## Derivation of 3D crown jewels

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

### Derivation of 3D crown jewels

Today I was thinking about various 3D crown jewels, and how they might be derived from "simpler" shapes.

J84 (snub disphenoid) - there are various different ways to analyze the snub disphenoid, but recently I noticed that it can be derived from a pentagonal bipyramid: imagine a paper model of a pentagonal bipyramid, where you select a pair of adjacent edges on the pentagonal cross-section. Use scissors to cut it up along those edges, then push out the now-split vertex between them, until you can fit in a new edge. The result is a snub disphenoid. I'm not sure if the icosahedral bipyramid in 4D has enough degrees of freedom to allow an analogous construction, but it might be worth trying?

J85 (sphenocorona) - if we start with a pentagonal bipyramid, and this time cut four consecutive edges along the pentagonal cross-section, then we can stretch the now cut vertices apart and fill in the gaps with a triangle-square-square-triangle patch to make a sphenocorona. This does require a great deal of freedom of deformation, though; I'm not sure if it's possible in 4D?

J86 (sphenomegacorona) - it seems to be an analogous derivation to J84, applied to an icosahedron instead of a pentagonal bipyramid.

J87 (hebesphenomegacorona) - seems to be an analogous derivation to J85, applied to an icosahedron instead of a pentagonal bipyramid, and the inserted patch has 3 squares instead of just two.

J88 (disphenocingulum) - appears to be a distorted hexagonal antiprism capped by two "spheno" wedges. Not sure what the 4D analogue would be; perhaps some kind of duoprism suitably deformed in order to fit odd-shaped augments with non-flat joining surfaces? Sorta like D4.8.x in the sense of having non-flat joining surface, except simpler.

The question of interest, of course, is whether the 4D analogues of the pentagonal bipyramid / icosahedron have enough degrees of freedom to allow similar kinds of cuttings and deformations? Thanks to the Blind couple's results, we know that any crown jewels in this direction cannot consist only of tetrahedra or other regular cells; in all likelihood the "pushing apart" stage of the derivation will introduce triangular prisms and maybe other kinds of prisms. But still, these derivations are a lot simpler than the currently-known BT polychora, so perhaps there's a chance we can find something here?

Of particular interest is, suppose we cut off a piece of the 600-cell's surface to get a bunch of linked tetrahedra. What is the maximum number of tetrahedra in the patch such that it will have at least 1 degree of freedom? When does a patch containing tetrahedra, 5 around each edge, become forced to take on the exact curvature of the 600-cell? Such "flexible patches" of tetrahedra may be useful in constructing 4D crown jewels, since their extra degree(s) of freedom may permit enough deformation to insert unusual cell shapes to close them up in a CRF way.

Furthermore, suppose we have a CRF that contains two conjoined icosahedral cells, with dichoral symmetry between them. Suppose we now perform the ike -> J86 derivation on these cells, in correponding positions. Will it be possible to insert CRF cells to close up the new shape? If so, this will produce a non-trivial CRF with two J86 cells. The idea behind this derivation is that if the higher interconnectivity of 4D polychora makes it hard for direct generalization of the ike->J86 deformation, maybe we can use the 3D ike->J86 deformation, but applies to ikes within a 4D polytope, then patch up the result with new cells to close it up.

Thoughts?
quickfur
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### Re: Derivation of 3D crown jewels

There is a piece of software ("Hedron" by Jim McNeill, cf. his website), which solves that problem quite nicely for polyhedra (i.e. for 3D): Provided a set of connectivities it sets up a random distribution of vertices and then uses random disturbations of those initial positions, subject to forces which represent the required incidences and coplanarities, to cool that shape down, always measuring the still contained deviations from that target figure (zero forces).

What you are asking here in your described transfer of building procedures of Johnson solid crown jewels from easier, i.e. readily constructable figures by means of cutting open and following insertions onto potential new 4D CRFs, would not only ask for ideas how to potentially construct those, but also would need for a similar piece of software which would allow to test such hypotheses.

That one could be set up completely in analogue to Hedron, I suppose. Just that one would have to use 4D coordinates instead of 3D. And one would have then 3 types of forces instead of only 2 so far:
a) Requiring unit edge lengths for the set of initially provided abstract edges, i.e. vertex pairs,
b) requiring coplanarity of 2D boundaries (including the required lengths of inner invisible "edges" / 2D-chords),
c) and now a third which requires corealmity of 3D cells (including achieving the required inner 3D-chords).

The setup of such a piece of software ought to be mostly straight forward. But the gaining of the required chords looks more difficult. Perhaps it might be useful here to have an external config file, providing all those informations for any allowed to use cells. (That externality would help to test earlier releases of that software with some restricted set of possible cells first. But not having to change its coding only for the reason of adding further allowed cells.)

- Anyone out there who would like to program such a software?

It furthermore ought to be easy to handle, running at least on any Windows engine (potentially also Linux?) - so perhaps in Java or even Web-based (e.g. PHP or JavaScript)? And, for sure, I'd like if it would become a freeware or a free accessible service.

--- rk
Klitzing
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### Re: Derivation of 3D crown jewels

I haven't used Windows in any non-trivial capacity for a looong time... I use Linux both at home and at work. So any software I write is likely to be specific to Linux. But I suppose I could hook it up to a web service that you can access with a browser. If I find the time to work on this, that is. I have been extremely busy lately -- I still haven't finished this month's Polytope of the Month! (And btw, it will be spidrox... I've only just finished the renders for the 12 rings of prisms/antiprisms yesterday, and now I have to do the renders for the 20 rings of pyramids.)

I was thinking more along the lines of analytic solutions, though. Suppose we are given some CRF that has two icosahedra sharing a face, preferably with a narrow dichoral angle. Would it be possible to, say, deform the icosahedra into J86's, while preserving most of the existing CRF structure, perhaps modifying them suitably, so that the result is CRF-able? This is somewhat similar to bilbiro'ing or thawro'ing, in that we start with an existing CRF structure and modify the icosahedra into bilbiro's/thawro's; except here, unlike bilbiro'ing or thawro'ing, dichoral angles may not be preserved.

