## CRF pyramids in n dimensions

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### CRF pyramids in n dimensions

Perhaps this is already well-known to everyone, but this morning I suddenly realized that 4D is the last dimension where you can have a CRF n-cube pyramid. The sequence goes:

1D: the point pyramid = line segment = the only possible 1-polytope. (CRF)
2D: the line segment pyramid = triangle (CRF)
3D: the square pyramid (CRF)
4D: the cubical pyramid (CRF)

You might think that in 5D you would have a tesseract pyramid, but the counterintuitive thing (at least to me) is that it cannot be CRF. Why? Because the radius of the tesseract's vertices is equal to its edge length, which means that a CRF tesseract pyramid has its apex lying in the same hyperplane as the base, i.e., it's flat, not a real pyramid. The only way to have a full-bodied tesseract pyramid is to make it non-CRF: the base-to-apex edge length must be greater than the edge lengths in the base, so the sides of the pyramid will have non-equilateral, isosceles triangles. In 6D, the radius of the 5-cube's vertices is larger than its edge length, so there is no CRF 5-cube pyramid either (not even a degenerate, flat one, like in 5D).

IOW, from 5D onwards, there are no more CRF n-cube pyramids(!). I find this very counterintuitive. This means that in 5D and onwards, it is no longer possible to get CRF polytopes by augmenting n-cube facets of uniform polytopes.
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### Re: CRF pyramids in n dimensions

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.
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### Re: CRF pyramids in n dimensions

wendy wrote:This happens. You can use longer edges for the pyramid.

Yes, I know that. But it won't be CRF, which is what surprised me, since I was unconsciously assuming that one can always make pyramids from an n-cube with regular polygon 2-faces. But this only works for n≤4.

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.

Hmm. Do these sphere arrangements correspond with any interesting polytopes?
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### Re: CRF pyramids in n dimensions

The cases of the 7d and 8d instances are 3_21 and 4_21. These are gosset's polytopes in those dimensions.

In 24 dimensions, it's the leech lattice, which has twe 13.7800 = dec 196560 vertices. (i had to get the calculator to find what the decimal value was - no one uses the decimal system) I don't know what this makes.
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### Re: CRF pyramids in n dimensions

Interesting! I looked up the leech lattice in wikipedia, and found references to sublattices in 12 and 16 dimensions. Do these correspond with interesting polytopes as well?
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### Re: CRF pyramids in n dimensions

wendy wrote:This happens. [...]

In this spirit I now considered the hypercubical antiprisms too.

Code: Select all
`2D: x || x :                 height = 13D: x4o || o4x :             height = 0.8408964D: x4o3o || o4o3x :         height = 0.6760975D: x4o3o3o || o4o3o3x :     height = 0.455096D: 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.

--- rk
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### Re: CRF pyramids in n dimensions

Klitzing wrote:[...]
In this spirit I now considered the hypercubical antiprisms too.

Code: Select all
`2D: x || x :                 height = 13D: x4o || o4x :             height = 0.8408964D: x4o3o || o4o3x :         height = 0.6760975D: x4o3o3o || o4o3o3x :     height = 0.455096D: 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?
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### Re: CRF pyramids in n dimensions

quickfur wrote:
Klitzing wrote:[...]
In this spirit I now considered the hypercubical antiprisms too.

Code: Select all
`2D: x || x :                 height = 13D: x4o || o4x :             height = 0.8408964D: x4o3o || o4o3x :         height = 0.6760975D: x4o3o3o || o4o3o3x :     height = 0.455096D: 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?

This gives start to a nice further research, indeed!

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.

--- rk
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### Re: CRF pyramids in n dimensions

Although it's been some time since John Leech discovered this lattice (in relation to gray codes), it's been some time before the nature of the 23 deep holes (ie cells), has been evaluated.

It is pretty obvious ( well, sort of obvoius), that these holes are not generally cross-polytope shape, tp the extent that one might use a centralising product, because in 120 dimensions, this exceeds the maximum bounds.

I don't hold much that these figures will yield crf (convex - uniform surhedra) elements. The main reason for this is that there are other lengths involved, like sqrt(2), and more likely, it is the result of several intersections.

I have managed to get something like the todd-coxeter lattice (the 12d one), but the 16-d one eludes me.
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### Re: CRF pyramids in n dimensions

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.
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### Re: CRF pyramids in n dimensions

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.

