Sharpohedron

For general discussion of polyhedra, not necessarily Stella-specific.
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marcelteun
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Sharpohedron

Post by marcelteun » Wed Mar 10, 2021 10:54 am

Hi

Some time ago George Hart posted a polyhedron on Instagram and some time late, since he got some questions, he made a little video, which can be found here:
https://www.youtube.com/watch?v=D1tAv3cfmvI&authuser=0

I forwarded this to a friend of mine, Don Romano, who quickly built a model. Then he combined two of them which looked like this:
Image

It reminded him of Wenninger's # 70
https://en.wikipedia.org/wiki/Small_dit ... decahedron

I confirmed but stated that the triangles aren't really coplanar. That lead me to the following "Swissohedron":
https://tunnissen.eu/polyh/local_off.ht ... istance=15

I am not sure whether this is a known polyhedron, I cannot imagine that I am the first, but I couldn't find anything with Google. Probably I using the wrong keywords. Anyway, since i thought it made for an interesting model, that is why I wanted to share it with you.

It can be seen as consisting of 8 cuboctahedra combined with six square pyramids (half octahedra). Together with octahedra you could fill a space. I used the colours that could remind one of a Swiss flag. Perhaps I should make a paper model of it and call it Swiss Armygedron. :P
Last edited by marcelteun on Sun Apr 11, 2021 5:14 pm, edited 4 times in total.

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Ulrich
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Re: Sharpohedron

Post by Ulrich » Wed Mar 10, 2021 4:01 pm

That‘s very nice! I never saw it before. It‘s a truncated octahedron augmented by 8 triangular cupolae (j3).
Switzerland should take a gif file of this rotating model as a flag on their website schweiz.ch.
Ulrich

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marcelteun
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Re: Crossohedra

Post by marcelteun » Wed Mar 10, 2021 7:57 pm

Yes, that is another way of looking at it!

It seems that in his book from 1900 Brückner did come a"cross" some polyhedra with the cross (as pointed out by my friend). See for instance on the right side of plate 8: row 1, #2 and row 3, #2. The former can easily be transformed by moving the crosses to the centre until the crosses touch and you only add squares. That one is quite similar to Wenniger's 73. Perhaps that way you can easily derive polyhedra with crosses from uniform polyhedra with pentagrams. I vaguely seem to remember someone (Grünbaum?) mentioned polyhdra with crosses in the early 2000's.

Perhaps I should look more into this (so much to explore and build, so little time)
Last edited by marcelteun on Mon Mar 15, 2021 7:32 am, edited 1 time in total.

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Ulrich
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Re: Sharpohedron

Post by Ulrich » Thu Mar 11, 2021 3:19 pm

I found your model on page 41 of Wenzel Jamnitzer‘s Perspectiva Corporum Regularium:

https://digital.slub-dresden.de/werkans ... 2830/41/0/

He drew a compound of it with an octahedron inside, the spikes of which are looking through the central squares of the crosses, building 4-fold pyramids on them.

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marcelteun
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Re: Sharpohedron

Post by marcelteun » Thu Mar 11, 2021 9:34 pm

Good found, thanks Ulrich, also from my friend Don! I was quite convinced this would be a rediscovery, since it was such a simple model. Apparently it was known at least in 1568. I think I will build a model of it in the Swiss colours. And I might look into more of these "positive" models.

Cheers!
Marcel

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Re: Crossohedra

Post by marcelteun » Sat Mar 13, 2021 10:46 am

marcelteun wrote:
Sat Mar 13, 2021 10:46 am
I found some more variations on this theme. So are super simple and some can be found in Brückner, some are quite neat, though they are probably rediscoveries, but that doesn't spoil the fun.

The following models were found by Don Romano, Ulrich Mikloweit and me (last updated 2021-03-17)

Tetrahedral Symmetry
https://tunnissen.eu/polyh/local_off.ht ... istance=13

Octahedral Symmetry
Snub:
https://tunnissen.eu/polyh/local_off.ht ... istance=25
https://tunnissen.eu/polyh/local_off.ht ... istance=20)
With cross on 2-fold axis:
https://tunnissen.eu/polyh/local_off.ht ... istance=35
https://tunnissen.eu/polyh/local_off.ht ... istance=35
https://tunnissen.eu/polyh/local_off.ht ... istance=33
https://tunnissen.eu/polyh/local_off.ht ... istance=33
https://tunnissen.eu/polyh/local_off.ht ... encil=true

