A pentagrammic polychoron with CRF cells

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

A pentagrammic polychoron with CRF cells

Postby quickfur » Wed Dec 25, 2013 2:10 am

While searching for CRF polychora, I stumbled upon a curious fact: the dichoral angle between the pentagonal cupola in the segmentochoron 5-prism||10-gon is exactly 144°. We can therefore glue 5 copies of this segmentochoron together at their pentagonal cupola cells, and it will close up into a polychoron (just like the 600-cell lunae, except that here it doesn't close up until it wraps around the circle twice, so it's non-convex). Sadly, the result is non-convex, but it does have all CRF cells.

The resulting polychoron has a pentagrammic arrangement around the plane of the 10-gon. All 5 copies of the segmentochoron share the same 10-gon, which becomes internal to the polychoron once it closes up; the cupolae likewise become internal to the polychoron. What's left on the surface are 5 pentagonal prisms, 25 triangular prisms, and 25 tetrahedra. It looks like some kind of pentagonal-pentagrammic duoprism augmented with some kind of pentagrammic tegum bridging augments (i.e., pentagrammic pyramid prisms bridged by pentagrammic tegums).
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Re: A pentagrammic polychoron with CRF cells

Postby wendy » Wed Dec 25, 2013 7:55 am

The pentagon-pentagram tegum in four dimensions, has 25 pentachoral faces. Is that interesting?
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Re: A pentagrammic polychoron with CRF cells

Postby Klitzing » Wed Dec 25, 2013 10:06 am

Hi Quickfur,
this polychoron is what is known as "stawros", (pentagrammic ({10} || pip-)wedge rosette, cf. http://bendwavy.org/klitzing/incmats/stawros.htm.
That one too could be derived from starpedip (5/2,5-dip) by augmenting the stips by line||stip segmentochora and filling in the gaps then by stappypies (line||ortho pentagram).
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Re: A pentagrammic polychoron with CRF cells

Postby quickfur » Wed Dec 25, 2013 3:57 pm

wendy wrote:The pentagon-pentagram tegum in four dimensions, has 25 pentachoral faces. Is that interesting?

The interest of this polychoron is that it can be cut up into five congruent decagon||5-prism pieces.
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Re: A pentagrammic polychoron with CRF cells

Postby quickfur » Wed Dec 25, 2013 3:59 pm

Klitzing wrote:Hi Quickfur,
this polychoron is what is known as "stawros", (pentagrammic ({10} || pip-)wedge rosette, cf. http://bendwavy.org/klitzing/incmats/stawros.htm.
That one too could be derived from starpedip (5/2,5-dip) by augmenting the stips by line||stip segmentochora and filling in the gaps then by stappypies (line||ortho pentagram).
--- rk

Right, that's what I was describing. :) I thought it was interesting that this shape can be cut into five congruent 10-gon||5-prism's, almost like the 600-cell can be cut into 10 lunae.
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Re: A pentagrammic polychoron with CRF cells

Postby Klitzing » Wed Dec 25, 2013 9:28 pm

quickfur wrote:... Right, that's what I was describing. :) I thought it was interesting that this shape can be cut into five congruent 10-gon||5-prism's, almost like the 600-cell can be cut into 10 lunae.


You then might like to consider quawros too (quadratic ({8} || cube-)wedge rosette), cf. http://bendwavy.org/klitzing/incmats/cube=8=cube.htm. It is made up similarily from 4 copies of the cube||{8} segmentochoron. That one moreover comes out to be convex.

In fact, quawros is a partial Stott contraction of sidpith (x3o3o4x).

--- rk
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Re: A pentagrammic polychoron with CRF cells

Postby quickfur » Thu Dec 26, 2013 5:18 am

Klitzing wrote:
quickfur wrote:... Right, that's what I was describing. :) I thought it was interesting that this shape can be cut into five congruent 10-gon||5-prism's, almost like the 600-cell can be cut into 10 lunae.


You then might like to consider quawros too (quadratic ({8} || cube-)wedge rosette), cf. http://bendwavy.org/klitzing/incmats/cube=8=cube.htm. It is made up similarily from 4 copies of the cube||{8} segmentochoron. That one moreover comes out to be convex.

In fact, quawros is a partial Stott contraction of sidpith (x3o3o4x).

--- rk

Wait, isn't that the same thing as the convex hull of a tesseract and an octagon, that I discovered in the Johnsonian polytopes thread? Yes, I did notice that it can be cut into 4 cube||8gon's.

In fact, I was looking for analogues of the 600-cell be constituted by 10 lunae, and looking at various CRF wedges to see if they have suitable dichoral angles that would allow them to be put together to close up a ball-like CRF shape. Unfortunately, I haven't found anything new in this direction yet.
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Re: A pentagrammic polychoron with CRF cells

Postby Klitzing » Thu Dec 26, 2013 7:44 pm

Ahh, finally you do remember :P

Yes this square version was indeed topic of the other thread, e.g. at http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&start=570#p18445 and followings.

And the star version, which you used for the opener of this thread, likewise was mentioned there already at http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&start=570#p18465.

---rk
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Re: A pentagrammic polychoron with CRF cells

Postby quickfur » Thu Dec 26, 2013 8:52 pm

Klitzing wrote:Ahh, finally you do remember :P

Yes this square version was indeed topic of the other thread, e.g. at http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&start=570#p18445 and followings.

Well, I remember that.

And the star version, which you used for the opener of this thread, likewise was mentioned there already at http://teamikaria.com/hddb/forum/viewtopic.php?f=25&t=1468&start=570#p18465.

---rk

Hehe, probably at that time I didn't realize what it was. :) I vaguely remembering skipping over something because I was only interested in convex shapes at the time.
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