higher-dimensional CRFs

[mr_e_man]

It appears that the smallest angles are in pyramid-like things.

If we can find a polytope with circumradius slightly less than 1, then its pyramid should have very small ditopal angles at the base. Here are the circumradii of regular polytopes:

n-simplex: √[n/(2(n+1))]; approaches √[1/2] as n increases

n-cube: √[n/4]; approaches ∞

n-orthoplex (n≥2): √[1/2]

First I tried the (n-simplex, n-simplex)-duoprism pyramid. The duoprism has circumradius √[n/(n+1)] (which approaches 1, as desired), so the pyramid based on the duoprism has height √[1/(n+1)]. But I calculate the relevant ditopal angle as arccos√[1/(2n+1)], which is actually increasing toward 90°, not decreasing toward 0°. I think this is related to the fact that the inradius of the simplex is decreasing toward 0.

Next I tried the (n-simplex, square)-duoprism pyramid; that is the n-simplex prism prism pyramid. The duoprism has circumradius √[(2n+1)/(2n+2)], so the pyramid has height √[1/(2n+2)]. The ditopal angle in an n-simplex is arccos(1/n). The ditopal angles in an n-simplex prism pyramid, at the base, are arccos√[2/((n+1)(n+2))] and arccos√[(n+1)/(2(n+2))]. The ditopal angles in an n-simplex prism prism pyramid, at the base, are arccos√[1/(n+1)] and arccos√[(n+1)/(n+3)]. The latter is just what we were looking for: it decreases toward 0° as the dimension n increases.

So it is true, the infimum of A_∞ is 0°.

But can we construct a sequence of, say, CRF 6-polytopes, whose angles approach 0° ? If so, that would make it much more difficult to enumerate CRF 7-polytopes, as there'd be no limit on the number of 6-faces around a 4-face.

[mr_e_man]

Thanks to Klitzing's list, I found another potential example: The rectified n-simplex has circumradius approaching 1, so its pyramid has height approaching 0 . But I haven't calculated the ditopal angles.

[quickfur]

Thanks to Klitzing's list, I found another potential example: The rectified n-simplex has circumradius approaching 1, so its pyramid has height approaching 0 . But I haven't calculated the ditopal angles.

Wouldn't it have at least the height of the n-simplex itself? Since the rectified n-simplex is a Stott expansion of the n-simplex, its height (given a fixed edge length) cannot be less than the n-simplex itself.

[mr_e_man]

I'm talking about the pyramid based on the rectified n-simplex, not the rectified n-simplex itself.

[quickfur]

Ohh, right.

Makes me wonder what the structure of the general rectified n-simplex would be. In 3D, the rectified triangle is the dual triangle, so its pyramid is just a tetrahedron, 3+1 triangle faces. In 4D, it's the octahedral pyramid: 8 tetrahedra and 1 octahedron. In 5D, the rectified 5-cell pyramid would have 5 5-cells, 5 octahedral pyramids, and 1 rectified 5-cell. In 6D, the rectified 5-simplex pyramid would have 6 5-simplexes, 6 rectified 5-cell pyramids, and 1 rectified 5-simplex.

Aha, I see it now. In each dimension n, the rectified (n-1)-simplex pyramid would have n (n-1)-simplexes, which have decreasing height with n, and n rectified (n-2)-simplex pyramids, also with decreasing height, and a rectified (n-1)-simplex as facets. Since the lateral facets have decreasing height with n, this series of pyramids in fact has decreasing height that would appear to converge on 0? Not 100% sure but it certainly seems to be so.

Fascinating! So as the dimension increases these pyramids become shallower and shallower, so they could potentially augment a *lot* of CRFs and still remain convex. This does require the augment to fit on a rectified (n-1)-simplex facet, though. How many polytopes would have such cells? The n-cube truncates would, and the number of those increases exponentially with dimension. Which leads to the question: at what dimension does the rectified n-simplex pyramid become shallow enough that it can augment a birectified (n+1)-cube and still remain convex? That would be an interesting question to answer.

[quickfur]

More precisely, polytopes with CD diagram .No3o3o...o3x3o for N=3,4 would have rectified n-simplex facets. At which dimension does it become convexly augmentable?

[Klitzing]

The rectified n-simplex pyramid is nothing but
the vertex figure pyramid of the n+1-demihypercube!

Thence the base angles of those pyramids are nothing but
the complements of the dihedrals of the truncated demihypercubes.

--- rk

[quickfur]

the rectified n-simplex pyramid is nothing but
the vertex figure pyramid of the n+1-demihypercube!

--- rk

[Klitzing]

...
Fascinating! So as the dimension increases these pyramids become shallower and shallower, so they could potentially augment a *lot* of CRFs and still remain convex. This does require the augment to fit on a rectified (n-1)-simplex facet, though. How many polytopes would have such cells? The n-cube truncates would, and the number of those increases exponentially with dimension. Which leads to the question: at what dimension does the rectified n-simplex pyramid become shallow enough that it can augment a birectified (n+1)-cube and still remain convex? That would be an interesting question to answer.

