Interesting 600-cell diminishings

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

Interesting 600-cell diminishings

Postby quickfur » Fri Jan 29, 2016 12:17 am

Today I started exploring diminishings of the 600-cell that feature dodecahedral cells. I started by cutting off two layers of cells to make a dodecahedral cell, then made two more similar cuts to obtain a linear chain of 3 dodecahedra along a great circle. Turns out that the dichoral angles between the dodecahedra is small enough that another dodecahedron would not fit along the same great circle; but the first and last dodecahedra can be connected to close the great circle by a pentagonal antiprism.

This sets the basic structure of this CRF. After that, I looked around the result a bit to see what else can be diminished to yield a more-or-less symmetrical structure. Around the pentagonal antiprism, it looked like deleting a circle of 10 vertices would produce a ring of teddies (tridiminished icosahedra) around it in alternating orientation. Furthermore, another 10 vertices can be deleted above 10 of the faces of the middle dodecahedron (where they are not shared with the adjacent dodecahedra), yielding two adjacent rings of icosahedral wedges (metabidiminished icosahedra) that encircle the two pentagonal faces shared between the middle dodecahedron and the two adjacent dodecahedra. The remaining gaps are filled by an alternating ring of 10 tetrahedra connected by their vertices. Each tetrahedron shares an edge with a great circle of edges that lies orthogonal to the plane of the 3-dodecahedron + antiprism ring.

The overall shape is a kind of blunt wedge, or trapezoidal shape with pentagonal symmetry. Here are some projections of it:

Side-view, showing the blunt-wedge shape:
Image

The "blunt edge" of the shape (projected to the top here) is the pentagonal antiprism; the other 3 sides are the 3 dodecahedra. The metabidiminished icosahedra lie between the red tetrahedra. There are actually 6 tetrahedra visible in this projection, but for clarity I didn't color the two that lie on the limb of the projection (they are at 90° to the 4D viewpoint). The far side of the polytope has the same structure but in skewed orientation, making a total of 10 tetrahedra.

Projection centered on middle dodecahedron:
Image

The big yellow dodecahedron is the one in the center. You can see parts of the other two dodecahedra as yellow pentagons on the top and bottom. The metabidiminished icosahedra are clearly seen, and the edges around outer faces of the tetrahedra trace out the bottom of the teddies that lie on the far side of the polytope from this viewpoint.

Projection centered on pentagonal antiprism:
Image

Here we see the pentagonal antiprism sandwiched between the other two dodecahedra. If you look carefully, you can see teddies attached to each of the triangular faces of the antiprism, with their bottoms pointing outwards.

This polytope has 60 vertices, 170 edges, 144 faces (100 triangles, 44 pentagons), and 34 cells (10 tetrahedra, 10 metabidiminished icosahedra, 10 tridiminished icosahedra, 3 dodecahedra, and 1 pentagonal antiprism).

Here are the coordinates of the 60 vertices (scaled such that the edge length is 2):
Code: Select all
# phi = (1+sqrt(5))/2 = golden ratio
<0, 0, ±2*phi, 0>
<0, ±2*phi, 0, 0>

<0, ±1, ±phi, -phi^2>
<0, ±phi, ±phi^2, -1>
<0, ±phi^2, ±1, phi>
<±1, 0, ±phi^2, phi>
<±1, ±phi, 0, -phi^2>
<±1, ±phi^2, ±phi, 0>
<-phi, 0, 1, -phi^2>
<phi, 0, -1, -phi^2>
<-phi, ±1, phi^2, 0>
<phi, ±1, -phi^2, 0>
<±phi, ±phi^2, 0, -1>
<-phi^2, 0, phi, -1>
<phi^2, 0, -phi, -1>
<±phi^2, ±1, 0, phi>
<-phi^2, ±phi, 1, 0>
<phi^2, ±phi, -1, 0>

<±phi, ±phi, ±phi, phi>
Last edited by quickfur on Fri Jan 29, 2016 10:07 pm, edited 1 time in total.
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Re: A 600-cell diminishing

Postby Klitzing » Fri Jan 29, 2016 4:07 pm

Ah, I understand what you are after here.

