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Symmetrohedra

 
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Mon Jul 06, 2015 5:24 am    Post subject: Symmetrohedra Reply with quote

Another newbie question!

I recently discovered Symmetrohedra. See http://archive.bridgesmathart.org/2001/bridges2001-21.pdf for a copy of the 2001 Kaplan/Hart paper on the topic, and Craig Kaplan's web pages at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/ for some graphics.

My question: is there any way that Great Stella can be used to produce Symmetrohedra? And if so, how?

I hope someone can help, IMHO many of the symmetrohedra have a lot of appeal from an aesthetic viewpoint.


Last edited by dabeard on Fri Jul 10, 2015 11:13 am; edited 1 time in total
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robertw
Site Admin


Joined: 10 Jan 2008
Posts: 395
Location: Melbourne, Australia

PostPosted: Thu Jul 09, 2015 12:32 pm    Post subject: Reply with quote

There's no general way in Stella to make these. I notice some are already in Stella's Near Miss library.

It may be possible to make some others with a bit of ingenuity.

You can make things along these lines sometimes by taking an intermediate point during dual-morphing.
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Fri Jul 10, 2015 9:44 am    Post subject: Symmetrohedra Reply with quote

Hi Rob and thanks for this

Since posting this query I’ve found that Robert Austin on his blog at https://robertlovespi.wordpress.com has several references to Symmetrohedra all of which he has found using Stella4D:

https://robertlovespi.wordpress.com/2013/11/29/a-bowtie-polyhedron-featuring-regular-nonagons-and-octagons/

https://robertlovespi.wordpress.com/2014/04/07/a-bowtie-symmetrahedron-featuring-twelve-decagons-and-twenty-equilateral-triangles/

https://robertlovespi.wordpress.com/2014/04/20/symmetrohedron-featuring-eighteen-regular-octagons-eight-equiangular-hexagons-and-twenty-four-isosceles-trapezoids/

https://robertlovespi.wordpress.com/2014/04/20/a-cube-based-symmetrohedron-featuring-six-regular-hexadecagons-eight-equilateral-triangles-and-two-types-of-icosceles-trapezoids/

https://robertlovespi.wordpress.com/2014/05/24/a-bowtie-symmetrohedron-featuring-twelve-regular-pentadecagons-twenty-regular-hexagons-and-thirty-pairs-of-isosceles-trapezoids/

An impressive collection, and an equally impressive amount of effort to produce them!

Robert was kind enough to respond to an email request asking how he produced the various Symmetrohedra shown on his blog and if there was a general methodology that could be followed to produce more.

His answer, reproduced here with his permission was:

‘Most of my "method" involves "random-walk" manipulations of polyhedra, in numerous ways, until I see a result I like. I'm sorry there's not a method which better lends itself to explanation, but that is the best I can do.”

Fair enough, and thanks to Great Stella and Stella4D we now have the tools to do the "random-walk"

Kind regards
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RobertLovesPi



Joined: 21 Jul 2015
Posts: 2
Location: Arkansas, USA

PostPosted: Wed Jul 22, 2015 3:08 am    Post subject: Reply with quote

Hi,

Checking those links, above, to posts I made long ago on my blog, I found I did include a description of how I made at least one of these symmetrohedra, using Stella. You can find this description in this post:

https://robertlovespi.wordpress.com/2013/11/29/a-bowtie-polyhedron-featuring-regular-nonagons-and-octagons/

Also, for most, or perhaps all, of these symmetrohedra on my blog, I didn't find them first. The general pattern goes like this:

1. I find one of these cool things with Stella, and think I'm the first person to ever see it.

2. I create a .gif file of the polyhedron, and blog it. (The .gif format lets the viewer see the polyhedron rotate.)

3. I then find out that the team of Craig Kaplan and George Hart beat me to the punch. This isn't surprising; after all, I think they coined the term "symmetrohedra."

4. My last step is to edit the blog-post(s) to give Kaplan and Hart the credit they deserve.
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You can see many of my creations with "Stella 4d" at http://www.robertlovespi.wordpress.com.
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