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Zonohedrification

 
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Tue Jun 16, 2015 7:09 am    Post subject: Zonohedrification Reply with quote

Hi

By way of introduction, I recently purchased Great Stella and am absolutely blown away by its capabilities, and by the highly professional way in which it has been designed and presented - I find the user interface generally makes the software very easy to use, even for a newcomer like myself. I bought Great Stella primarily as a tool to assist me with my prime interest, which is in making polyhedral models using modular origami. I definitely would not class myself as a mathematician although along the way have picked up a working knowledge of some of the basic maths underpinning polyhedra.

I recently discovered Zonohedra/Zonish polyhedra, and one of my many aims with Great Stella is to use it to generate Zonohedra which I can then replicate using Origami (sorry Rob, no plans to use your nets at this stage, although I might be tempted with some models that can't be constructed using my preferred modular origami).

I've now spent some time using Great Stella to generate a number of Zonohedra/Zonish polyhedra. Brilliant!

I've been using George Hart's website as a reference point, which I've found particularly useful in gaining some understanding of the principles involved.

However I now have a few questions:

1) George Hart on his Zonish Polyhedra page - see http://www.georgehart.com/virtual-polyhedra/zonish_polyhedra.html, refers to the use of an icosidodecahedron as a seed to produce various polyhedra, starting with the addition of one zone,then two etc. Addition of a third zone produces two results - an oblate and a prolate version. Great Stella generates the oblate version, but I can't find a way to generate the prolate version. Can it be done and if so , how?

As an aside George Hart has made these into a couple of sculptures - see http://www.georgehart.com/sculpture/fat-and-skinny.html.

2) My second question relates to a massive Zonohedron that George Hart shows at the bottom of his Zonohedrification page -see http://www.georgehart.com/virtual-polyhedra/zonohedrification.html. He states that it is a 24 zone solid based on the truncated cuboctahedron. However the latter only has 13 zones, so I presume I need to further populate my "star" using some combination of vertices, edges or faces. But what combination? And how do I determine what combination to use?

3) Finally, I have now generated so many different Zonohedra/Zonish polyhedra that I badly need some form of numbering or naming system so that I can keep them organised and readily accessible. I don't want to reinvent the wheel, but my limited research to date has failed to reveal any commonly used system. Is there such a thing?

I would really welcome some help, so if anyone has any experience in using the Zonohedrify function and could share their knowledge, or can point me in the direction of any material that might help that would be very useful.

Thanks in advance
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robertw
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Joined: 10 Jan 2008
Posts: 394
Location: Melbourne, Australia

PostPosted: Sat Jun 20, 2015 8:36 am    Post subject: Reply with quote

Thanks for your feedback!

To answer your questions:

(1) Not sure how to do this off-hand. When you tell it to only use a limited number of zones, the order is random. But I'm sure there'd be a way to do it.

(2) Start with the truncated cuboctahedron, then create a zonohedron (not a zonish polyhedron) with just "All vertices" ticked. I just found this by trial and error, and just using vertices is probably the most common way.

(3) I don't know if there's any kind of naming convention for zonohedra. Maybe invent your own!

Rob.
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Sun Jun 21, 2015 4:29 am    Post subject: Reply with quote

Hi Rob and thanks for your response

robertw wrote:


(1) Not sure how to do this off-hand. When you tell it to only use a limited number of zones, the order is random. But I'm sure there'd be a way to do it.
.

I'll keep trying with this one.

robertw wrote:


(2) Start with the truncated cuboctahedron, then create a zonohedron (not a zonish polyhedron) with just "All vertices" ticked. I just found this by trial and error, and just using vertices is probably the most common way.


Great and thanks! I'd tried trial and error also, but when you are unsure of what you are doing, and there are so many permutations and combinations....!

robertw wrote:


(3) I don't know if there's any kind of naming convention for zonohedra. Maybe invent your own!


Maybe I will, but I'll see if there are any more suggestions from others first.

Thanks again
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dekay5555555



Joined: 21 Jun 2015
Posts: 2

PostPosted: Mon Jun 22, 2015 6:56 pm    Post subject: Zonohedra Reply with quote

Just to chime in and it may not be of any help, but I would like to point out that Pascal's Triangle can be useful in predicting Zonohedra.

The third row of the triangle, 1,3,3,1.

A regular hexahedron or cube is a zonohedron, there are three unique directions and it makes 3 sets of parallel rhombi.

The fourth row 1,4,6,4,1

A rhombic dodecahedron has 4 unique directions, 6 pairs of rhombi, and 4 hexahedral cells call be formed inside the polyhedron.

The sixth row 1,6,15,20,15,6,1

A rhombic triacontahedron has 6 unique directions, 15 pair of parallel rhombi, 20 hexahedral cells (10 fat and 10 thin), those cells can be formed 15 different way to make rhombic dodecahedra (2 fat + 2 thin), 6 different rhombic icosahedron and of course only 1 rhombic triacontahedron.