The first step to such analysis, of course, is to find out how many degrees of freedom we have in 4D, given some CRF patch of cells. Does the extra dimension of cell-connectivity increase or decrease the number of degrees of freedom in deforming a given (open) cell complex? What are the conditions for maximizing/minimizing the flexibility of a given cell complex? For example, in 3D, if we remove the pentagonal face from a pentagonal pyramid, the remaining 5 tetrahedra show a considerable amount of flexibility in their dihedral angles. In fact, if I'm not wrong, there are 2 degrees of freedom in how you can deform this net without breaking the triangles apart. In 4D, suppose we remove the icosahedron from an icosahedral pyramid. Can the remaining 20 tetrahedra vary in their dichoral angles freely? If so, how many degrees of freedom do you have in deforming this net? And if not, what are the restrictions on what kind of 4D nets confer flexibility of deformation?
quickfur
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### Re: Derivation of 3D crown jewels

Let's look at a general 5-triangle vertex.
We put the vertex in [0,0,0]. Then the endpoints of edges will lie on the unit sphere. First vertex will be at [1,0,0]. Second will be still somewhere in xy plane, at [x2,y2,0]. The position of remaining three is general: [x3,y3,z3], [x4,y4,z4] and [x5,y5,z5].
Now, we have 11 unknowns and we can set constraints:

Unit sphere constraints:
1. x2^2 + y2^2 = 1
2. x3^2 + y3^2 + z3^2 = 1
3. x4^2 + y4^2 + z4^2 = 1
4. x5^2 + y5^2 + z5^2 = 1

Edge constraints:
5. (x2 - 1)^2 + y2^2 = 1
6. (x3 - x2)^2 + (y3 - y2)^2 + z3^2 = 1
7. (x4 - x3)^2 + (y4 - y3)^2 + (z4 - z3)^2 = 1
8. (x5 - x4)^2 + (y5 - y4)^2 + (z5 - z4)^2 = 1
9. (x5 - 1) ^2 + y5^2 + z5^2 = 1

So we have 9 equation for 11 unknowns, and so we see there are two degrees of freedom. Adding coplanarity condition would add two more equations (you make an equation of plane passing through first three points and then force 4th and 5th point to lie in it as well), so we'd get no degrees of freedom then.

Now look at 4D case. In 4D, vertices typically have significantly more faces than in 3D. A tetrahedral vertex has 6 faces -- that is 6 chords whose lengths must be precisely set to create a CRF figure.

Let's look at an icosahedral vertex. It has 12 edges and 30 faces. Each of the adjacent vertices will have 4 coordinates in 4D:
[1,0,0,0]
[x2,y2,0,0]
[x3,y3,z3,0]
[x4,y4,z4,w4]
...
[x12,y12,z12,w12]

So we have 41 unknowns altogether.

Constraints are:
11 equations enforcing unit hypersphere
30 equations enforcing chord lengths (corresponding to 30 faces at the central vertex).

So unless some of these equations are dependent (and I don't know whether they are], it would mean that vertex of icosahedral pyramid actually has no degrees of freedom at all.

Let's look at the general case. In 3D, you generally have an n-gon with 3*n coordinates. 4 of these are fixed ([1,0,0] for first vertex and 0 z coordinate for second), so you have 3n - 4. Then you have n - 1 unit sphere constraints and n chords. This translates in n-3 degrees of freedom (0 for triangles, 1 for quadrangles and 2 for pentagons -- no higher polygons can occur as vertex figures). In 4D, the number of coordinates to find is 4n - 7, n - 1 unit sphere constraints, and then the number of chords varies.

What's the maximum number of chords? Well, that would be reached if the vertex polyhedron was deltahedron, i.e. composed solely of triangles. If it's not a deltahedron, then you could split a non-triangular face by a diagonal.

So, how many edges does a deltahedron have? If it has n triangles, then it has 3n/2 edges, as every triangle has 3 and every edge belongs to 2 triangles. Tetrahedron (4 triangles) has 6 edges, triangular bipyramid (6 triangles) has 9 edges, icosahedron (20 triangles) has 30 edges, etc.

How many vertices does a deltahedron have? That can be obtained from Euler's formula V + F - E = 2. If F is n and E is 3n/2, we get

V + n - 3n/2 = 2
V - n/2 = 2
V = 2 + n/2

So tetrahedron has 4 (2 + 4/2) vertices, triangular bipyramid has 5 (2 + 6/2) and icosahedron has 12 (2 + 20/2).

We'll use 2n for number of faces from now on to make it simpler (deltahedra must have even number of faces, of course).
So, for a deltahedron-like vertex figure with 2n triangles, you have 2 + n vertices. These have 4n + 1 coordinates to determine (4*(2 + n) - 7).
Now, we have n + 1 unit hypersphere constraints and 3n chords.

See the problem? Together, they add to 4n + 1 -- in other words, we get 0 degrees of freedom!

Tetrahedral vertex -- 2*2 triangles, 9 coordinates, 3 + 6 constraints = 0 degrees of freedom.
Triangular bipyramidal vertex -- 2*3 triangles, 13 coordinates, 4 + 9 constraints = 0 degrees of freedom.
Octahedral vertex -- 2*4 triangles, 17 coordinates, 5 + 12 cponstraints = 0 degrees of freedom.
Icosahedral vertex -- 2*10 triangles, 41 coordinates, 11 + 30 constraints = 0 degrees of freedom!