- Provided the angles would match for those gyrated ones. (Latteral facets, i.e. lacings, might have different angles.) -
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.

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.

--- rk
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### Re: CRF pyramids in n dimensions

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.) -

You're right, the other facets (those introduced by the Stott expansion) have a different dichoral angle, so some of the gyrated augments might not be CRF. So those have to be checked.

[...]
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.
[...]

Sorry, I didn't explain myself clearly. What I meant was that given a fundamental lace prism, you can add a ring to the i'th node in the top face and a ring to the i'th node in the bottom face, and the result is still CRF. (For any i for which the i'th nodes in the top/bottom are not already ringed.) You can add any number of rings to the fundamental lace prism, as long as rings are always added to both the top and bottom nodes at the same position.

So if there are m nodes that are unringed in both the top and bottom facets (in the same positions), then that fundamental lace prism will give rise to (2^m)-1 additional non-fundamental lace prisms. If the fundamental lace prism is CRF, then so are these (2^m)-1 others. By enumerating just the fundamental lace prisms, we've saved ourselves the effort of going through (2^m)-1 non-fundamental cases for each fundamental.
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### Re: CRF pyramids in n dimensions

There is an interesting cluster of 'teddi's like this xfo3ooxPooo&t, for P=3, 4, 5.

What these give, is an x3oPo (T, O, I), then 4, 8 or 20 teddies, the lone triangle pointing at the top face. The base is an o3xPo, ie O, CO, ID. The balance of faces are oxPoo&t, which are pyramids of base 3, 4, 5.

You can apiculate (raise pyramids on the faces) many of the faces of these. An apiculation on a teddi gives a tri-diminished icosahedral pyramid, consists of pentagon-prisms × 3, and tetrahedra × 5.
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### Re: CRF pyramids in n dimensions

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?
[...]

This morning I decided to do a quick calculation to determine whether x4ooo... || o4xoo... will always be CRF-able. Turns out that this is only CRF-able up to 19 dimensions.

Proof. Given edge length 2, the coordinates of x4ooo... are apacs<1,1,1,1,...,1>, and the coordinates of o4xoo... are apacs<0,sqrt(2),sqrt(2),sqrt(2),...,sqrt(2)>. If we construct an n-dimensional segmentotope from these (n-1)-dimensional facets, then its coordinates will be <1,1,1,...,1,H> and <0,sqrt(2),sqrt(2),...,sqrt(2),0> for some H>0. Since the edge length is 2, the difference between these two vectors must have a magnitude equal to 2, since otherwise the result will not be CRF.

So, ||<1,1,1,...,H> - <0,sqrt(2),sqrt(2),...,0>||^2 < 2^2, which gives us: 1 + (n-2)(sqrt(2)-1)^2 + H^2 = 4. A little algebra gives us: H^2 = (9-4*sqrt(2)) + n(2*sqrt(2)-3). Since we require H>0, so we also have H^2>0, which means:

(9-4*sqrt(2)) + n(2*sqrt(2)-3) > 0
n(2*sqrt(2)-3) > -(9-4*sqrt(2))

Now, note that since 8 < 9, so 2*sqrt(2) < 3, and so 2*sqrt(2) - 3 < 0. So we have to invert the sense of the inequality when we divide by 2*sqrt(2)-3:

n < -(9-4*sqrt(2)) / (2*sqrt(2)-3)

A little more algebra gives:

n < 11 + 6*sqrt(2)

The RHS works out to be about 19.49, and therefore we conclude that n≤19. QED.

This is interesting, because it means that in 20D, n-cube || rectified n-cube cannot be CRF! However, n-cross || rectified n-cross is always CRF-able. Why? Because n-cross is o4oo...ox, which has coordinates <0,0,0,...,0,sqrt(2)> and rectified n-cross is o4o...oxo, which has coordinates <0,0,0,...0,sqrt(2),sqrt(2)>. Note that the number of non-zero elements is independent of dimension. So to construct the segmentotope, we simply append H to the n-cross's coordinates and 0 to the rectified n-cross's coordinates: <0,0,0,...0,sqrt(2),H> and <0,0,...0,sqrt(2),sqrt(2),0>, respectively. Then given the requirement that the difference between these two points must have magnitude 2, we get:

||<0,0,0,...0,sqrt(2),H> - <0,0,...sqrt(2),sqrt(2),0>||^2 = 2^2
2 + H^2 = 4
H^2 = 2
H = sqrt(2)

Since H is independent of dimension, the segmentotope exists in all dimensions ≥3.
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### Re: CRF pyramids in n dimensions

Just revisited the distinction of a bipyramid (or dipyramid) and the pyramid of a pyramid.
If you've some object X and you take the bipyramid, then this is nothing but pt || pseudo X || pt, i.e. you'll add one dimension.
But if you build the pyramid of X (so far you also get into one dimension plus), and build thereon a further pyramid, you'd obviously result within 2 dimensions plus!

