Squap verf

[mr_e_man]

First, some definitions:

By "vertex figure" (or "verf") I mean the intersection of the polytope with a small sphere centred on the vertex. This includes information about the dihedral/dichoral angles, but not necessarily about the identities of the cells. In the picture below, for example, an octahedron could be replaced with a square pyramid, or an augmented sphenocorona, and the vertex figure would be the same.

By "vertex configuration" (or "verc") I mean the topological arrangement of faces/cells around the vertex. This includes information about their identities and orientations, but not necessarily about the dihedral/dichoral angles. (Formally, I could define it in terms of the abstract polytope, which is a partially ordered set: the vertex configuration is all facets greater than the vertex, and all k-faces (of any dimension k) less than these facets.)

For 2D polygons, vercs are trivial; any vertex has two edges around it. But verfs are not trivial; they tell the angle at the vertex.

For 3D polyhedra, in general, verfs and vercs are independent; neither determines the other. But for CRF polyhedra, the verf determines the verc, since a face's angle at one vertex uniquely determines the type of face.

For 4D polychora, this is reversed: the verf does not determine the verc, as I explained with the octahedron; but the verc determines the verf, which is equivalent to saying that 4D vertices are rigid. (This would follow from the rigidity of convex polyhedra in spherical space, which I think has been proven....)

So, I was calculating the possible vertex figures of the Blind polychora (convex regular-celled polychora). There's a bunch of pentagonal things like mibdi or pip, which appear in ex diminishings or in rox; there's the augmented triangular prism, which appears in aurap; but there's also the square antiprism, which does not appear in any Blind polychoron.

squapVerf.png (40.92 KiB) Viewed 21998 times

Does this vertex configuration appear in any CRF polychoron?

If not, does this vertex figure appear in any CRF polychoron?

[Mercurial, the Spectre]

Let me see... I need to clarify what you mean.
Let's take a tetrahedral prism, which can be made uniform. You get a verf of an equilateral-triangular pyramid, with the lateral faces being isosceles triangles of length 1:sqrt(2). Now denote the number n representing the side length of 2cos(pi/n) (this follows from the regular n-gon in 2D). So that verf would have a base of 3.3.3 (a unit equialteral triangle) and lateral faces of 3.4.4 (the isosceles triangles). Notice that these are the vertex configurations of the tetrahedron and the triangular prism, which are the cells of the tetrahedral prism. That's the same idea you proposed earlier.

If you're talking about square antiprism verfs, the cube-octahedron segmentochoron (which is an orbiform CRF) contains the square antiprism in one of its two vertex types. To illustrate, take a vertex belonging to the octahedral cell, then the neighboring cells are an octahedron, a square pyramid (the vertex is its apex), and eight tetrahedra connecting the square pyramid's lateral faces to the octahedron. Remember that the octahedron is 3.3.3.3, a square pyramid is 3.3.3.3 (apex) / 3.3.4 (base), and a tetrahedron is 3.3.3. Taking all of these it isn't hard to find a polyhedron with two squares and eight triangles, which is the square antiprism.

In conclusion, yes.

[mr_e_man]

I would say that the tet prism's verf is an equilateral triangular pyramid with base length 60° and lateral length 90°. It's a spherical polyhedron.

Yes, cube || oct answers my second question (the verf is a square antiprism with all edge lengths 60°). Thanks.

My first question remains. Does this particular arrangement, with precisely 2 octs and 8 tets, appear anywhere?

[mr_e_man]

The spherical squap can be diminished, revealing a trapezoid face. This trapezoid is the verf of a (flat, not spherical) squap.

dimSquapVerf.png (71.66 KiB) Viewed 21862 times

However, if this new verf is used to complete one of the vertices with 3 tets and 1 oct in the first image, I think an adjacent vertex could not be completed... but I need to double check. I'm finding Marek's list of dihedral angles useful.