So, what about vertex figures that are NOT deltahedra? Actually, there's no such thing! If the vertex figure contains a polygon with more than three sides, that polygon must correspond to one of the finite list of valid polyhedra. That means that its chords CANNOT vary freely, they must take one of a discrete set of values. For example, you can have a quadrangle with unit edges corresponging to a octahedron's square, a quadrangle corresponding to equatorial vertex of triangular bipyramid or quadrangle corresponding to a 4-triangle vertex of snub disphenoid, but not a generic, flexible quadrangle! So every time you have a non-deltahedron vertex figure, you still have to add additional chords to ensure the big polygon will work out correctly, and that will, in effect, turn it into a deltahedron!

So... unless there are prevalent dependencies between these equations, I think that this shows that ANY 4D vertex of CRF polychoron must have degree of freedom 0, and therefore is rigid.
Marek14
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### Re: Derivation of 3D crown jewels

But if 4D vertices are always rigid, then doesn't that mean we can just brute-force enumerate all possible vertex configurations, and thereby find all 4D CRFs? We don't even need to precompute any verfs; just enumerate all possible combinations of 3D CRFs around a vertex, and discard those that don't close up locally.
quickfur
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### Re: Derivation of 3D crown jewels

quickfur wrote:But if 4D vertices are always rigid, then doesn't that mean we can just brute-force enumerate all possible vertex configurations, and thereby find all 4D CRFs? We don't even need to precompute any verfs; just enumerate all possible combinations of 3D CRFs around a vertex, and discard those that don't close up locally.

Haven't I been mentioning the possibility of something like that for quite some time? But it was just a hunch, only today I actually tried to calculate the conditions. The result was surprising

Of course, it IS possible that for some vertices the equations will turn out to be dependent and they won't be rigid, but it should be the exception, not the rule.
Marek14
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### Re: Derivation of 3D crown jewels

Hmm. So it appears that while edges are flexible, vertices are not (at least, not in the general case). So it seems that the condition for a 4D cell complex (non-closed net) to be flexible, all of its vertices must be open (i.e., lie on the unconnected boundary of the net).

This is a very interesting result. In 3D, this doesn't hold -- a closed vertex (completely surrounded by faces) may still be flexible, e.g., the vertex surrounded by 4 or 5 triangles. Because of this, a large-enough 3D net may have transitive flexibility. For example, if you cut a series of edges on an icosahedron, the closed vertices (not adjacent to any of the cut edges) are still flexible, and they can deform enough to accomodate the cut edges to widen into a gap where more faces can be filled in, thus producing the sphenocoronas. It's because of this transitive flexibility that you can end up with some complicated dependencies in dihedral angles: as you widen the gap where the cut was made, the deformation propagates transitively throughout the remainder of the polyhedron, modifying dihedral angles accordingly, so in the general case, to compute the coordinates of the deformed shape you need to solve a whole series of interdependent equations.

But in 4D, if this is indeed true that all vertices are rigid, then there cannot be any transitive flexibility; the only flexible parts of a partially-constructed 4D CRF must lie on its unconnected boundary! Which means that once a vertex in the net becomes closed, it forces that part of the net to become rigid, and so the closed portion of the net is always rigid, and any dependencies of dichoral angles must only lie on the open (unconnected) boundary of the net.

This seems to suggest that maybe the only 4D CVP3 CRFs are those that contain CVP3 polyhedra as cells... because the boundary of the net is, at most, a 3-manifold, so if there are any CVP3 relationships between vertices (which arise from flexibility, since otherwise they can be solved with just independent quadratic equations and so can't be CVP3), it must arise from the 3D boundary of the net at some point during its construction. Which in turn suggests that such relationships must come from a CVP3 cell somewhere on the boundary of the net. So without a CVP3 cell somewhere in the net during the construction of the polytope, its coordinates will not be CVP3.
quickfur
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### Re: Derivation of 3D crown jewels

Marek14 wrote:
quickfur wrote:But if 4D vertices are always rigid, then doesn't that mean we can just brute-force enumerate all possible vertex configurations, and thereby find all 4D CRFs? We don't even need to precompute any verfs; just enumerate all possible combinations of 3D CRFs around a vertex, and discard those that don't close up locally.

Haven't I been mentioning the possibility of something like that for quite some time? But it was just a hunch, only today I actually tried to calculate the conditions. The result was surprising
Indeed. Formerly, the absence of such calculations made that I didn't consider such an approach as easy. However, now that you have made these, I think we can finally investigate some CVP3-polytopes (or prove their non-existence)
Of course, it IS possible that for some vertices the equations will turn out to be dependent and they won't be rigid, but it should be the exception, not the rule.
I think you don't have to worry about that much. let's take a central vertex. now we add some vertices. let's say these vertices are all part of a surtope. This means the added vertices should make the verf of the central vertex. This verf can be seen in matrix context: vertex 1 has distance x to vertex 2, vertex 2 has distance y to vertex 3, vertex 1 has distance z to vertex 3, vertex 3 has distance w to vertex 2, etc, nicely placed in a matrix.
This verf is (clearly) rigid (though you can turn it around in space, but it stays congruent all the time). If we place some more vertices now, of a surtope which shares vertices 1 and 2 and the central vertex, these vertices make another verf, and thus should have another matrix. Of course, these verfs are independent, though they have an overlapping distance between 1 and 2. When we build the rest of the verf, all surtopal verfs stay independent, with occasional overlap. This means all distances between vertices are independent, if the surtopal verfs don't have internal independencies themselves. You shouldn't worry about those either, as they only occur with verfs with more than 3 vertices, and you already proved these differently.

