Pyramid antiprisms

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Pyramid antiprisms

Postby quickfur » Mon Jul 16, 2012 7:07 pm

Question: is the square pyramid antiprism CRF?

The square pyramid is self-dual, so we may place a square pyramid and its dual in parallel hyperplanes and connect them with two more square pyramids (connecting apex of one pyramid to base of the other) and 8 tetrahedra (4 tetrahedra connecting each face of one pyramid to a corner of the other pyramid's base; another 4 tetrahedra to connect the side edges of one pyramid to the other). Topologically, I don't see why it shouldn't be possible to make all edges of equal length, and therefore the result should be CRF, but when I try to compute the coordinates, I can't seem to get the right edge lengths!

So, can the square pyramid antiprism be made CRF? Or does it only work with the triangular pyramid antiprism (aka the 16-cell)?
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Re: Pyramid antiprisms

Postby wendy » Tue Jul 17, 2012 6:44 am

It works with the square pyramid antiprism. The result is a cubic tegum, the tegum-product of a line of edge 1, and a cube. It's the sort of thing you stick on a tesseract to make a 24-choron, but you just paste the pyramids back to back.

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Re: Pyramid antiprisms

Postby quickfur » Tue Jul 17, 2012 2:56 pm

wendy wrote:It works with the square pyramid antiprism. The result is a cubic tegum, the tegum-product of a line of edge 1, and a cube. It's the sort of thing you stick on a tesseract to make a 24-choron, but you just paste the pyramids back to back.

W

Oh? I'm not sure if we're talking about the same object. The one I have in mind has 4 square antiprisms and 8 tetrahedra. Its cross-section with the hyperplane parallel to two opposite square pyramids is a kind of truncated square pyramid (i'm not sure truncated how far, maybe a rectified square pyramid of some sort?).
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Re: Pyramid antiprisms

Postby wendy » Wed Jul 18, 2012 7:39 am

It's actually a square antiprism tegum. It's back-to-back antiprism pyramids.

The thing works for triangles (16-chor), square and pentagons (with 4n tetrahedra and 4 p-pyramids.
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Re: Pyramid antiprisms

Postby wintersolstice » Wed Jul 18, 2012 12:04 pm

wendy wrote:It's actually a square antiprism tegum. It's back-to-back antiprism pyramids.


that would mean it's one of the pyramid forms :D (bipyramid)
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Re: Pyramid antiprisms

Postby quickfur » Wed Jul 18, 2012 2:04 pm

Hmm. Apparently there is more than one class of shapes that have 4 n-pyramids and 4n tetrahedra. The one I have in mind has coordinates that look like:

(±sqrt(2), 0, 0, 0)
(0, ±sqrt(2), 0, 0)
(0, 0, sqrt(2), 0)
(±1,±1, G, H)
(0, 0, G-sqrt(2), H)

for some unknown positive values of G and H.
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Re: Pyramid antiprisms

Postby quickfur » Thu Jul 19, 2012 6:12 pm

Gah, I'm an idiot. The square pyramid antiprism is definitely CRF. I couldn't find the right coordinates before because I accidentally set the height of the dual square pyramid to 1 instead of sqrt(2), so the numbers wouldn't work out. Anyway, I found the right coordinates now. The coordinates are as above, with G=1 and H=sqrt(2*sqrt(2)-1). Sorry for wasting everybody's time.

Here's a render of it:

Image

The highlighted (inverted) square pyramid is the dual of the square pyramid outlined in red. There are two more square pyramids, one on top of the dual, one below the red-outlined. There are 16 tetrahedra that fit into the gaps between these 4 square pyramids.

Anyway, I'm not sure why this CRF isn't in Klitzing's list. Is it because it doesn't inscribe a 3-sphere properly? I recently realized that Klitzing's segmentochora only covers CRF polychora that can be inscribed in a 3-sphere; not all of the Johnson solid prisms are included in his list. This means that the segmentotope-like CRFs are a superset of Klitzing's list. So the search isn't over yet!! There are yet more CRFs with vertices in two parallel hyperplanes to be found. It would appear that the square pyramid antiprism is one of them (unless I missed its listing in Klitzing's paper?).
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Re: Pyramid antiprisms

Postby quickfur » Fri Jul 20, 2012 12:26 am

Hahaha I've been making a fool of myself. :oops: Wendy was right all along; the square pyramid antiprism is the same as the square antiprism bipyramid. The pentagonal pyramid antiprism is also the same thing as the pentagonal antiprism bipyramid. Both are CRF. :)
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Re: Pyramid antiprisms

Postby Klitzing » Wed Sep 19, 2012 11:32 am

quickfur wrote:Hahaha I've been making a fool of myself. :oops: Wendy was right all along; the square pyramid antiprism is the same as the square antiprism bipyramid. The pentagonal pyramid antiprism is also the same thing as the pentagonal antiprism bipyramid. Both are CRF. :)


This CRF is a nice find! The equivalence is clearly understood in terms of lace cities: the n-gonal pyramid can be given as a lace prism ox-n-oo&#x, i.e. as o-n-o (point) || x-n-o (n-gon). The according pyramid in dual positioning clearly would be o-n-x (dual n-gon) || o-n-o (point at the opposite side). So you would get for lace city (= stack of towers):
Code: Select all
o-n-o   o-n-x

x-n-o   o-n-o

And this very city, rotated by 45 degrees, shows: you would have 3 towers: o-n-o (point), o-n-x || x-n-o (n-antiprism), o-n-o (point). Thus indeed, it is the n-antiprism-dipyramid.