Just consider the lace city display of ex (600-cell) wrt. o5o2o5o subsymmetry. There you'll have:
Code: Select all
                 o5o           o5o                 
                        o5x                       
                                                  
            x5o                     x5o           
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                  
     o5x                                   o5x     
            f5o                     f5o           
                                                  
o5o                     x5x                     o5o
                                                  
            o5f                     o5f           
     x5o                                   x5o     
                                                  
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                  
                        x5o                       
                 o5o           o5o                 

Then you cut off 3 times a 2-segmental cap, each uderneath an ex vertex, providing as sefa (sectioning facet underneath) these does (dodecahedra):
Code: Select all
                                    x5o
                                      
                        f5o           
                 o5f           o5f     
                                      
     o5x                               
            f5o                     f5o
                                      
o5o                     x5x           
                                      
            o5f                     o5f
     x5o                               
                                      
                 f5o           f5o     
                        o5f           
                                      
                                    o5x

Finally you cut off the remaining single vertex from that former "outer circuit" (great circle) of 10 vertices, this time just a single segment of depth. - Without the former diminishings this then would result in a cut off of an ikepy (icosahedral pyramid) only. But now you just cut off a pappy (pyramid above a 5-antiprism). - Thus your final lace city here becomes:
Code: Select all
                               x5o
                                  
                   f5o           
            o5f           o5f     
                                  
o5x                               
       f5o                     f5o
                                  
                   x5x           
                                  
       o5f                     o5f
x5o                               
                                  
            f5o           f5o     
                   o5f           
                                  
                               o5x

This projection then additionally provides that this multi-wedge has dihedral angles of 72° at the pentagons between the does and that it has dihedral angles of 108° at the pentagons between doe and pap (5-antiprism).

Nice find of yours, indeed!

--- rk
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Re: A 600-cell diminishing

Postby quickfur » Fri Jan 29, 2016 4:22 pm

I also considered a different way of proceeding after the initial cuts that produce the 3 dodecahedra and antiprism. Instead of trimming the vertices around these cells to make diminished icosahedra, one could instead delete the great circle of 10 vertices that lie orthogonal to the plane of these cells, thereby creating a ring of 10 pentagonal antiprisms. In that case, we find that we obtain a diminishing of the grand antiprism containing 3 dodecahedra!
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Re: A 600-cell diminishing

Postby Klitzing » Fri Jan 29, 2016 4:48 pm

Oops, it just occured to me, that you used (in that first mentioned one with pics) also several mibdis (metabidim. ikes) and teddis (tridim. ikes). That is, my just posted lace city is still wrong. As that one still contains their circumcenters as additional vertices. When those additionally are to be erased, then we are left with

Code: Select all
                               x5o
                                  
                   f5o           
            o5f                   
                                  
o5x                               
                               f5o
                                  
                   x5x           
                                  
                               o5f
x5o                               
                                  
            f5o                   
                   o5f           
                                  
                               o5x

which then provides truely your vertex count (60).

So your CRF is a bit different from the mere multi-wedge, I'd just described (having 80 vertices instead).

--- rk
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Re: Interesting 600-cell diminishings

Postby quickfur » Fri Jan 29, 2016 11:05 pm

I found another interesting diminishing of the 600-cell today, along similar lines, sporting dodecahedral cells. This time, it's a more symmetrical one: it has 3 dodecahedra in a ring, sharing vertices, and 6 icosahedra in an orthogonal ring, sharing faces. Thus, it has a kind of 3,6-duoprism symmetry.

The construction again starts by diminishing two layers of vertices from the 600-cell to produce a dodecahedron. Next, instead of finding another cut that produces a dodecahedron sharing faces with the first one, we find a cut that produces a dodecahedron that only touches the first one at a vertex. This is uniquely determined, since at any other angle the second dodecahedron will share at least an edge. The angle between the two dodecahedra turns out to be exactly 60°, so a third cut is possible, with the third dodecahedron touching the first two at two of its antipodal vertices. This gives us the trigonal symmetry.

Each pentagonal face of the dodecahedra connect with pentagonal pyramids, with adjacent pentagonal pyramids interlocking in interesting ways.

Next, looking at the remaining surface of the 600-cell between these dodecahedra, we find that there's enough space squeeze in icosahedra along an orthogonal ring, sharing edges with the dodecahedra. It turns out that it's possible to delete 6 more vertices along this ring, producing 6 icosahedra sharing faces with each other. This gives us the hexagonal symmetry.

