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Sharpohedron

Posted: Wed Mar 10, 2021 10:54 am
by marcelteun
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

Re: Sharpohedron

Posted: Wed Mar 10, 2021 4:01 pm
by Ulrich
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

Re: Crossohedra

Posted: Wed Mar 10, 2021 7:57 pm
by marcelteun
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)

Re: Sharpohedron

Posted: Thu Mar 11, 2021 3:19 pm
by Ulrich
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.

Re: Sharpohedron

Posted: Thu Mar 11, 2021 9:34 pm
by marcelteun
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

Re: Crossohedra

Posted: Sat Mar 13, 2021 10:46 am
by marcelteun
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

Re: Sharpohedron

Posted: Sun Mar 14, 2021 4:35 pm
by Ulrich
The penultimate is the best.

Re: Crossohedra

Posted: Sun Mar 14, 2021 6:54 pm
by marcelteun
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

Crossoshedra

Posted: Mon Mar 15, 2021 7:31 am
by marcelteun

Re: Sharpohedron

Posted: Mon Mar 15, 2021 7:37 am
by Ulrich
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!

Re: Crossohedra

Posted: Mon Mar 15, 2021 7:49 am
by marcelteun
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.

Re: Crossohedra

Posted: Mon Mar 15, 2021 9:37 am
by marcelteun
Added one with prism symmetry to the list below:
https://tunnissen.eu/polyh/local_off.ht ... istance=20 (by Don Romano)

Re: Sharpohedron

Posted: Mon Mar 15, 2021 9:07 pm
by marcelteun
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.

Re: Sharpohedron

Posted: Wed Mar 17, 2021 10:32 pm
by marcelteun
Updated the list below

Re: Sharpohedron

Posted: Mon Mar 29, 2021 7:34 am
by marcelteun
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