wendy wrote:This happens. You can use longer edges for the pyramid.
What's interesting is this. You can make a cubic arrangement of spheres in N dimensions. In four dimensions, you can have two of these in the same space. By seven dimensions, you can have eight of them. In eight dimensions, this jumps to 16. By 24 dimensions, you can havd 16777216 of them.
Cubics are horibly ineffective for packing spheres.
wendy wrote:This happens. [...]
2D: x || x : height = 1
3D: x4o || o4x : height = 0.840896
4D: x4o3o || o4o3x : height = 0.676097
5D: x4o3o3o || o4o3o3x : height = 0.45509
6D: x4o3o3o3o || o4o3o3o3x : height is imaginary (no unit lacing possible)
Klitzing wrote:[...]
In this spirit I now considered the hypercubical antiprisms too.
- Code: Select all
2D: x || x : height = 1
3D: x4o || o4x : height = 0.840896
4D: x4o3o || o4o3x : height = 0.676097
5D: x4o3o3o || o4o3o3x : height = 0.45509
6D: x4o3o3o3o || o4o3o3o3x : height is imaginary (no unit lacing possible)
In fact, this behaviour was to estimate, as the circumradius of the orthoplex is 1/sqrt(2) for every dimension, while that of the hypercubes is sqrt(D)/2 for dimension D.
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quickfur wrote:Klitzing wrote:[...]
In this spirit I now considered the hypercubical antiprisms too.
- Code: Select all
2D: x || x : height = 1
3D: x4o || o4x : height = 0.840896
4D: x4o3o || o4o3x : height = 0.676097
5D: x4o3o3o || o4o3o3x : height = 0.45509
6D: x4o3o3o3o || o4o3o3o3x : height is imaginary (no unit lacing possible)
In fact, this behaviour was to estimate, as the circumradius of the orthoplex is 1/sqrt(2) for every dimension, while that of the hypercubes is sqrt(D)/2 for dimension D.
[...]
Interesting! I didn't realize that as low as 6D, the n-cube antiprism is no longer possible (or no longer CRF).
This brings up an interesting topic related to what we've been discussing recently. I came up with the idea of defining "cupola" as a segmentotope in which the bottom cell is produced by adding a ring to the CD diagram of the top cell. I didn't intend for it to be taken seriously, but in any case, this discussion made me consider the following:
Let's say we start with the CD diagram of some top cell, and make some changes to it (adding rings/removing rings). What are the conditions in which such ringings will admit CRF segmentotopes? I do not have proof, but I believe that the case of adding a single ring to the CD diagram can always be made CRF, regardless of dimension. By the same token, removing a ring should also be CRF-able. But the case of removing 1 ring and adding 1 ring on another node seems to be more complicated. As we see above, in 6D you can no longer do this with x4ooooo || o4oooox. In 4D, we already can't have o5ox || o5xo be made CRF, although o4ox || o4xo is still possible (I think!). And I think x5oo || o5xo is still possible, too.
So the question is, what are the conditions in which removing and adding a ringed node to the CD diagram of the top cell will produce a bottom cell that can form a unit lace prism (i.e. a CRF segmentotope)? It seems that the relative positions of the unringed/ringed nodes play an important role here -- I believe that x4ooooo || o4xoooo should still be CRF-able; I could be wrong, but if I'm right, then where is the "breaking point"? Can x4oooo || o4oxooo be CRF-able? What about x4oooo || o4ooxoo? Or x4oooo || o4oooxo?
Another interesting thought: even though o4ox || x4ox is non-CRF, it can be made CRF by inserting an intermediate layer: o4ox || o4oq || x4ox, which is the bisected o4oxx. This essentially amounts to interfacing the top cell with the bottom cell by inserting a bunch of square pyramids so that we "bridge" the overly-long distance between the top cell's vertices and the bottom cell's vertices. What intermediate layers can be inserted into, say, x4ooooo || o4oooox so that the resulting lace tower will be CRF?
Klitzing wrote:[...]
Just to provide a small input to that. Probably you heard about the Stott-addition. This is kind of pulling things apart and introducing new stuff inbetween. E.g. you take an octahedron, pull the triangles out untill the tips are exactly 1 unit apart, so you could insert squares between the vertices (infact the vertex figure), and further squares between formerly incident edges. So you get the sirco. In terms of Dynkin diagrams this is just the ringing of a further node: x3o4o -> x3o4x.
In conjunction with lace prisms this is even more interesting. Consider 3d first. The trianle lacings of a tricup are exactly parallel to those of the triangular pyramid (tet). So esp. the size of the lacing edges, resp. the height of the lace prism will be exactly te same.
Putting together both informations, you could consider the lace prism in its Dynkin-type description, and un-ring all nodes where both, the top-node and the bottom-node is ringed simultanuously. This would not affect the lace prism height at all! Or, taken the other way round: if one exists, the other would too, or vice versa.
[...]
quickfur wrote:Klitzing wrote:[...]
Just to provide a small input to that. Probably you heard about the Stott-addition. This is kind of pulling things apart and introducing new stuff inbetween. E.g. you take an octahedron, pull the triangles out untill the tips are exactly 1 unit apart, so you could insert squares between the vertices (infact the vertex figure), and further squares between formerly incident edges. So you get the sirco. In terms of Dynkin diagrams this is just the ringing of a further node: x3o4o -> x3o4x.