So this CRF has a nice 3,6-duoprism symmetry, and seems to be somewhat related to the snub 24-cell, which also has 6-membered rings of icosahedra sharing faces. It isn't a true diminishing of the snub 24-cell, because some of the dodecahedra's vertices do not lie on an inscribed snub 24-cell. Nevertheless, it does share a similar structure of one of the rings of 6 icosahedra.

Anyway, here are some pictures:

Image

This is an orthogonal projection looking straight at one of the icosahedral cells. You can see how the 3 dodecahedra, in yellow, wrap tightly around the ring of icosahedra. Interfacing these two rings are pentagonal pyramids and tetrahedra. The tetrahedra form quite an intricate structure; there are 90 tetrahedra in total, with 30 tetrahedra in a ring system between each pair of dodecahedra. Tetrahedra from different rings are disconnected, except for a few that touch at their vertices. Within each ring, not all 30 tetrahedra are equivalent; there are at least two classes: those that share faces with two adjacent icosahedra, and those that snake around the pentagonal pyramids. I made a graph of their face connectivity, and it showed up as a 24-membered ring with 6 branches to isolated nodes (probably the tetrahedra that sit between two icosahedra).

Here's another projection, looking at one of the dodecahedral cells:

Image

Due to the large number of visually-conflicting edges, I colored the edges of the dodecahedron red to make them easier to discern. You can see how the ring of icosahedra wrap around the 3-fold symmetry of the dodecahedron, sharing edges of alternating orientation.

Here's a look at the joint between two dodecahedra:

Image

You can see how 3 pentagonal pyramids attached to one dodecahedron straddle 3 pentagonal pyramids attached to the second dodecahedron in an interesting alternating fashion, with a band of tetrahedra wrapping around them and interfacing with the outer ring of icosahedra.

In total, there are 75 vertices, 306 edges, 366 faces (330 triangles, 36 pentagons), and 135 cells (90 tetrahedra, 36 pentagonal pyramids, 6 icosahedra, and 3 dodecahedra).

Here are the coordinates:
Code: Select all
<0, 0, 0, -(1+sqrt(5))>
<0, 0, ±1+sqrt(5), 0>
<0, ±1+sqrt(5), 0, 0>
<±1+sqrt(5), 0, 0, 0>

<0, -1, (1+sqrt(5))/2, -(3+sqrt(5))/2>
<0, 1, -(1+sqrt(5))/2, -(3+sqrt(5))/2>
<0, -(1+sqrt(5))/2, ±(3+sqrt(5))/2, 1>
<0, -(1+sqrt(5))/2, (3+sqrt(5))/2, -1>
<0, (1+sqrt(5))/2, -(3+sqrt(5))/2, ±1>
<0, (1+sqrt(5))/2, (3+sqrt(5))/2, 1>
<0, -(3+sqrt(5))/2, ±1, (1+sqrt(5))/2>
<0, -(3+sqrt(5))/2, 1, -(1+sqrt(5))/2>
<0, (3+sqrt(5))/2, -1, ±(1+sqrt(5))/2>
<0, (3+sqrt(5))/2, 1, (1+sqrt(5))/2>
<-1, 0, ±(3+sqrt(5))/2, (1+sqrt(5))/2>
<-1, 0, (3+sqrt(5))/2, -(1+sqrt(5))/2>
<1, 0, -(3+sqrt(5))/2, ±(1+sqrt(5))/2>
<1, 0, (3+sqrt(5))/2, (1+sqrt(5))/2>
<-1, (1+sqrt(5))/2, 0, -(3+sqrt(5))/2>
<1, -(1+sqrt(5))/2, 0, -(3+sqrt(5))/2>
<-1, ±(3+sqrt(5))/2, (1+sqrt(5))/2, 0>
<1, ±(3+sqrt(5))/2, -(1+sqrt(5))/2, 0>
<-(1+sqrt(5))/2, 0, 1, -(3+sqrt(5))/2>
<(1+sqrt(5))/2, 0, -1, -(3+sqrt(5))/2>
<-(1+sqrt(5))/2, 1, ±(3+sqrt(5))/2, 0>
<(1+sqrt(5))/2, -1, ±(3+sqrt(5))/2, 0>
<-(1+sqrt(5))/2, ±(3+sqrt(5))/2, 0, 1>
<-(1+sqrt(5))/2, (3+sqrt(5))/2, 0, -1>
<(1+sqrt(5))/2, -(3+sqrt(5))/2, 0, ±1>
<(1+sqrt(5))/2, (3+sqrt(5))/2, 0, 1>
<-(3+sqrt(5))/2, 0, ±(1+sqrt(5))/2, 1>
<-(3+sqrt(5))/2, 0, (1+sqrt(5))/2, -1>
<(3+sqrt(5))/2, 0, -(1+sqrt(5))/2, ±1>
<(3+sqrt(5))/2, 0, (1+sqrt(5))/2, 1>
<-(3+sqrt(5))/2, ±1, 0, (1+sqrt(5))/2>
<-(3+sqrt(5))/2, 1, 0, -(1+sqrt(5))/2>
<(3+sqrt(5))/2, -1, 0, ±(1+sqrt(5))/2>
<(3+sqrt(5))/2, 1, 0, (1+sqrt(5))/2>
<±(3+sqrt(5))/2, -(1+sqrt(5))/2, 1, 0>
<±(3+sqrt(5))/2, (1+sqrt(5))/2, -1, 0>