https://tunnissen.eu/polyh/local_off.ht ... stencil=on
https://tunnissen.eu/polyh/local_off.ht ... stencil=on
https://tunnissen.eu/polyh/local_off.ht ... stencil=on
With cross on 4-fold axis (type A, isogonal):
https://tunnissen.eu/polyh/local_off.ht ... istance=15 (The original Swissohedron :wink: also found in Jamnitzer, Wenzel 1568)
https://tunnissen.eu/polyh/local_off.ht ... istance=15 (note Stella Octangula inside)
https://tunnissen.eu/polyh/local_off.ht ... istance=15 (note self-intersecting hexagons, also Stella Octangula inside)
https://tunnissen.eu/polyh/local_off.ht ... istance=15 (note self-intersecting hexagons, also Stella Octangula inside)
https://tunnissen.eu/polyh/local_off.ht ... istance=15
https://tunnissen.eu/polyh/local_off.ht ... istance=20
https://tunnissen.eu/polyh/local_off.ht ... stencil=on
https://tunnissen.eu/polyh/local_off.ht ... istance=13
With cross on 4-fold axis (type B, isogonal):
https://tunnissen.eu/polyh/local_off.ht ... istance=25
https://tunnissen.eu/polyh/local_off.ht ... istance=18
https://tunnissen.eu/polyh/local_off.ht ... istance=30

Icosahedral Symmetry
https://tunnissen.eu/polyh/local_off.ht ... istance=28
https://tunnissen.eu/polyh/local_off.ht ... istance=70
https://tunnissen.eu/polyh/local_off.ht ... istance=90
https://tunnissen.eu/polyh/local_off.ht ... encil=true

Prism Symmetry
https://tunnissen.eu/polyh/local_off.ht ... istance=20
https://tunnissen.eu/polyh/local_off.ht ... istance=20
https://tunnissen.eu/polyh/local_off.ht ... istance=23
https://tunnissen.eu/polyh/local_off.ht ... istance=27
Isogonal:
https://tunnissen.eu/polyh/local_off.ht ... istance=20
https://tunnissen.eu/polyh/local_off.ht ... istance=20 (can be stretched so that rectangles <-> squares)

Some Alternatives
These have facelets lying in a plane, but that plane doesn't form a face in the polyhedron.
https://tunnissen.eu/polyh/local_off.ht ... istance=15
https://tunnissen.eu/polyh/local_off.ht ... istance=70
Last edited by marcelteun on Thu Apr 01, 2021 8:59 am, edited 32 times in total.

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Ulrich
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Re: Sharpohedron

Post by Ulrich » Sun Mar 14, 2021 4:35 pm

The penultimate is the best.

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marcelteun
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Re: Crossohedra

Post by marcelteun » Sun Mar 14, 2021 6:54 pm

Ulrich wrote:
Sun Mar 14, 2021 4:35 pm
The penultimate is the best.
Which one is that?

I updated the list from before with
https://tunnissen.eu/polyh/local_off.ht ... istance=42 (by Don Romano)
https://tunnissen.eu/polyh/local_off.ht ... istance=35
Last edited by marcelteun on Mon Mar 15, 2021 7:32 am, edited 1 time in total.

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Crossoshedra

Post by marcelteun » Mon Mar 15, 2021 7:31 am

Last edited by marcelteun on Mon Mar 15, 2021 7:50 am, edited 1 time in total.

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Ulrich
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Re: Sharpohedron

Post by Ulrich » Mon Mar 15, 2021 7:37 am

Up to now the best one imho was this:

https://tunnissen.eu/polyh/local_off.ht ... istance=15

but now I think that snub one beats all!

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marcelteun
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Re: Crossohedra

Post by marcelteun » Mon Mar 15, 2021 7:49 am

Ulrich wrote:
Mon Mar 15, 2021 7:37 am
Up to now the best one imho was this:
https://tunnissen.eu/polyh/local_off.ht ... istance=15
Yes, that one I like too. I built a model of the first one, but that one is going to be the next one. The snub, I didn't like so much, though.

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Re: Crossohedra

Post by marcelteun » Mon Mar 15, 2021 9:37 am

Added one with prism symmetry to the list below:
https://tunnissen.eu/polyh/local_off.ht ... istance=20 (by Don Romano)

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Re: Sharpohedron

Post by marcelteun » Mon Mar 15, 2021 9:07 pm

I updated the list below. Almost all names were updated. I will try to keep that list up to date. the links from the other posts might become broken later.

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Re: Sharpohedron

Post by marcelteun » Wed Mar 17, 2021 10:32 pm

Updated the list below

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Re: Sharpohedron

Post by marcelteun » Mon Mar 29, 2021 7:34 am

Updated the list below
Added one by Ulrich:
https://tunnissen.eu/polyh/local_off.ht ... stencil=on
Also did some renaming and added this one
https://tunnissen.eu/polyh/local_off.ht ... istance=15

The faceted part should be seen as a self-intersecting isosceles hexagon, not as a concave one, because in that case there will be loose edges. The off-files cheats a bit here, since it uses the concave hexagons.
The same is true for
https://tunnissen.eu/polyh/local_off.ht ... istance=15

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