I actually independently (re)discovered Stott-addition while trying to find a general scheme for deriving coordinates for n-cube uniform truncations. So thanks, now I know what it's called.In conjunction with lace prisms this is even more interesting. Consider 3d first. The trianle lacings of a tricup are exactly parallel to those of the triangular pyramid (tet). So esp. the size of the lacing edges, resp. the height of the lace prism will be exactly te same.
Yeah I noticed that recently. Which has far-reaching consequences for CRFs, because that means that if some given polychoron P can be CRF-augmented with some polyhedral pyramid Q, then a suitably expanded version of P can also be CRF-augmented with a suitably expanded version of Q. One consequence that I've noticed is that the 1633 m,n-duoprism augmentations with n-prism pyramids have corresponding counterparts in m,2n-duoprism augmentations with 2n-prism||n-gon, so there are at least another 1633 CRFs in the latter category. Of course, in the case of duoprisms, because of the additional symmetry of the 2n-prism cells, the augments can have two distinct orientations, so this would lead to even more CRF augmented duoprisms in the latter category.
I think this larger set of augmented duoprisms should be easily reachable via computer enumeration (the limited symmetry of duoprisms makes exhaustive enumeration relatively easy). I should work on this, one of these days, to count the exact number of CRF augmentations possible.Putting together both informations, you could consider the lace prism in its Dynkin-type description, and un-ring all nodes where both, the top-node and the bottom-node is ringed simultanuously. This would not affect the lace prism height at all! Or, taken the other way round: if one exists, the other would too, or vice versa.
[...]
Yes, that is indeed very interesting. So that means if oPoQo...oRxSo... || oPoQo...oTxUo... is CRF in n dimensions, then any of the remaining 2^(n-2) ringings of corresponding top/bottom nodes should also be CRF.
So actually, this gives us a nice handle on enumerating lace prisms of this sort. We may say that the lace prisms with no common ringed nodes are the "basic" or "fundamental" lace prisms. Once we have fully enumerated all of the fundamental lace prisms, all the others can be generated by adding rings to corresponding nodes in the top/bottom cells. So then the problem of enumerating CRF lace-prisms in n dimensions is reduced to enumerating all fundamental lace prisms, which are those where the top/bottom cells have no common ringed nodes in their CD diagrams.
Klitzing wrote:quickfur wrote:[...] that means that if some given polychoron P can be CRF-augmented with some polyhedral pyramid Q, then a suitably expanded version of P can also be CRF-augmented with a suitably expanded version of Q. One consequence that I've noticed is that the 1633 m,n-duoprism augmentations with n-prism pyramids have corresponding counterparts in m,2n-duoprism augmentations with 2n-prism||n-gon, so there are at least another 1633 CRFs in the latter category. Of course, in the case of duoprisms, because of the additional symmetry of the 2n-prism cells, the augments can have two distinct orientations, so this would lead to even more CRF augmented duoprisms in the latter category.
- Provided the angles would match for those gyrated ones. (Latteral facets, i.e. lacings, might have different angles.) -
[...]
Yes, that is indeed very interesting. So that means if oPoQo...oRxSo... || oPoQo...oTxUo... is CRF in n dimensions, then any of the remaining 2^(n-2) ringings of corresponding top/bottom nodes should also be CRF.
So actually, this gives us a nice handle on enumerating lace prisms of this sort. We may say that the lace prisms with no common ringed nodes are the "basic" or "fundamental" lace prisms. Once we have fully enumerated all of the fundamental lace prisms, all the others can be generated by adding rings to corresponding nodes in the top/bottom cells. So then the problem of enumerating CRF lace-prisms in n dimensions is reduced to enumerating all fundamental lace prisms, which are those where the top/bottom cells have no common ringed nodes in their CD diagrams.
Not too fast! What I meant is, provided oPoQo...oRxSo... || oPoQo...oTxUo... would exist, then too does xPoQo...oRxSo... || xPoQo...oTxUo..., and oPxQo...oRxSo... || oPxQo...oTxUo..., and xPxQo...oRxSo... || xPxQo...oTxUo..., etc. But nothing can be deduced therefrom about xPoQo...oRxSo... || oPxQo...oTxUo... ! - Nonetheless that observation already simplifies your new research topic a lot.
[...]
quickfur wrote:[...] Let's say we start with the CD diagram of some top cell, and make some changes to it (adding rings/removing rings). What are the conditions in which such ringings will admit CRF segmentotopes? I do not have proof, but I believe that the case of adding a single ring to the CD diagram can always be made CRF, regardless of dimension. By the same token, removing a ring should also be CRF-able. But the case of removing 1 ring and adding 1 ring on another node seems to be more complicated. As we see above, in 6D you can no longer do this with x4ooooo || o4oooox. In 4D, we already can't have o5ox || o5xo be made CRF, although o4ox || o4xo is still possible (I think!). And I think x5oo || o5xo is still possible, too.
So the question is, what are the conditions in which removing and adding a ringed node to the CD diagram of the top cell will produce a bottom cell that can form a unit lace prism (i.e. a CRF segmentotope)? It seems that the relative positions of the unringed/ringed nodes play an important role here -- I believe that x4ooooo || o4xoooo should still be CRF-able; I could be wrong, but if I'm right, then where is the "breaking point"? Can x4oooo || o4oxooo be CRF-able? What about x4oooo || o4ooxoo? Or x4oooo || o4oooxo?
[...]
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