<-(1+sqrt(5))/2, -(1+sqrt(5))/2, ±(1+sqrt(5))/2, (1+sqrt(5))/2>
<(1+sqrt(5))/2, -(1+sqrt(5))/2, ±(1+sqrt(5))/2, ±(1+sqrt(5))/2>
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Re: Interesting 600-cell diminishings

Postby Klitzing » Sat Jan 30, 2016 11:44 am

And here follows now the incidence matix of that first fellow.
I'll use the following vertex type descriptions (provided at the right end of the matrix) as are shown now in this recap of its lace city:
Code: Select all
                               x5o
                               A 
                   f5o           
            o5f    C             
            E                     
o5x                               
F                              f5o
                               B 
                   x5x           
                   D             
                               o5f
x5o                               
                                  
            f5o                   
                   o5f           
                                  
                               o5x
                                  
 |           |      |           +- doe
 |           |      +------------- non-CRF para-bidim. id
 |           +-------------------- f-pap
 +-------------------------------- pap

then it runs as follows:
Code: Select all
10  *  *  *  *  * |  2  1  1  0  0  0  0  0  0  0  0  0  0 | 1  2  2  1  0  0  0  0  0  0  0  0 0  0 | 1 1  2  0  0 0 A
 * 10  *  *  *  * |  0  1  0  2  1  2  0  0  0  0  0  0  0 | 0  3  0  1  2  2  1  0  0  0  0  0 0  0 | 1 0  3  1  0 0 B
 *  * 10  *  *  * |  0  0  1  0  1  0  2  2  0  0  0  0  0 | 0  0  2  1  0  2  0  1  2  1  0  0 0  0 | 0 1  2  1  1 0 C
 *  *  * 10  *  * |  0  0  0  0  0  2  2  0  2  2  0  0  0 | 0  0  0  0  1  2  2  2  2  0  2  1 0  0 | 0 0  2  2  2 0 D
 *  *  *  * 10  * |  0  0  0  0  0  0  0  2  0  2  1  0  0 | 0  0  1  0  0  0  0  0  2  2  1  2 0  0 | 0 1  1  0  3 0 E
 *  *  *  *  * 10 |  0  0  0  0  0  0  0  0  0  0  1  2  2 | 0  0  0  0  0  0  0  0  0  2  0  2 1  3 | 0 1  0  0  3 1 F
------------------+----------------------------------------+-----------------------------------------+---------------
 2  0  0  0  0  0 | 10  *  *  *  *  *  *  *  *  *  *  *  * | 1  1  1  0  0  0  0  0  0  0  0  0 0  0 | 1 1  1  0  0 0
 1  1  0  0  0  0 |  * 10  *  *  *  *  *  *  *  *  *  *  * | 0  2  0  1  0  0  0  0  0  0  0  0 0  0 | 1 0  2  0  0 0
 1  0  1  0  0  0 |  *  * 10  *  *  *  *  *  *  *  *  *  * | 0  0  2  1  0  0  0  0  0  0  0  0 0  0 | 0 1  2  0  0 0
 0  2  0  0  0  0 |  *  *  * 10  *  *  *  *  *  *  *  *  * | 0  2  0  0  1  0  0  0  0  0  0  0 0  0 | 1 0  2  0  0 0
 0  1  1  0  0  0 |  *  *  *  * 10  *  *  *  *  *  *  *  * | 0  0  0  1  0  2  0  0  0  0  0  0 0  0 | 0 0  2  1  0 0
 0  1  0  1  0  0 |  *  *  *  *  * 20  *  *  *  *  *  *  * | 0  0  0  0  1  1  1  0  0  0  0  0 0  0 | 0 0  2  1  0 0
 0  0  1  1  0  0 |  *  *  *  *  *  * 20  *  *  *  *  *  * | 0  0  0  0  0  1  0  1  1  0  0  0 0  0 | 0 0  1  1  1 0
 0  0  1  0  1  0 |  *  *  *  *  *  *  * 20  *  *  *  *  * | 0  0  1  0  0  0  0  0  1  1  0  0 0  0 | 0 1  1  0  1 0
 0  0  0  2  0  0 |  *  *  *  *  *  *  *  * 10  *  *  *  * | 0  0  0  0  0  0  1  1  0  0  1  0 0  0 | 0 0  1  1  1 0