Now this result is very useful for determining CVP-3 polytopes, as this means dichoral angles are fixed as well. As quickfur already noted, vertices are only CVP3 if they have CVP3-surtopes themselves. Now what I'm hoping is that these CVP3-surtopal verfs make the dichoral angles horrible, so that we can only combine CVP3 verfs with CVP3 verfs. This means the number of CVP3-polytopes stays conveniently small, and thus they won't combinatorically explode, allowing a computer search. of course we first have to proof my first hypothesis.
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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student91
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### Re: Derivation of 3D crown jewels

First step here might be enumeration of dichoral angles. I enumerated those about two years ago:

Code: Select all
`So, let's look at dihedral angles. Stella is a big help here since its net mode computes dihedral angles automatically.I include all prisms and antiprisms up to 10, for completeness.BTW, you shouldn't automatically assume that there can be only 5 faces to an edge -- dihedral angles of some Johnson solids can get quite small.Now, let's talk "possibilities". Basically, in some Johnson solids, the same dihedral angle exists at several nonequivalent places. I note this. Some of those places might be also assymetrical (i.e. joining two identical polygons, but without an axis of symmetry passing through the edge, so you have to try to fit it in both orientations).The exact number of possiblities for CRF purposes will have to be checked, some possibilities might be chiral.For the weird solids (sphenocorona and up) I've given up on describing the edges and counting possibilities...31.7175 (dpc4)4-10 in pentagonal cupola37.3774 (dpp)3-5 in pentagonal pyramid3-10 in pentagonal cupola454-8 in square cupola54.7356 (90 - d4/2)3-4 in square pyramid4-6 in triangular cupola3-8 in square cupola604-4 in triangular prism4-4 in elongated triangular pyramid (asymmetrical)4-4 in elongated triangular dipyramid4-4 in augmented triangular prism63.4349 (2*dpc4)5-10 in pentagonal rotunda4-4 in pentagonal orthobicupola5-5 in metabidiminished icosahedron5-5 in tridiminished icosahedron (asymmetrical)5-5 in augmented tridiminished icosahedron (asymmetrical)5-5 in bilunabirotunda69.0948 (dpc4 + dpp = d20/2)3-4 in pentagonal gyrobicupola (equatorial)70.5288 (let's call it d4 because it's an important number)3-3 in tetrahedron6-6 in truncated tetrahedron3-6 in triangular cupola3-3 in elongated triangular pyramid (asymmetrical)3-3 in triangular dipyramid (apex) (asymmetrical)3-3 in elongated triangular dipyramid (asymmetrical)3-3 in augmented tridiminished icosahedron (augment) (asymmetrical)6-6 in augmented truncated tetrahedron72.9734-4 in sphenomegacorona74.7547 (2*dpp)3-3 in pentagonal dipyramid (equatorial)3-3 in pentagonal orthobicupola79.1877 (dpr3; dpr3 + 2*dpc4 = did)3-10 in pentagonal rotunda86.72683-3 in sphenomegacorona904-4 in cube8-8 in truncated cube3-4 in triangular prism4-5 in pentagonal prism4-6 in hexagonal prism4-7 in heptagonal prism4-8 in octagonal prism4-9 in enneagonal prism4-10 in decagonal prism3-4 in elongated triangular pyramid (base)4-4 in elongated square pyramid (2 possibilities) (both asymmetrical)4-5 in elongated pentagonal pyramid4-4 in elongated square dipyramid4-6 in elongated triangular cupola (2 possibilities)4-8 in elongated square cupola (2 possibilities)4-10 in elongated pentagonal cupola (2 possibilities)4-10 in elongated pentagonal rotunda (2 possibilities)3-4 in gyrobifastigium (individual prisms)4-4 in square orthobicupola (equatorial)3-4 in augmented triangular prism (base/side)3-4 in biaugmented triangular prism (base/side)4-5 in augmented pentagonal prism (2 possibilities)4-5 in biaugmented pentagonal prism (2 possibilities)4-6 in augmented hexagonal prism (3 possibilities)4-6 in parabiaugmented hexagonal prism4-6 in metabiaugmented hexagonal prism (3 possibilities)4-6 in triaugmented hexagonal prism8-8 in augmented truncated cube (2 possibilities) (both asymmetrical)8-8 in biaugmented truncated cube95.1524 (3*dpc4)4-5 in pentagonal gyrocupolarotunda (equatorial)95.2466 (da10)3-10 in decagonal antiprism3-10 in gyroelongated pentagonal cupola3-10 in gyroelongated pentagonal rotunda95.8433-9 in enneagonal antiprism96.19833-3 in snub disphenoid (type 1) (2 possibilities) (1 asymmetrical)96.5945 (da8)3-8 in octagonal antiprism3-8 in gyroelongated square cupola97.45553-4 in sphenocorona3-4 in augmented sphenocorona97.57233-7 in heptagonal antiprism98.8994 (da6)3-6 in hexagonal antiprism3-6 in gyroelongated triangular cupola99.7356 (135 - d4/2)3-4 in square gyrobicupola (equatorial)100.1944-4 in disphenocingulum100.812 (d20 - dpp = dpp + 2*dpc4)3-5 in pentagonal antiprism3-5 in gyroelongated pentagonal pyramid3-5 in pentagonal orthocupolarotunda (equatorial)3-5 in metabidiminished icosahedron (2 possibilities)3-5 in tridiminished icosahedron (2 possibilities)3-5 in augmented tridiminished icosahedron (main body)3-5 in bilunabirotunda3-5 in triangular hebesphenorotunda102.5244-4 in hebesphenomegacorona103.836 (da4)3-4 in square