you have. Therefore, in such an approach you do need to include enough beforehand to derive different sets of polytopes (non-abstract, constructible, ones with a defineable dimension, etc.).
So actually, I want a generalization to be used to prove things, rather than a representation that does not encompass all possible things that I think should be called "polytopes".
I was thinking of defining a polytope as a set of vertices S, together with a distance-function f:S×S -> R, that gives the distance of two points s1 and s2.
However, this does not yet seem to be enough to define a polytope. I think when we take (S,f,r), where S and f are as before, and r is some as-of-yet-ill-defined notion of curvature (with r=0 for Euclidean space), any polytope given as such is unique up to isomorphism. (where an isomorphism is a function g:S->S with f(s1,s2)=f(g(s1),g(s2)))
quickfur wrote:The idea did occur to me, though, that rather than use the crown jewel Johnsons as cells, perhaps a more likely place to find 4D crown jewels would be to look for analogous constructions using simpler 4D cells. That's why yesterday I was musing about whether it's possible to insert a triangular prism into an icosahedral dipyramid such that the resulting deformation will close up in a CRF way. In some abstract, ill-defined sense, one might think of it as using the 4D geometry itself to produce a crown jewel shape, rather than start with a 3D crown jewel already "using up" some degrees of freedom, then try to close it up in 4D which requires even more degrees of freedom which there may not be "enough" of.
Nevertheless, I still can't help wondering if all of this is just a figment of our imagination (or lack thereof!), and that perhaps there is a way to close up a CRF containing snub disphenoid cells, just that we haven't thought of it yet.
o o | o (o) | o
x A | x x | x o
where A>f>B
o
x f o
o
x B
o
o o o
x f o x A
o o o
o | x
o C o | o D o
o o | o o
o o | o
f | B o
x (x)| x
f | B o
o o | o
(o) (o) |
f | x
x x | x x
f | D
o o | o o
o o
f
x x
becomes:
o o
C x
x
becomes:
o o
x x x which is coplanar
o o | o o | x x
f | C x | B
x x | x | x x
f | C x | A
o o | o o | o o
o | o o | o //squippy -> triangular bipyramid
o u o | o x | o D
o
o | o o | o oct -> pentagonal bipyramid
o u o | o C | o f x
o | o o | o
o4o
o4o x4o o4o
o4o o4o
o4o A4o as A approaches o, this becomes a square :S
o4o
o4o B4o
o4o as B approaches some value, the distance between the leftmost and rightmost o4o might become one, which may result in something cool?
o4o | o4o o4o
o4o x4o o4o | o4o A4o
o4o | o4o o4o
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