 0  0  0  1  1  0 |  *  *  *  *  *  *  *  *  * 20  *  *  * | 0  0  0  0  0  0  0  0  1  0  1  1 0  0 | 0 0  1  0  2 0
 0  0  0  0  1  1 |  *  *  *  *  *  *  *  *  *  * 10  *  * | 0  0  0  0  0  0  0  0  0  2  0  2 0  0 | 0 1  0  0  3 0
 0  0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 10  * | 0  0  0  0  0  0  0  0  0  1  0  0 1  1 | 0 1  0  0  1 1 pap-base-edges
 0  0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  *  * 10 | 0  0  0  0  0  0  0  0  0  0  0  1 0  2 | 0 0  0  0  2 1 pap-lacings
------------------+----------------------------------------+-----------------------------------------+---------------
 5  0  0  0  0  0 |  5  0  0  0  0  0  0  0  0  0  0  0  0 | 2  *  *  *  *  *  *  *  *  *  *  * *  * | 1 1  0  0  0 0
 2  3  0  0  0  0 |  1  2  0  2  0  0  0  0  0  0  0  0  0 | * 10  *  *  *  *  *  *  *  *  *  * *  * | 1 0  1  0  0 0
 2  0  2  0  1  0 |  1  0  2  0  0  0  0  2  0  0  0  0  0 | *  * 10  *  *  *  *  *  *  *  *  * *  * | 0 1  1  0  0 0
 1  1  1  0  0  0 |  0  1  1  0  1  0  0  0  0  0  0  0  0 | *  *  * 10  *  *  *  *  *  *  *  * *  * | 0 0  2  0  0 0
 0  2  0  1  0  0 |  0  0  0  1  0  2  0  0  0  0  0  0  0 | *  *  *  * 10  *  *  *  *  *  *  * *  * | 0 0  2  0  0 0
 0  1  1  1  0  0 |  0  0  0  0  1  1  1  0  0  0  0  0  0 | *  *  *  *  * 20  *  *  *  *  *  * *  * | 0 0  1  1  0 0
 0  1  0  2  0  0 |  0  0  0  0  0  2  0  0  1  0  0  0  0 | *  *  *  *  *  * 10  *  *  *  *  * *  * | 0 0  1  1  0 0
 0  0  1  2  0  0 |  0  0  0  0  0  0  2  0  1  0  0  0  0 | *  *  *  *  *  *  * 10  *  *  *  * *  * | 0 0  0  1  1 0
 0  0  1  1  1  0 |  0  0  0  0  0  0  1  1  0  1  0  0  0 | *  *  *  *  *  *  *  * 20  *  *  * *  * | 0 0  1  0  1 0
 0  0  1  0  2  2 |  0  0  0  0  0  0  0  2  0  0  2  1  0 | *  *  *  *  *  *  *  *  * 10  *  * *  * | 0 1  0  0  1 0
 0  0  0  2  1  0 |  0  0  0  0  0  0  0  0  1  2  0  0  0 | *  *  *  *  *  *  *  *  *  * 10  * *  * | 0 0  1  0  1 0
 0  0  0  1  2  2 |  0  0  0  0  0  0  0  0  0  2  2  0  1 | *  *  *  *  *  *  *  *  *  *  * 10 *  * | 0 0  0  0  2 0
 0  0  0  0  0  5 |  0  0  0  0  0  0  0  0  0  0  0  5  0 | *  *  *  *  *  *  *  *  *  *  *  * 2  * | 0 1  0  0  0 1
 0  0  0  0  0  3 |  0  0  0  0  0  0  0  0  0  0  0  1  2 | *  *  *  *  *  *  *  *  *  *  *  * * 10 | 0 0  0  0  1 1
------------------+----------------------------------------+-----------------------------------------+---------------
10 10  0  0  0  0 | 10 10  0 10  0  0  0  0  0  0  0  0  0 | 2 10  0  0  0  0  0  0  0  0  0  0 0  0 | 1 *  *  *  * * doe
 5  0  5  0  5  5 |  5  0  5  0  0  0  0 10  0  0  5  5  0 | 1  0  5  0  0  0  0  0  0  5  0  0 1  0 | * 2  *  *  * * doe
 2  3  2  2  1  0 |  1  2  2  2  2  4  2  2  1  2  0  0  0 | 0  1  1  2  2  2  1  0  2  0  1  0 0  0 | * * 10  *  * * mibdi
 0  1  1  2  0  0 |  0  0  0  0  1  2  2  0  1  0  0  0  0 | 0  0  0  0  0  2  1  1  0  0  0  0 0  0 | * *  * 10  * * tet
 0  0  1  2  3  3 |  0  0  0  0  0  0  2  2  1  4  3  1  2 | 0  0  0  0  0  0  0  1  2  1  1  2 0  1 | * *  *  * 10 * teddi
 0  0  0  0  0 10 |  0  0  0  0  0  0  0  0  0  0  0 10 10 | 0  0  0  0  0  0  0  0  0  0  0  0 2 10 | * *  *  *  * 1 pap