Next. You don't find that fellow within my list of segmentochorons. In the (unrotated) representation it would well be a lace prism (monostratic), not a true tower. So, why it isn't a segmentochoron?

In fact, it is the restriction having the edge length, connecting the (nearer) base vertices of either pyramid the same as those of the pyramid edges them selves, and further the edge length connecting the base vertices of one pyramid to the tip of the other as the same length as well. This is what defines the values of G and H. H just is the height of that lace prism (in fact twice the height, as your chosen edge length is 2). But G represents an shift of the pyramids out of their (vertex) circumcenter. Therefore the very alignment (shift) of the base polytopes disallows a single vertex distance, and so too the overall arangement of the 2 antiparallel, gyrated pyramids does not allow for one. This is why it is not a segmentochoron for n=4 and n=5. - For n=3 it would be, in fact, it just is a different description of the hexadecachoron!

But there is a small spin-off! If you would diminish one of the pyramids down to its base polygon, you would be allowed to shift that degenerate (flat) polytope out of its circumcenter without leaving the restriction of having a unique 3d circumcenter (of that base). So you migh, as in your coordinates) attach that total shift G to the diminished pyramid, and the other one would get shift-free. Accordingly you re-enter the realm of segmentochorons:
3g || gyro tet,
4g || gyro 4pyr,
5g || gyro 5pyr
all are listed in my enumeration!

By virtue of that lace city you even could understand the there being found equivalences:
3g || gyro tet = point || oct,
4g || gyro 4pyr = point || 4ap,
5g || gyro 5pyr = point || 5ap.

--- rk
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Re: Pyramid antiprisms

Postby Klitzing » Thu Sep 20, 2012 9:40 am

BTW, it just occured to me, as pyramids and cupolae would have parallel triangular faces and same heights, that the same trick would be possible with cupolae as well:
    n-cupola || inv gyro n-cupola.
Facet counts being: 4 n-cupolae, 2n squippies, and 2n trips (n=2, 3, 4, 5). Here spuippies (=4pyr) are achieved by elongating one base vertex of a tet to an own edge, and trips likewise are achieved by pulling one edge of a tet apart getting a square. (I.e. the former 4n tets of your fellows fall apart into 2 classes here.) Those are the induced effects onto the lacings for the Stott expansion of those bases.

The according general lace city would be:
Code: Select all
x-n-o   x-n-x

x-n-x   o-n-x


Sure, for the same reasons only the case n=2 would be a segmentochoron (which is number 4.13 of my listings - so we got that exceptual fellow into some broader context finally, hehe). Or, the other way round, we just push up the count of known CRF by 3 more members, which even can be considered monostratic (in one of their possible orientations)!

And again, the diminishing by the omission of one of those n-gons clearly re-enters the realm of segmentochora for all n, by applying the necessary shift to the remaining degenerate base (the other n-gon) only. (This furthermore shows an additional, so far not mentioned connection between 4.12 and 4.13 of my listings!)

--- rk
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Re: Pyramid antiprisms

Postby Klitzing » Thu Sep 20, 2012 9:53 am

wendy wrote:It's actually a square antiprism tegum. It's back-to-back antiprism pyramids.

The thing works for triangles (16-chor), square and pentagons (with 4n tetrahedra and 4 p-pyramids.


Having just spoken of its expansion to cupolae instead of pyramids, it becomes obvious that we can use the case n=2 as well. In the case of pyramids, the 2-pyramids themselves clearly become degenerate, i.e. could be omitted. Accordingly we then are left with the lacing tetrahedra only.

By virtue of your introductory coment or the already provided lace city it becomes obvious, that this n=2 fellow is a long-known polychoron as well: it happens to be nothing but tedpy (=tet-dipyramid)!

--- rk
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Re: Pyramid antiprisms

Postby wendy » Fri Sep 21, 2012 6:49 am

The actual rule here is: Ap( pyr (A, B, ...)) = tegum (Ap(A), Ap(B), ...) . The antiprism of a pyramid product, is the tegum product of the antiprisms. Likewise, the antitegum of a pyramid product is the prism product of the antiprisms. When the AP of a point is a line, one sees that the Ap(Pyr(point, square) = tegum(Ap (point), Ap(square)).
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Re: Pyramid antiprisms

Postby Klitzing » Sat Sep 29, 2012 7:49 am

Klitzing wrote:BTW, it just occured to me, as pyramids and cupolae would have parallel triangular faces and same heights, that the same trick would be possible with cupolae as well:
    n-cupola || inv gyro n-cupola.
Facet counts being: 4 n-cupolae, 2n squippies, and 2n trips (n=2, 3, 4, 5). Here spuippies (=4pyr) are achieved by elongating one base vertex of a tet to an own edge, and trips likewise are achieved by pulling one edge of a tet apart getting a square. (I.e. the former 4n tets of your fellows fall apart into 2 classes here.) Those are the induced effects onto the lacings for the Stott expansion of those bases.