antiprism3-4 in gyroelongated square pyramid1084-4 in pentagonal prism4-4 in elongated pentagonal pyramid (asymmetrical)4-4 in elongated pentagonal dipyramid4-4 in augmented pentagonal prism (2 possibilities) (1 asymmetrical)4-4 in biaugmented pentagonal prism109.471 (180 - d4)3-3 in octahedron3-6 in truncated tetrahedron6-6 in truncated octahedron3-3 in square pyramid3-3 in elongated square pyramid (asymmetrical)3-3 in gyroelongated square pyramid (apex) (asymmetrical)3-3 in elongated square dipyramid (asymmetrical)3-3 in gyroelongated square dipyramid (apex) (asymmetrical)4-4 in triangular orthobicupola3-3 in square orthobicupola3-3 in augmented triangular prism (augment) (asymmetrical)3-3 in biaugmented triangular prism (augment) (2 possibilities) (both asymmetrical)3-3 in triaugmented triangular prism (augment) (asymmetrical)3-3 in augmented pentagonal prism (asymmetrical)3-3 in biaugmented pentagonal prism (2 possibilities) (both asymmetrical)3-3 in augmented hexagonal prism (asymmetrical)3-3 in parabiaugmented hexagonal prism (asymmetrical)3-3 in metabiaugmented hexagonal prism (2 possibilities) (both asymmetrical)3-3 in triaugmented hexagonal prism (asymmetrical)3-6 in augmented truncated tetrahedron (main body) (2 possibilities)3-3 in augmented sphenocorona109.5243-4 in sphenocorona3-4 in augmented sphenocorona110.905 (dpc4 + dpr3)3-4 in pentagonal orthocupolarotunda (equatorial)3-4 in bilunabirotunda4-6 in triangular hebesphenorotunda3-4 in triangular hebesphenorotunda111.7353-3 in hebesphenomegacorona114.6453-3 in snub square antiprism (middle edges) (2 possibilities)114.736 (150 - d4/2)3-4 in augmented triangular prism (augment/side)3-4 in biaugmented triangular prism (augment/side)116.565 (d12 = dpp + dpr3)5-5 in dodecahedron10-10 in truncated dodecahedron3-3 in pentagonal gyrocupolarotunda5-5 in augmented dodecahedron (4 possibilities) (all asymmetrical)5-5 in parabiaugmented dodecahedron (2 possibilities) (1 asymmetrical)5-5 in metabiaugmented dodecahedron (7 possibilities) (5 asymmetrical)5-5 in triaugmented dodecahedron (4 possibilities) (all asymmetrical)10-10 in augmented truncated dodecahedron (4 possibilities) (all asymmetrical)10-10 in parabiaugmented truncated dodecahedron (2 possibilities) (1 asymmetrical)10-10 in metabiaugmented truncated dodecahedron (7 possibilities) (5 asymmetrical)10-10 in triaugmented truncated dodecahedron (4 possibilities) (all asymmetrical)5-10 in diminished rhombicosidodecahedron5-10 in paragyrate diminished rhombicosidodecahedron5-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)5-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)5-10 in parabidiminished rhombicosidodecahedron5-10 in metabidiminished rhombicosidodecahedron (3 possibilities)5-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)5-10 in tridiminished rhombicosidodecahedron (3 possibilities)117.019 (dsp)4-4 in sphenocorona117.3563-3 in sphenomegacorona118.8923-3 in sphenocorona3-3 in augmented sphenocorona1204-4 in hexagonal prism4-4 in elongated triangular cupola (prism) (asymmetrical)4-4 in elongated triangular orthobicupola (prism) (asymmetrical)4-4 in elongated triangular gyrobicupola (prism)4-4 in augmented hexagonal prism (2 possibilities) (both asymmetrical)4-4 in parabiaugmented hexagonal prism4-4 in metabiaugmented hexagonal prism (asymmetrical)121.717 (90 + dpc4)4-4 in elongated pentagonal cupola (cupola/prism) (asymmetrical)4-4 in elongated pentagonal orthobicupola (cupola/prism) (asymmetrical)4-4 in elongated pentagonal gyrobicupola (cupola/prism) (asymmetrical)4-4 in elongated pentagonal orthocupolarotunda (cupola/prism) (asymmetrical)4-4 in elongated pentagonal gyrocupolarotunda (cupola/prism) (asymmetrical)4-10 in diminished rhombicosidodecahedron4-10 in paragyrate diminished rhombicosidodecahedron4-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)4-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)4-10 in parabidiminished rhombicosidodecahedron4-10 in metabidiminished rhombicosidodecahedron (3 possibilities)4-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)4-10 in tridiminished rhombicosidodecahedron (3 possibilities)121.7433-3 in snub disphenoid (type 2) (2 possibilities) (both asymmetrical)124.7023-3 in disphenocingulum125.264 (90 + d4/2)3-4 in cuboctahedron4-6 in truncated octahedron3-8 in truncated cube6-8 in truncated cuboctahedron3-4 in triangular cupola (2 possibilities)3-4 in elongated triangular cupola (cupola) (2 possibilities)3-4 in gyroelongated triangular cupola (cupola) (2 possibilities)3-4 in triangular orthobicupola (2 possibilities)3-4 in elongated triangular orthobicupola (cupola) (2 possibilities)3-4 in elongated triangular gyrobicupola (cupola) (2 possibilities)3-4 in gyroelongated triangular bicupola (cupola) (2 possibilities)3-4 in augmented truncated tetrahedron (augment) (2 possibilities)3-8 in augmented truncated cube (3 possibilities)3-8 in biaugmented truncated cube126.87 (4*dpc4)5-5 in pentagonal orthobirotunda126.964 (da10 + dpc)3-4 in gyroelongated pentagonal cupola (cupola/antiprism)3-4 in gyroelongated pentagonal bicupola (cupola/antiprism)3-4 in gyroelongated pentagonal cupolarotunda (cupola/antiprism)127.377 (90 + dpp)3-4 in elongated pentagonal pyramid3-4 in elongated pentagonal dipyramid3-4 in elongated pentagonal cupola (cupola/prism)3-4 in elongated pentagonal orthobicupola (cupola/prism)3-4 in elongated pentagonal gyrobicupola (cupola/prism)3-4 in elongated pentagonal orthocupolarotunda (cupola/prism)3-4 in elongated pentagonal gyrocupolarotunda (cupola/prism)127.5523-3 in square antiprism3-3 in gyroelongated square pyramid (antiprism band) (asymmetrical)3-3 in gyroelongated square dipyramid (antiprism band)128.4963-3 in hebesphenomegacorona128.5714-4 in heptagonal prism129.4453-3 in sphenomegacorona131.4423-3 in augmented sphenocorona132.624 (da10 + dpp)3-3 in gyroelongated pentagonal cupola (cupola/antiprism) (asymmetrical)3-3 in gyroelongated pentagonal bicupola (cupola/antiprism) (asymmetrical)3-3 in gyroelongated pentagonal cupolarotunda (cupola/antiprism) (asymmetrical)133.5913-3 in disphenocingulum133.9733-4 in hebesphenomegacorona1354-4 in rhombicuboctahedron (asymmetrical)4-8 in truncated cuboctahedron4-4 in octagonal prism4-4 in square cupola4-4 in elongated square cupola (3 possibilities) (all asymmetrical)4-4 in gyroelongated square cupola (asymmetrical)4-4 in square orthobicupola (cupola) (asymmetrical)4-4 in square gyrobicupola (asymmetrical)4-4 in elongated square gyrobicupola (3 possibilities) (2 of them asymmetrical)4-4 in gyroelongated square bicupola (asymmetrical)4-4 in augmented truncated cube (asymmetrical)4-4 in biaugmented truncated cube (asymmetrical)135.9923-3 in sphenocorona3-3 in augmented sphenocorona136.3363-4 in disphenocingulum137.243-4 in sphenomegacorona138.19 (d20 = 2*(dpc4 + dpp)3-3 in icosahedron6-6 in truncated icosahedron3-3 in pentagonal antiprism3-3 in pentagonal pyramid3-3 in elongated pentagonal pyramid (asymmetrical)3-3 in gyroelongated pentagonal pyramid (3 possibilities) (all asymmetrical)3-3 in pentagonal dipyramid (apex) (asymmetrical)3-3 in elongated pentagonal dipyramid (asymmetrical)3-3 in augmented dodecahedron (asymmetrical)3-3 in parabiaugmented dodecahedron (asymmetrical)3-3 in metabiaugmented dodecahedron (3 possibilities) (all asymmetrical)3-3 in triaugmented dodecahedron (3 possibilities) (all asymmetrical)3-3 in metabidiminished icosahedron (4 possibilities) (3 asymmetrical)3-3 in tridiminished icosahedron (asymmetrical)3-3 in augmented tridiminished icosahedron (main body) (asymmetrical)3-6 in triangular hebesphenorotunda3-3 in triangular hebesphenorotunda1404-4 in enneagonal prism141.058 (2*d4)3-3 in triangular dipyramid (equatorial)3-3 in triangular orthobicupola3-6 in augmented truncated tetrahedron (augment/main body)141.313-3 in hebesphenomegacorona141.595 (45 + da8)3-4 in gyroelongated square cupola (cupola/antiprism)3-4 in gyroelongated square bicupola (cupola/antiprism)142.623 (did)3-5 in icosidodecahedron5-6 in truncated icosahedron3-10 in truncated dodecahedron6-10 in truncated icosidodecahedron3-5 in pentagonal rotunda (3 possibilities)3-5 in elongated pentagonal rotunda (3 possibilities)3-5 in gyroelongated pentagonal cupola (rotunda) (3 possibilities)3-5 in pentagonal orthocupolarotunda (rotunda) (3 possibilities)3-5 in pentagonal gyrocupolarotunda (3 possibilities)3-5 in pentagonal orthobirotunda (3 possibilities)3-5 in elongated pentagonal orthocupolarotunda (3 possibilities)3-5 in elongated pentagonal gyrocupolarotunda (3 possibilities)3-5 in elongated pentagonal orthobirotunda (3 possibilities)3-5 in elongated pentagonal gyrobirotunda (3 possibilities)3-5 in gyroelongated pentagonal cupolarotunda (3 possibilities)3-5 in gyroelongated pentagonal birotunda (3 possibilities)3-10 in augmented truncated dodecahedron (main body) (7 possibilities)3-10 in parabiaugmented truncated dodecahedron (main body) (3 possibilities)3-10 in metabiaugmented truncated dodecahedron (main body) (14 possibilities)3-10 in triaugmented truncated dodecahedron (main body) (9 possibilities)3-5 in bilunabirotunda3-5 in triangular hebesphenorotunda142.9833-4 in snub cube143.4793-3 in sphenocorona3-3 in augmented sphenocorona143.7383-3 in sphenomegacorona1444-4 in decagonal prism4-4 in elongated pentagonal cupola (prism) (asymmetrical)4-4 in elongated pentagonal rotunda (asymmetrical)4-4 in elongated pentagonal orthobicupola (prism) (asymmetrical)4-4 in elongated pentagonal gyrobicupola (prism)4-4 in elongated pentagonal orthocupolarotunda (prism) (asymmetrical)4-4 in elongated pentagonal gyrocupolarotunda (prism) (asymmetrical)4-4 in elongated pentagonal orthobirotunda (asymmetrical)4-4 in elongated pentagonal gyrobirotunda144.1443-3 in snub square antiprism (other edges of triangles adjacent to squares) (2 possibilities) (both asymmetrical)144.736 (180 - d4/2)3-4 in rhombicuboctahedron4-6 truncated cuboctahedron3-4 in square cupola3-4 in elongated square pyramid3-4 in elongated square dipyramid4-4 in elongated triangular cupola (cupola/prism) (asymmetrical)3-4 in elongated square cupola (2 possibilities)3-4 in gyroelongated square cupola (cupola)3-4 in square orthobicupola3-4 in square gyrobicupola (cupola)4-4 in elongated triangular orthobicupola (cupola/prism) (asymmetrical)4-4 in elongated triangular gyrobicupola (cupola/prism) (asymmetrical)3-4 in elongated square gyrobicupola (2 possibilities)3-4 in gyroelongated square bicupola (cupola)3-3 in augmented triangular prism (augment/base) (asymmetrical)3-3 in biaugmented triangular prism (augment/base) (asymmetrical)3-3 in triaugmented triangular prism (augment/base) (asymmetrical)3-5 in augmented pentagonal prism3-5 in biaugmented pentagonal prism3-6 in augmented hexagonal prism3-6 in parabiaugmented hexagonal prism3-6 in metabiaugmented hexagonal prism3-6 in triaugmented hexagonal prism3-4 in augmented truncated cube (augment)3-8 in augmented truncated cube (augment/main body)3-4 in biaugmented truncated cube (augment)3-8 in biaugmented truncated cube (augment/main body)145.2223-3 in hexagonal antiprism3-3 in gyroelongated triangular cupola (antiprism) (2 possibilities) (both asymmetrical)3-3 in in gyroelongated triangular bicupola (antiprism) (3 possibilities) (1 asymmetrical)145.4413-4 in snub square antiprism148.2834-5 in rhombicosidodecahedron4-10 in truncated icosidodecahedron4-5 in pentagonal cupola4-5 in elongated pentagonal cupola4-5 in gyroelongated pentagonal cupola4-5 in pentagonal orthobicupola4-5 in pentagonal gyrobicupola4-5 in pentagonal orthocupolarotunda4-5 in pentagonal gyrocupolarotunda (cupola)4-5 in elongated pentagonal orthobicupola4-5 in elongated pentagonal gyrobicupola4-5 in elongated pentagonal orthocupolarotunda (cupola)4-5 in elongated pentagonal gyrocupolarotunda (cupola)4-5 in gyroelongated pentagonal bicupola4-5 in augmented truncated dodecahedron4-5 in parabiaugmented truncated dodecahedron4-5 in metabiaugmented truncated dodecahedron (3 possibilities)4-5 in triaugmented truncated dodecahedron (3 possibilities)4-5 in gyrate rhombicosidodecahedron (7 possibilities)4-5 in parabigyrate rhombicosidodecahedron (3 possibilities)4-5 in metabigyrate rhombicosidodecahedron (15 possibilities)4-5 in trigyrate rhombicosidodecahedron (10 possibilities)4-5 in diminished rhombicosidodecahedron (6 possibilities)4-5 in paragyrate diminished rhombicosidodecahedron (5 possibilities)4-5 in metagyrate diminished rhombicosidodecahedron (22 possibilities)4-5 in bigyrate diminished rhombicosidodecahedron (21 possibilities)4-5 in parabidiminished rhombicosidodecahedron (2 possibilities)4-5 in metabidiminished rhombicosidodecahedron (11 possibilities)4-5 in gyrate bidiminished rhombicosidodecahedron (20 possibilities)4-5 in tridiminished rhombicosidodecahedron (6 possibilities)148.4343-3 in disphenocingulum149.5653-3 in hebesphenomegacorona1503-4 in gyrobifastigium (blend)150.2223-3 in heptagonal antiprism151.33 (90 + da8 - d4/2)3-3 in gyroelongated square cupola (cupola/antiprism) (asymmetrical)3-3 in gyroelongated square bicupola (cupola/antiprism) (asymmetrical)152.1913-3 in augmented sphenocorona152.933-5 in snub dodecahedron152.9763-4 in hebesphenomegacorona153.2353-3 in snub cube (2 possibilities) (1 asymmetrical)153.435 (90 + 2*dpc4)4-5 in elongated pentagonal rotunda4-5 in elongated pentagonal orthocupolarotunda (rotunda/prism)4-5 in elongated pentagonal gyrocupolarotunda (rotunda/prism)4-5 in elongated pentagonal orthobirotunda4-5 in elongated pentagonal gyrobirotunda4-4 in gyrate rhombicosidodecahedron (asymmetrical)4-4 in parabigyrate rhombicosidodecahedron (asymmetrical)4-4 in metabigyrate rhombicosidodecahedron (3 possibilities) (all asymetrical)4-4 in trigyrate rhombicosidodecahedron (3 possibilities) (all asymmetrical)4-4 in paragyrate diminished rhombicosidodecahedron (asymmetrical)4-4 in metagyrate diminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)4-4 in bigyrate diminished rhombicosidodecahedron (5 possibilities) (all asymmetrical)4-4 in gyrate bidiminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)153.635 (180 + da6 - d4/2)3-4 in gyroelongated triangular cupola (cupola/antiprism)3-4 in gyroelongated triangular bicupola (cupola/antiprism)153.942 (dpp + d12)3-5 in augmented dodecahedron3-5 in parabiaugmented dodecahedron3-5 in metabiaugmented dodecahedron (3 possibilities)3-5 in triaugmented dodecahedron (3 possibilities)3-10 in augmented truncated dodecahedron (augment/main body)3-10 in parabiaugmented truncated dodecahedron (augment/main body)3-10 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)3-10 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)3-5 in gyrate rhombicosidodecahedron3-5 in parabigyrate rhombicosidodecahedron3-5 in metabigyrate rhombicosidodecahedron (3 possibilities)3-5 in trigyrate rhombicosidodecahedron (3 possibilities)3-5 in paragyrate diminished rhombicosidodecahedron3-5 in metagyrate diminished rhombicosidodecahedron (3 possibilities)3-5 in bigyrate diminished rhombicosidodecahedron (5 possibilities)3-5 in gyrate bidiminished rhombicosidodecahedron (3 possibilities)153.9623-3 in octagonal antiprism3-3 in gyroelongated square cupola (antiprism) (2 possibilities) (both asymmetrical)3-3 in gyroelongated square bicupola (antiprism) (3 possibilities) (1 asymmetrical)154.4193-4 in disphenocingulum154.7223-4 in sphenomegacorona156.8663-3 in enneagonal antiprism157.1483-3 in hebesphenomegacorona158.375 (2*dpr3)3-3 in pentagonal orthobirotunda158.572 (90 + da4 - d4/2)3-3 in