Hence that latter one truely deserves an own name. Hedrondude here already proposed "scalene" as the generalisation of the 2D example, the general or scalene triangle. That then becomes a point scalene, i.e. a pyramid on a pyramid of a point. (That first pyramid just results in a line segment, and so the scalene here indeed is a general triangle. - But a scalene of a line then already becomes a tetrahedron, etc.

In this post I now want to deal with the scalene of a cube. (Well, Hedrondude would then abbreviate that fellow to "cubasc".)
For those of the toratopic threads one could rewrite that figure also into "|||>>".

Thus we consider first "|||>" = "cubpy", the mere pyramid on the cube. It can be done with unit edges throughout and its height then evaluates to exactly 1/2 of an edge length. Its cells clearly are one cube ("|||") and 6 square pyramids ("||>" = "squippy").

From that height it becomes obvious that the corresponding cubical bipyramid would have a tip to tip distance of one edge unit. - Nice coincidence, ain't it? - And that fellow then would have no cube any more, but now 12 square pyramids for cells.

Now let's consider one half of a "cubdipy" (cubical bipyramid). That spot within 5D, which has 1 unit distance both from the cubical vertices and from the tip of the cubpy (cubical monopyramid), then would be exactly the other tip of cubdipy! - Thus we have seen in a quite elementary way that "cubasc" in fact becomes degenerate, i.e. would have zero height only.

But still cubasc is a (even so degenerate) object of 5D! That is, its boundaries are still polychora: At the bottom we have one cubpy, and the lacing cells are one further cubpy (joining the cube to the new tip) and 6 "squascs" (square scalenes or square-pyramid-pyramids, i.e. "||>>"). - But from ist degeneracy on the other hand we derive that this set of boundaries becomes a double cover of the flat hull.

Thus we are interested first in that hull, a flat polyteron, thus a mere polychoron. That one clearly is nothing but our previous cubdipy, the cubical bipyramid. Thus we have found thereby 2 truely 4D decompositions of cubdipy into smaller parts (in the sense of an external blend): first we clearly have the antipodal decomposition into 2 cubpies. But on the other hand we likewise have the decomposition into 6 squascs. That latter one is kind of analogue to a 4D orange: narrow pieces connecting both tips in a way of a thickend up meridian.

And, for sure, from these decompositions in turn we get further flat (= degenerate) polytera: just adjoin either one on the one side to the cubdipy on the other. And our cubasc then becomes nothing but the blend of these 2, just by blending out right that additional cubdipy each.

--- rk
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### Re: CRF pyramids in n dimensions

Note that this topic on the rather tiny cubasc has an even larger counterpart!

Consider the tesseract ("tes" = "||||"). The regular, unit edged version also has a circumradius of 1 edge unit. Accordingly the pyramid of the tesseract ("tespy" = "||||>") will be degenerate as well. And moreover, the lacing polychora here are 8 cubpies.

On the other hand it also is known, that the augmentation of the tesseract by 8 cubpies will result in the icositetrachoron ("ico").

Thereby we get a further decomposition of ico for free: it can be glued up from 8 cubdipies! - and by means of the in the former mail mentioned decomposition of that one into 6 squascs each, we now get also a decomposition of ico into 48 squascs. Here the squares would outline the pseudo cubes (of the inscribed tes), and the 2 tips would connect the body center to the 8 "additional" vertices of ico.

And what, if we glue these squasc not around the diagonal of a cubdipy, but instead by two at their squares?

Well a squasc ("||>>") clearly is the pyramid of a squippy ("||>"). By adjoining these at the square, the 2 squippies recombine to an octahedron (or square dipyramid) again. Therefore we get a further decomposition of ico into 24 octahedral pyramids ("octpy"). That one is trivial, for sure. But once more it emphasises that the ico itself also has unit circumradius. And that thus a unit tes is truely inscribable into the vertex set of ico (in 3 different ways). - Well this leads astray, towards the idea of compounds...

--- rk
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