Hy quickfur,
would you like to generate a pic of those?
As those are kind of Stott expanded versions of your pyramid antiprisms, the coordinates of case n=4 should be readily deducable from your former pic (resp. those coordinates already listed within this thread)...
--- rk
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Re: Pyramid antiprisms

Postby quickfur » Sat Sep 29, 2012 4:59 pm

Klitzing wrote:
Klitzing wrote:BTW, it just occured to me, as pyramids and cupolae would have parallel triangular faces and same heights, that the same trick would be possible with cupolae as well:
    n-cupola || inv gyro n-cupola.
Facet counts being: 4 n-cupolae, 2n squippies, and 2n trips (n=2, 3, 4, 5). Here spuippies (=4pyr) are achieved by elongating one base vertex of a tet to an own edge, and trips likewise are achieved by pulling one edge of a tet apart getting a square. (I.e. the former 4n tets of your fellows fall apart into 2 classes here.) Those are the induced effects onto the lacings for the Stott expansion of those bases.


Hy quickfur,
would you like to generate a pic of those?
As those are kind of Stott expanded versions of your pyramid antiprisms, the coordinates of case n=4 should be readily deducable from your former pic (resp. those coordinates already listed within this thread)...
--- rk

Actually, it appears that due to the non-orbiform-ness of the square pyramid antiprism, the last two coordinates are drastically different after the expansion (if one assumes equal edge lengths). Regardless, after re-calculating the last two coordinates, I found the CRF solution as:

Code: Select all
# Base cupola
< 1,  (1+sqrt(2)), -sqrt(2)/2, -sqrt(2)/2>
< 1, -(1+sqrt(2)), -sqrt(2)/2, -sqrt(2)/2>
<-1,  (1+sqrt(2)), -sqrt(2)/2, -sqrt(2)/2>
<-1, -(1+sqrt(2)), -sqrt(2)/2, -sqrt(2)/2>
< (1+sqrt(2)),  1, -sqrt(2)/2, -sqrt(2)/2>
< (1+sqrt(2)), -1, -sqrt(2)/2, -sqrt(2)/2>
<-(1+sqrt(2)),  1, -sqrt(2)/2, -sqrt(2)/2>
<-(1+sqrt(2)), -1, -sqrt(2)/2, -sqrt(2)/2>

< 0,  sqrt(2),  sqrt(2)/2, -sqrt(2)/2>
< 0, -sqrt(2),  sqrt(2)/2, -sqrt(2)/2>
< sqrt(2),  0,  sqrt(2)/2, -sqrt(2)/2>
<-sqrt(2),  0,  sqrt(2)/2, -sqrt(2)/2>

# Gyrated inverted cupola
< 1,  (1+sqrt(2)),  sqrt(2)/2, sqrt(2)/2>
< 1, -(1+sqrt(2)),  sqrt(2)/2, sqrt(2)/2>
<-1,  (1+sqrt(2)),  sqrt(2)/2, sqrt(2)/2>
<-1, -(1+sqrt(2)),  sqrt(2)/2, sqrt(2)/2>
< (1+sqrt(2)),  1,  sqrt(2)/2, sqrt(2)/2>
< (1+sqrt(2)), -1,  sqrt(2)/2, sqrt(2)/2>
<-(1+sqrt(2)),  1,  sqrt(2)/2, sqrt(2)/2>
<-(1+sqrt(2)), -1,  sqrt(2)/2, sqrt(2)/2>

< 1,  1,  -sqrt(2)/2, sqrt(2)/2>
< 1, -1,  -sqrt(2)/2, sqrt(2)/2>
<-1,  1,  -sqrt(2)/2, sqrt(2)/2>
<-1, -1,  -sqrt(2)/2, sqrt(2)/2>


It turns out that this is identical to square||octagonal_prism||gyrated_square, which is two copies of square||octagonal_prism attached to each other in gyrated orientation. Here's a projection centered on a square face between two cupolae:

Image

I turned off visibility clipping so that both square ridges are visible. If visibility clipping is turned on, then this particular viewpoint produces exactly the same projection as a single square||octagonal_prism (since the other one would just be on the far side which gets culled).

Here's a projection from another viewpoint, centered on one of the square cupola cells:

Image

The nearest cell is outlined in red, and the farthest cell is rendered in yellow. Here, the alternating triangular prisms/square pyramids are much clearer.
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Re: Pyramid antiprisms

Postby quickfur » Sat Sep 29, 2012 5:11 pm

Here's another render:

Image

This is a projection centered on an octagonal face. I've colored the farthest two cupolae in green and yellow. This viewpoint gives the best view of how the triangular prisms and square pyramids are attached to each other in an alternating formation.

EDIT: Here's the converted .off file for those Stella4D users among you. (Are you reading this thread, Marek? ;) )
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