gyroelongated square pyramid (apex/antiprism) (asymmetrical)3-3 in in gyroelongated square dipyramid (apex/antiprism) (asymmetrical)158.682 (da10 + 2*dpc4)3-5 in gyroelongated pentagonal rotunda (rotunda/antiprism)3-5 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)3-5 in gyroelongated pentagonal birotunda (rotunda/antiprism)159.0953-4 in rhombicosidodecahedron4-6 in truncated icosidodecahedron3-4 in pentagonal cupola3-4 in elongated pentagonal cupola (cupola)3-4 in gyroelongated pentagonal cupola (cupola)3-4 in pentagonal orthobicupola3-4 in pentagonal gyrobicupola (cupola)3-4 in pentagonal orthocupolarotunda (cupola)3-4 in pentagonal gyrocupolarotunda3-4 in elongated pentagonal orthobicupola (cupola)3-4 in elongated pentagonal gyrobicupola (cupola)3-4 in elongated pentagonal orthocupolarotunda (cupola)3-4 in elongated pentagonal gyrocupolarotunda (cupola)3-4 in gyroelongated pentagonal bicupola (cupola)3-4 in gyroelongated pentagonal cupolarotunda (cupola)3-4 in augmented truncated dodecahedron (augment)3-4 in parabiaugmented truncated dodecahedron (augment)3-4 in metabiaugmented truncated dodecahedron (augment) (5 possibilities)3-4 in triaugmented truncated dodecahedron (augment) (5 possibilities)3-4 in gyrate rhombicosidodecahedron (7 possibilities)3-4 in parabigyrate rhombicosidodecahedron (3 possibilities)3-4 in metabigyrate rhombicosidodecahedron (14 possibilities)3-4 in trigyrate rhombicosidodecahedron (9 possibilities)3-4 in diminished rhombicosidodecahedron (6 possibilities)3-4 in paragyrate diminished rhombicosidodecahedron (5 possibilities)3-4 in metagyrate diminished rhombicosidodecahedron (21? possibilities -- I'm honestly not quite sure here)3-4 in bigyrate diminished rhombicosidodecahedron (17? possibilities)3-4 in parabidiminished rhombicosidodecahedron (2 possibilities)3-4 in metabidiminished rhombicosidodecahedron (9 possibilities)3-4 in gyrate bidiminished rhombicosidodecahedron (13 possibilities)3-4 in tridiminished rhombicosidodecahedron (4 possibilities)3-4 in bilunabirotunda3-4 in triangular hebesphenorotunda159.1873-3 in decagonal antiprism3-3 in gyroelongated pentagonal cupola (antiprism) (2 possibilities) (both asymmetrical)3-3 in gyroelongated pentagonal rotunda (antiprism) (2 possibilities) (both asymmetrical)3-3 in gyroelongated pentagonal bicupola (antiprism) (3 possibilities) (1 asymmetrical)3-3 in gyroelongated pentagonal cupolarotunda (antiprism) (4 possibilities) (all asymmetrical)3-3 in gyroelongated pentagonal birotunda (3 possibilities) (1 asymmetrical)159.8923-3 in sphenocorona3-3 in augmented sphenocorona160.529 (90 + d4)3-4 in elongated triangular pyramid (apex)3-4 in elongated triangular dipyramid3-4 in elongated triangular cupola (cupola/prism)3-4 in elongated triangular orthobicupola (cupola/prism)3-4 in elongated triangular gyrobicupola (cupola/prism)161.4833-3 in sphenomegacorona162.736 (198 - d4/2)3-4 in augmented pentagonal prism (augment/side)3-4 in biaugmented pentagonal prism (2 possibilities)164.1723-3 in snub dodecahedron (2 possibilities) (1 asymmetrical)164.207   (270 - 3*d4/2)3-4 in augmented truncated tetrahedron (augment/main body)164.2573-3 in snub square antiprism (vertical edges) (asymmetrical)166.4413-3 in snub disphenoid (type 3) (2 possibilities)166.8113-3 in disphenocingulum169.188 (90 + dpr3)3-4 in elongated pentagonal rotunda3-4 in elongated pentagonal orthocupolarotunda (rotunda/prism)3-4 in elongated pentagonal gyrocupolarotunda (rotunda/prism)3-4 in elongated pentagonal orthobirotunda3-4 in elongated pentagonal gyrobirotunda169.428 (d4 + da6)3-3 in gyroelongated triangular cupola (cupola/antiprism) (asymmetrical)3-3 in gyroelongated triangular bicupola (cupola/antiprism) (asymmetrical)169.471 (240 - d4)3-3 in biaugmented triangular prism (augment/augment)3-3 in triaugmented triangular prism (augment/augment)170.264 (135 + d4/2)3-4 in augmented truncated cube (augment/main body)3-4 in biaugmented truncated cube (augment/main body)171.341 (dpp + 2*dpc4 + d4)3-5 in augmented tridiminished icosahedron (augment/main body)171.6463-3 in sphenomegacorona171.755 (90 + dsp - d4/2)3-4 in augmented sphenocorona174.34 (dpc4 + did)3-4 in augmented truncated dodecahedron (augment/main body)3-4 in parabiaugmented truncated dodecahedron (augment/main body)3-4 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)3-4 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)174.434 (da10 + dpr3)3-3 in gyroelongated pentagonal rotunda (rotunda/antiprism) (asymmetrical)3-3 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)3-3 in gyroelongated pentagonal birotunda (rotunda/antiprism)174.736 (210 - d4/2)3-4 in augmented hexagonal prism3-4 in parabiaugmented hexagonal prism3-4 in metabiaugmented hexagonal prism (2 possibilities)3-4 in triaugmented hexagonal prismOf course, you must also take into account chiral polyhedra:snub cube, snub dodecahedron, gyroelongated triangular bicupola, gyroelongated square bicupola, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda. Those might count twice in all cases.`

From these, it should be possible to enumerate 4D edges.
Marek14
Pentonian

Posts: 1102
Joined: Sat Jul 16, 2005 6:40 pm