--- rk
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Re: Interesting 600-cell diminishings

Postby Klitzing » Sat Jan 30, 2016 1:32 pm

quickfur wrote:I found another interesting diminishing of the 600-cell today, along similar lines, sporting dodecahedral cells. This time, it's a more symmetrical one: it has 3 dodecahedra in a ring, sharing vertices, and 6 icosahedra in an orthogonal ring, sharing faces. Thus, it has a kind of 3,6-duoprism symmetry.

...

Image

...

And here is the according lace city. The 3 dodecahedra are spotted immediately as the sides of the trigon. Moreover it is evident that those connect at vertices only. The central vertices (of the former ex) in this projection likewise are rejected. These were the tips of the 6 icosahedral pyramids.
Code: Select all
               o3o               
                                
             o3f f3o             
                                
                                
          f3x       x3f         
                                
            o3F   F3o           
                                
                                
     x3f F3o         o3F f3x     
                                
                                
  f3o       o3F   F3o       o3f 
                                
o3o o3f   f3x       x3f   f3o o3o

(as usual: |x| = 1, |f| = 1.618, |F| = |f|² = 2.618)

--- rk
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Re: Interesting 600-cell diminishings

Postby Klitzing » Tue Aug 16, 2016 2:46 pm

After being rather quiet in this forum these days, I tried to bring up a further interesting CRF to wake you up again.

To that reason I started with a dodecahedron (doe). So we have pentagons to adjoin. Here a good idea would be to use either gyroelongated pentagonal pyramids (gyepip, J11), metabidiminished icosahedra (mibdi, J62), or tridiminished ones (teddi, J63). The first ones would be not much interesting, as we would end in a way too simple hexacosachoron (ex) diminishing.

Therefore I tried the other ones. Those, when applied, then would break the symmetries of the pentagons. This is possible to be done systematically, when using the pyritohedral subsymmetry of doe.