The lacing facets here will be simplices and just those very rectified simplex pyramids - of one dimension less.

While the base size of a simplex-pyramid increases slowly and the height decreases faster, it is indeed that the base dihedral between the simplex facets and the base runs down to zero - just as already mentioned. On the other hand, the base size of the other lacing pyramids however increases much faster, that is those lacing facets dig much deeper into the body. Thence there the dihedral between the base and those lacings would have the counterintuitive behaviour to increase with the dimension instead. In fact the base dihedrals have size arccos[(n-2)/n] at the simplex and arccos[1/sqrt(n)] for the facetal rectified simplex pyramids.

So indeed the dihedrals between lacing simplex and base rectified simplex would allow for arbitrary many such components around an ridge, however the neighbouring ridge would prohibite that at the same time!

--- rk

[quickfur]

[...]
Thence there the dihedral between the base and those lacings would have the counterintuitive behaviour to increase with the dimension instead.
[...]
So indeed the dihedrals between lacing simplex and base rectified simplex would allow for arbitrary many such components around an ridge, however the neighbouring ridge would prohibite that at the same time!
[...]

Aha, so it can be understood as the analogue to the oblong pyramid situation: take a long rectangle and erect over it a shallow pyramid. The dihedral angle between the narrow triangles and the base will decrease as the long edge of the rectangle lengthens, but the dihedral angle between the wide triangles and the base will increase as the short edge of the rectangle narrows. Thanks to the two dimensions of the rectangle these two things happen simultaneously.

Of course, in our case it's a matter of multiple dimensions in place of the long edge, but the analogy holds.

[Klitzing]

Yes, exactly, Quickfur.

The most counterintuitive then would be: you always could consider a rectified simplex bipyramid.
For if you'd take the limit of n towards infinity of that shape, then some dihedrals at its equator would run towards 0°,
while the other ones at the same time would run towards 180°!

--- rk

[mr_e_man]

For a fixed dimension n, the total number of CRF n-polytopes with small 2-faces (e.g. 50-sided polygons at the largest) is finite. Any finite set of positive numbers cannot approach 0 . So, if we want to find ditopal angles approaching 0, without increasing the dimension, we must consider polytopes with large 2-faces.


The 3D ones are very familiar: prisms and antiprisms. Given edge length 1, an n-gon prism has height 1, and an n-gon antiprism has height

h = √3/2 √[(2 + 4 cos 180°/n)/(3 + 3 cos 180°/n)]
= √[(1/2 + cos 180°/n)/(1 + cos 180°/n)]
= √[1 - 1/(2 cos 90°/n)²]

which increases with n, approaching √3/2.

higher-dimensional CRFs 1.png (29.41 KiB) Viewed 28056 times

The 4D ones are also known. Let's consider their "poke sections", perpendicular to an n-gon. The section of an (n,m)-duoprism is just an m-gon, with edge length 1. The section of an antiprismatic prism is a rectangle, with lengths 1,h,1,h. The section of a biantiprismatic ring, or antifastegium, is an isosceles triangle, with lengths 1,h,h.

higher-dimensional CRFs 2.png (13.76 KiB) Viewed 28056 times

Here I claim (tentatively) that these are the only possibilities for CRF 4-polytopes with large 2-faces.


Now to 5D. Clearly, any CRF 3-polytope with edge length 1 will be a section of a valid CRF 5-polytope; consider the prism product of an n-gon and that polyhedron. Also the polygons shown above, extruded into prisms with height 1, will be valid sections. Many of the CRF polyhedra have variants where some edges have length h; these may or may not be valid sections.

higher-dimensional CRFs 3.png (114.73 KiB) Viewed 28056 times

How exactly do these correspond to 5-polytopes? Well, each vertex (x,y,z) represents an n-gon with vertices at either

(R cos (2k)180°/n, R sin (2k)180°/n, x, y, z)

or

(R cos (2k+1)180°/n, R sin (2k+1)180°/n, x, y, z)

for various integers k, where R = 1/(2 sin 180°/n) is the circumradius of an n-gon. If two vertices in 3D are connected by an edge with length 1 (depicted blue), then the corresponding n-gons should have the same orientation; both use '2k', or both use '2k+1'. If two vertices in 3D are connected by an edge with length h (depicted orange), then the corresponding n-gons should have opposite orientations; one is the dual of the other.

Is a polyteron made in this way guaranteed to be CRF? It's not clear to me.

...Isn't this construction essentially a "lace hyper-city"?

Anyway, it doesn't look like any of these have angles approaching 0 (though I don't claim to have found all of them, and that pentagonal cupola has a very small angle, from 6.1655° to 10.8123° depending on n).