[Klitzing]

You even could do that on both sides.
Taken as a verf this then asks for chains (rings) of pink squaps …

--- rk

[mr_e_man]

Yes, the squap can be diminished twice; actually in two different ways:

bidimSquapVerf.png (33.04 KiB) Viewed 21708 times

I have been thinking about rings of antiprisms. I want to prove that large (n > 20 perhaps) prisms or antiprisms can only appear in "simple" polychora such as duoprisms, antiprism prisms, or your segmentochora (n-prism || dual n-gon). But that's a different topic.

And the bidiminished squap cannot be used in the same way that I meant to use the diminished squap. In the top half of the image below, I've shown the partially constructed polychoron, and labelled vertices A, B, C, D. In the bottom half, I've shown the vertex figures. The verf at A is an undiminished squap. The verf at B is an incomplete polyhedron, with 3 triangles and 1 square around a vertex; this corresponds to the 3 tets and 1 oct around edge AB. This can be completed into a diminished squap, but not a bidiminished squap, which doesn't have 3 triangles and 1 square around any vertex.

squapVerf4.png (60.76 KiB) Viewed 21708 times

Now, with a square pyramid on the triangle BCD (with a dichoral angle determined by the verf at B), the edge CD cannot be completed; whatever polyhedra we try to fit there (with faces matched properly), the dihedral angles don't fit, or the result is not convex.

If, instead, the verf at B is completed into an undiminished squap, then on the triangle BCD there's an octahedron, or a square pyramid, or something augmented with a square pyramid. In any case, again the edge CD cannot be completed.

So the verf at B must be completed in a different way. It has part of a squap, but it cannot be a squap nor a diminished squap.

[Klitzing]

What I was after, was to use the bidiminished squap to be used as a verf at A.
This then provides as local configuration around A of 1 oct + 2 tets + 2 squaps.
Here the squaps will be joining at a square face.
That way I was dreaming up potential rings of squaps…
--- rk

[mr_e_man]

I knew that.

Well, with the bidiminished squap verf, the dichoral angle between the two squaps is 72.97°; or, the angle of turning through 4D from one squap to the next is 107.03°, which does not divide 360°; so we can't use this to make a ring of squaps. But we could have 2 squaps, and see if anything else fits in the gaps.... (By "gaps" I mean "empty spaces", not "grand antiprisms" . That's one problem with Bowers' acronyms.)

Aha! This verf appears in your segmentochoron which I mentioned: square prism || dual square. The verc has 1 square pyramid, 2 tets, and 2 squaps.

In fact, that's just a bidiminished cube || oct.

[mr_e_man]

The square antiprismatic prism (squap || squap) has one verf type. It's a trapezoidal pyramid, with base lengths 60°, 60°, 60°, 90°, and rising edge lengths all 90°.

The square antiprismatic pyramid (squap || point) has two verf types. One is the (spherical) squap. The other is the piece that I cut off to make the diminished squap; it's a trapezoidal pyramid with rising edge lengths all 60°.

The segmentochoron (square prism || dual square) has two verf types. One is the bidiminished squap. The other is a trapezoidal pyramid, with 90° edges rising from the 90° base edge to make an equilateral triangle (which is the verf of a cube), and all other edge lengths 60°.

The surprising thing is that this last pyramid can be joined to the diminished squap, and all joining edges vanish! In other words, the dihedral angles add up to 180°.

squapVerf5.png (22.75 KiB) Viewed 21522 times

(There is some visible distortion, because I'm drawing these as flat polyhedra, while they're supposed to be in spherical space.)

The first figure is the diminished squap. In the second figure, the cyan triangle is the verf of a cube, the yellow triangle (hidden in the lower right) is the verf of a tet, and the red triangles are the verfs of square pyramids. For the third figure, we could have 3 yellow tets, 1 cyan elongated square pyramid, 1 green (hidden on the bottom) truncated tetrahedron, and 2 red triangular cupolas. (The 60° edges on the bottom are joined to form 120° edges.)

This is interesting, though it's not what I was looking for.