When trying teddies this fails, because the omitted icosahedral (ike) vertices of one teddi would then still be required from the neighbouring one. But when using mibdies instead, we will get V-shaped gaps of pentagons at the 6 mutually orthogonal special edges. There then can be inserted one more mibdi each. That one then settles the overall curvature too. Thus we get back again to an ex diminishing. But that one looks not too familliar, as it uses pyritohedral symmetry in axial direction!

Therefore, the remainder surely can be filled with tetrahedra (tet) only. But on the other hand, the second set of mibdies just reaches as far as the opposite doe facet of ex. So we could fill the remainder with correspondingly fewer tets and 12 pentagonal pyramids (peppy) instead.


While thus having described that fellow already constructively, I furthermore wanted to understand that one by means of vertex layers. To that end we know already that ex can be described as  point || ike || doe || f-ike || id || f-ike || doe || ike || point.
Thus we will have to chop off both outer layers down to the doe facet, that is use deep diminishings there. Next we want to place mibdies. Mibdies are in turn diminished ikes and thus further shallow diminishings of ex. Thus each such mibdi deletes one further vertex of ex directly above eauch pentagon of one of those does. In sum this deletes the full next layer (f-ike). Thereafter we aimed to place a further set of 6 mibdies atop those special edges of doe. This then deletes again one vertex each in the next layer above, that is, we delete an octahedral subset of those id vertices.

When applying this to the lace city of ex (here for sure that one in o2o2o2o symmetry)
Code: Select all
                    o2o                          -- point
                                          
          o2x f2o   x2f   f2o o2x                -- ike
                                          
                                          
    x2o   f2f o2F   F2x   o2F f2f   x2o          -- doe
                                          
    o2f   F2o       f2F       F2o   o2f          -- f-ike
                                          
                                          
o2o f2x   x2F F2f  Vo2oV  F2f x2F   f2x o2o      -- id
                                          
                                          
    o2f   F2o       f2F       F2o   o2f          -- f-ike
                                          
    x2o   f2f o2F   F2x   o2F f2f   x2o          -- doe
                                          
                                          
          o2x f2o   x2f   f2o o2x                -- ike
                                          
                    o2o                          -- point

we would come out with the following lace city of the here described 2+12+6-diminishing of ex:
Code: Select all
x2o   f2f o2F   F2x   o2F f2f   x2o      -- doe
                                  
                                  
                                  
                                  
f2x   x2F F2f         F2f x2F   f2x      -- id \ fq-oct
                                  
                                  
o2f   F2o       f2F       F2o   o2f      -- f-ike
                                  
x2o   f2f o2F   F2x   o2F f2f   x2o      -- doe

Last but not least that resulting rectangular shape of the lace city, featuring one mibdi each on either side (as vertical column), is quite pleasing. Ain't it?

--- rk
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Re: Interesting 600-cell diminishings

Postby Klitzing » Thu Aug 18, 2016 5:21 pm

Thought about replacing the "doe + 3 mibdi around an edge" theme of the above figure by some "x3xPo + 3 tut", i.e. of replacing base / lacing pentagons by hexagons.

The relevant starting sequence of such a tower then is  xuxo...-3-xoop...-P-oxux...-&#xt,  where p = x(P,2) = 2 cos(pi/P).

  • For P=2 we then can finish right at that level, because of p=o here.
    Sadly the outcome is already a known uniform, it just happens to be tip = x3x3o3o.
  • For P=3 we also can finish right there as well, because of p=x then.
    Sadly that outcome also is known already to be uniform, as it then results in thex = x3x3o4o.
  • The case P=4 leads to a contradiction to CRF, because p=q here, and we don't have some xo..3oq..&#xt 3D-CRF.
  • The final case P=5 then has to be expanded way beyond that level.
    My current estimate is a still un-closed tower xuxoooxf...-3-xoofxuxu...-5-oxuxuxfo...-&#xt,
    which, if I got it right, so far incorporates 20+30+20 tut + 12 pap + 1+12 ti + 20 teddi + ...
Would anyone like to take the chance and try to close that P=5 fellow?
At least, when it could be closed somehow, it would be a true CRF, right because of those already contained teddies!

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Re: Interesting 600-cell diminishings

Postby Klitzing » Thu Aug 18, 2016 5:34 pm

Forgot to mention first, that the recently described axially pyrohedral ex diminishing amounts in a total of 132 cells:
2 does + 18 mibdies (J62) + 12 peppies (J2) + 100 tets.