For 6D, I thought of taking the prism product of an n-gon and a cube, {n}×{4,3} = {n}×{}×{}×{}, and making a segmentotope by aligning vertices (as much as possible) over the centres of the tesseracts: {n}×{4,3} || dual{n}. This continues the dimensional analogy:

{n}||dual{n} = antiprism, with height near √[3/4]
{n}×{}||dual{n} = biantiprismatic ring, with height near √[2/4]
{n}×{4}||dual{n} = polyteron produced by the square pyramid shown above, with height near √[1/4]

But it turns out that {n}×{4,3}||dual{n} has its squared height approaching 0 from the wrong direction: it's negative! Of course that's because the tesseract has circumradius 1, and can't have a CRF pyramid.

[mr_e_man]

I guess that antiprism with orange lacing edges represents the (n,m)-duoantifastegiaprism.

And "projection" is the better notion than "section", here.

[mr_e_man]

What about a tetrakis cube?

The cube's edges have length 1, and the augmenting edges have length h, the height of the N-gon antiprism. (Note that the augmentations are nearly coplanar when N is large.)

Put 'xNo' at each of the 8 vertices of the cube, and put 'oNx' at each of the 6 augmenting vertices.

Does this make a valid CRF polyteron?

[mr_e_man]

But can we construct a sequence of, say, CRF 6-polytopes, whose angles approach 0° ? If so, that would make it much more difficult to enumerate CRF 7-polytopes, as there'd be no limit on the number of 6-faces around a 4-face.

...
Fascinating! So as the dimension increases these pyramids become shallower and shallower, so they could potentially augment a *lot* of CRFs and still remain convex. This does require the augment to fit on a rectified (n-1)-simplex facet, though. How many polytopes would have such cells? The n-cube truncates would, and the number of those increases exponentially with dimension. Which leads to the question: at what dimension does the rectified n-simplex pyramid become shallow enough that it can augment a birectified (n+1)-cube and still remain convex? That would be an interesting question to answer.

You were thinking of augmenting existing CRFs, which requires the sum of angles around a ridge to be less than 180°.
I was thinking of constructing completely new CRFs, which requires the sum of angles around a peak to be less than 360°.
These are different tasks.

[Klitzing]

Just to state that I recently got the idea for a full series of 7D scaliform CRFs, the "(N,M)-duoantifastegiaprismatic alterprisms":

Consider first the 4D (N,M)-duoprism xNo xMo, which clearly can occur, while keeping the symmetry, within 3 further orientations: xNo oMx, oNx xMo, and oNx oMx. Now place these 4 in a further 3D position space (orthogonally to that 4D structure space) within the vertex positions of a simplex. The respective pairwise distance clearly has to be taken such that it remains CRF, esp. all lacing edges still are unity too. To that end the distance between xNo xMo and xNo oMx clearly ought be the height of an M-antiprism, while the distance between xNo xMo and oNx xMo will be that of an N-antiprism. These four pairwise distances line out a skew tetragon similar to the zig-zag of an antiprism (which is slightly chiral if N and M would be different). The 2 bases of that digonal "antiprism" of position space then would be the lacing prism between xNo xMo and its bidual oNx oMx, i.e. the 5D (N,M)-duoantifastegium.

Because the lacing skew tetragon of position space generally has different side lengths whenever N and M is not the same, it happens that the 2 digonal bases (i.e. those 2 duoantifastegia) would not be exactly orthogonal aligned ahead of each other. This is what made me to put "antiprism" into quotes above, respectively to use "alterprism" within its general name instead.

For sure, as such it is just an ordinary member of the set of general lace simplices, in fact a lace simplex with 3D position space. But on the other hand it somehow is rather special as all 4 "vertices" (of position space) are occupied by essentially the same polytope, but then again all 4 of them are oriented differently! This as well then is a further reason for being named as an "alterprism". (Or here rather "altersimplex" if you'd like.)

Four of its facets are clearly obvious: those would be the four faces of position space, i.e. the trigonics from e.g. xNo xMo, xNo oMx, and oNx oMx. Surely those are no longer scaliform themselves. Additionally those "(N,M)-duoantifastegiaprismatic alterprisms" would have 2M lace simplices consisting of xNo o, oNx o, xNo x, and oNx x, plus further 2N similar ones using the M-symmetry instead.

As it turns out this series exists for any combination of N>2 and M>2 and never becomes somehow degenerate. In fact the side lengths of that lacing simplex of position space not only remain strictly positive always, the lacing skew tetragon in addition will never become flat either. As long as N and M are chosen integral, the whole resulting polyexon will clearly be convex, in fact by means of construction even a CRF. And because the structures situated at the "vertices" of the position space simplex are oriented orthogonal to that position space and further all 4 have the same shape each, just oriented differently (but in a symmetrically indistinguishable way), the whole thingy becomes scaliform. Finally, none of the facet types, mentioned above, themselves would qualify as scaliform. Therefore those such constructed polyexa cannot be uniform for any N and M, hence they are purely scaliform.

--- rk