And it continues . . .
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Tue Jun 23, 2015 8:52 am    Post subject: Re: Zonohedra Reply with quote

dekay5555555 wrote:
Just to chime in and it may not be of any help, but I would like to point out that Pascal's Triangle can be useful in predicting Zonohedra


Thanks so much for pointing this out. Fascinating and I wasn't aware of the connection which I'll now pursue. You've also renewed my interest in Pascal's triangle which I originally learnt about many years ago from the recreational maths author Martin Gardner who in Mathematical Carnival says "it contains such inexhaustible riches, and links with so many seemingly unrelated aspects of mathematics, that it is surely one of the most elegant of all number arrays".
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dekay5555555



Joined: 21 Jun 2015
Posts: 2

PostPosted: Tue Jun 23, 2015 4:00 pm    Post subject: Pascal's Triangle and Zonohedra Reply with quote

Thanks for reading my post and responding, dabeard. I thought that there may be some basis for categorizing and subsequently naming zonohedra along these lines. At least group them.
I found out about the relationship between the Zonohedra and the Triangle, in a circuitous way.
I was introduced to a enneacontahedron, a 90 faced zonohedra with 30 oblate and 60 acute rhombi. Steve Baer, an interesting geometer in his own right (inspiration for Zometool), had decomposed the enneacontahedron into 120 hexahedral cells and I was trying to understand how to fit these all together. I now know that there are many ways, thanks to Russel Towle whom I had the honor to email dialog with just prior to his unfortunate passing.

http://blog.wolfram.com/2008/10/10/russell-towle-1949-2008/

Later I found this webpage, which made me ponder the first three missing parts to this wonderful accounting. There seemed to be a symmetry to the actual number of unique combinations. On the page it lists the forms in groups of "axes" from 3 to 10. Interestingly, at 10 axes there is only one form, the enneacontahedron itself. 9 axes also had only 1 form.

http://www.orchidpalms.com/polyhedra/rhombic/rh90/rh90.htm

I noted the pattern:

3 axes - 5 different Baer cells, 4 axes - 7 different rhombic dodecahedron and so it goes 3a-5,4a-7,5a-8,6a-7,7a-5,8a-2,9a-1,10a-1

Or 5,7,8,7,5,2,1,1
Seeing a crescendo/decrescendo of symmetry I postulated that there would be 0,1 & 2 axes and then the patter would be complete.

1,1,2,5,7,8,7,5,2,1,1

The 0 axes, represent a point, 1 axes, a line, There is only possibility for these, the same as for 9 and 10 axes.

2 axes make 2 different areas or rhombi. Acute and oblate

Later I somehow recalled that Pascal's Triangle's rows always started and ended with a 1 and that is how I figured that connection.

The enneacontahedron has 10 axes and the 10th row has 11 numbers.

Some rough videos that I have made pertaining to the subject:

Building an Enneacontahedron Part 1
https://www.youtube.com/watch?v=nicprWKT588

Building an Enneacontahedron Part 2
https://www.youtube.com/watch?v=9wZbodnqKkc
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Wed Jun 24, 2015 11:36 am    Post subject: Zoohedrification Reply with quote

Hi dekay5555555

Thanks for sharing your learning journey, and your videos. You've saved me a lot of time and provided some very useful pointers.

I've now got a lof of information to process and hopefully make sense of!

Another question for all members of the forum - does anyone have any .stel or .off files that they can share for polar zonohedra?

Thanks again

dabeard
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robertw
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Joined: 10 Jan 2008
Posts: 394
Location: Melbourne, Australia

PostPosted: Sun Jun 28, 2015 11:47 am    Post subject: Reply with quote

I asked around and have the other answers for you.

(1) This can be done in a couple of ways, but here's the easiest:
  • Load an icosidodecahedron
  • Select a pentagon
  • Poly->Zonohedrify
  • Select "Add Zones" (zonish polyhedron)
  • In the bottom part, tick only Current/Face.
  • A zone perpendicular to the selected pentagon will be added.
  • Select another pentagon and repeat.
  • You have a choice of pentagons on the third step, so use trial and error to find the right one.

(3) George Hart says he's not aware of any systematic naming.

Rob.
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dabeard



Joined: 13 Jun 2015
Posts: 12

PostPosted: Fri Jul 03, 2015 8:30 am    Post subject: Zonohedrification Reply with quote

Hi Rob

Have been a little busy, hence this belated response. Many thanks for all your efforts on the two issues I raised. Greatly appreciated. All going well I'll find some time this weekend to revisit my earlier attempts to reproduce Fat and Skinny, and will also start thinking seriously about a numbering/naming system.

Kind regards
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