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Re: Interesting 600-cell diminishings

Postby Klitzing » Tue Feb 14, 2017 10:57 pm

Half a year after the last post in this thread! And indeed quite silent within this forum in general in the last time. :\
So I thought it would be good to be revived.

Thought about placing triples of mibdies (J62's) onto neighbouring of pentagons of an id (icosidodecahedron). And atop the central triangle a teddi (J63). Continue with that with the remainder of id (which works) and then fill up this partial complex with tets (tetrahedra) up to the level of the tops of the teddies. There then a final ike (icosahedron) could be placed to close that CRF.

Even so this CRF just happens to be a special heptadeca-diminishing of the rotunda (hemiglome) of ex (hexacosachoron), it is kind of interesting as it implements a directed axially tetrahedral subsymmetry of the former directed axially icosahedral symmetry of that rotunda. And as such it then happens to be a chiral one as well. This chirality also can be seen from its following lace city display:

Code: Select all
        x3o   o3f f3o   o3x       
                                  
                                  
     o3f   f3x  demi(x3f)  f3o o3o
                                  
                                       (F=ff)
                                  
                                  
o3x   x3f F3o   f3f   o3F f3x   x3o

i.e. it also could be described as a diminished bistratic segment of ex, as   ike || tet-dim-doe || id.
That tet-dim-doe (tetrahedrally diminished dodecahedron) in turn rasps 4 vertices of a doe (in tetrahedrally arrangement) down to the neighbouring vertices. Accordingly its faces are (fxxx) = fx&#x -trapezia and f3o triangles. Even so this medial section itself is not CRF (here: not Johnsonian), the total polychoron is CRF. The trapezia there happen to be according sections of the mibdies and the large regular triangles are sections of the teddies. The longer f-edges of that section in fact just occur as sections of the lacing pentagons.

Within the above lace city one vertex at the left end of the medial section is missing and every second vertex of that semiregular triangle x3f. As this could be chosen in either of 2 alternating ways, this shows also the chirality. On the other hand it is known that the vertices of doe could be used for an inscribed chiral compound of 5 tets. One of these here has been selected for the diminishings. And in fact, the 4 missing vertices of that layer represent the centers of the teddies. While the centers of the 12 mibdies would represent a further layer, which in the above lace city was completely erased (when compared to that of ex).

The total count of cells of this CRF will be:
1 ike + 50 tets + 4 teddies + 12 mibdies + 1 id

Perhaps someone wants to render some nice pics of that polychoron? :D

--- rk
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Re: Interesting 600-cell diminishings

Postby Klitzing » Thu Feb 16, 2017 11:35 pm

... and then, there would be the possibility too, to diminish that fellow even further: to omit the opposite vertices of that already tetrahedrally diminished dodecahedronal section. That would make then a cubically diminished dodecahedronal section.

That is we could well consider:   ike || cube-dim-doe || id,   or in terms of lace cities either
Code: Select all
        x3o   o3f f3o   o3x       
                                  
         demi      demi           
     o3f  (f3x)     (x3f)  f3o     
                                  
                                       (F=ff)
                                  
                                  
o3x   x3f F3o   f3f   o3F f3x   x3o
or alternatively that orientation
Code: Select all
          o2x f2o   x2f   f2o o2x         
                                          
                                          
    x2o       o2F   F2x   o2F       x2o   
                                               (F=ff)
                                               (V=2f)
                                          
                                          
o2o f2x   x2F F2f  Vo2oV  F2f x2F   f2x o2o


Again these diminishings at this layer will produce teddies. That is 8 in our new scenario. But this then would also further diminish the cells, which occured by the diminishings at the already fully omitted lower medial layer (the f-scaled icosahedral one). That is, the 12 mibdies there would become teddies too. - While the former 8 ones are vertical ones, these 12 now would be lying ones.

And, for sure, the count of remaining tetrahedra also decreases once more. In fact, the total count of cells now becomes:
1 ike,
20 teddies (J63),
30 tets,
1 id

And the overall directed axially chiral tetrahedral symmetry of the last time polychoron here would become a directed axially pyritohedral symmetry. And, for sure, that new polychoron is still a CRF!

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