Isohedra with non-convex faces

For general discussion of polyhedra, not necessarily Stella-specific.
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oxenholme
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Isohedra with non-convex faces

Post by oxenholme » Wed Sep 23, 2009 7:16 am

My mathematics are sadly lacking. Has anybody here got the necessary mathematical ability to translate the equations and values in the paper referenced below into exact x-y co-ordinates or whatever to enable me to draw the faces necessary to make the polyhedra illustrated in Figs 10 thru 13 of the following paper?http://www.math.washington.edu/~grunbau ... 0faces.pdf
Last edited by oxenholme on Wed Sep 23, 2009 5:38 pm, edited 1 time in total.

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robertw
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Post by robertw » Wed Sep 23, 2009 10:59 am

I'm not going to attempt the maths, but some of those look similar to stellations of the pentagonal icositetrahedron. See here:

http://www.software3d.com/Stellations.php

These are all easily made in Stella.
Rob.

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oxenholme
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Post by oxenholme » Wed Sep 23, 2009 5:44 pm

They do indeed look similar. I love "chiral" polyhedra. My next will be "one of our tetrahedra is missing" - the compound of five minus one.

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oxenholme
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Post by oxenholme » Fri Oct 16, 2009 6:57 pm

Herewith some pics of the isohedra with tetrahedral symmetry...

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They aren't especially photogenic, but they look quite nice in the display cabinet.

The third one was a pig to make - I tried making trihedral vertices at the first attempt but found it virtually impossible to join them together. The second attempt was successful using trihedral dimples, but it didn't exactly go smoothly.

I'm now having a go at one with octahedral symmetry.

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oxenholme
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Post by oxenholme » Sat Oct 17, 2009 3:27 pm

First one with octahedral symmetry - in progress, then a couple of views complete...

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oxenholme
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Post by oxenholme » Sun Oct 25, 2009 4:29 pm

Image

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oxenholme
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Post by oxenholme » Sat Oct 31, 2009 4:52 pm

I suspect everybody has flown, but just in case...

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It was a pig to make...

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robertw
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Post by robertw » Sun Nov 01, 2009 7:36 am

I'm still here! It's nice to see new models being made.

Yeah that one looks like a challenge! How did you get the last piece in? I see you put a five-part section in last, but I can't imagine the concave tabs could be squashed together very easily.

If it were me, I'd probably have put a 15-part section in last. Your five, plus the surrounding ten parts, and then also have partial parts in place already on the rest of the model for those extra ten parts. Then the final section could be slid over the partial structure. The partial structure would already be rigid, unlike tabs, so the final section would be glued to it without any further need to squash any internal tabs together.

I've finished a few models that way, although not sure anyone will follow my explanation!

Rob.

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Post by oxenholme » Sun Nov 01, 2009 9:47 am

White tab upper right to orange tab upper right, then I worked anti-clockwise. It was slow going, but with the aid of tooth picks and hat pins to manipulate bits into place it wasn't too bad.

I'm dreading the last two with icosahedral symmetry. The penultimate will be easy to assemble using triangular dimples, but the colours will be difficult to get right.

The last one I suspect I will make with five sided spikes - the dihedral angles look fairly shallow, so it might work out OK.

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oxenholme
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Post by oxenholme » Sun Nov 08, 2009 3:46 pm

I've begun the last of the Icosahedral Isohedra. It is ridiculously difficult to assemble - a wonderful collection of linkages that move the tabs away from each other when the cement has been applied. It's two thirds of the way there. I've had to walk away from it to avoid the temptation of crushing it.

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Image

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Peter Kane
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Post by Peter Kane » Sat Nov 28, 2009 2:39 pm

I took a look at the paper, but didn't get very far. I couldn't see a definition of the parameters a, b and c. Perhaps these are widely understood in the context - not by me, alas.

Have you considerd cheating ? I don't know how successful this would be, but you could try printing the faces at a suitable magnification (600%, maybe). If you really want the coordinates, you could try copying the images into a graphics program that allows you to see the pixel coordinates. Success will depend upon the acuracy of the images.

That looks like a decent pair of compasses you have there in that image of Oct 16th. Where did you get them ?

PeteK

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oxenholme
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Post by oxenholme » Sun Nov 29, 2009 8:32 am

Peter Kane wrote:I took a look at the paper, but didn't get very far. I couldn't see a definition of the parameters a, b and c. Perhaps these are widely understood in the context - not by me, alas.

Have you considerd cheating ? I don't know how successful this would be, but you could try printing the faces at a suitable magnification (600%, maybe). If you really want the coordinates, you could try copying the images into a graphics program that allows you to see the pixel coordinates. Success will depend upon the acuracy of the images.

That looks like a decent pair of compasses you have there in that image of Oct 16th. Where did you get them ?

PeteK
Hi Peter

I don't know whether you can access the following link, but it was on a music forum where I had the maths explained to me!

http://www.stevehoffman.tv/forums/showt ... p?t=196036

I bought the Rotring compasses many years ago - it would have been from somewhere in the West Riding of Yorkshire, but where I cannot remember.

I finished this one yesterday - it took over two hours to add the last five pieces...

Image

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oxenholme
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Post by oxenholme » Sun Nov 29, 2009 8:35 am

Peter Kane wrote:I took a look at the paper, but didn't get very far. I couldn't see a definition of the parameters a, b and c. Perhaps these are widely understood in the context - not by me, alas.
PeteK
The parameter values are shown by each line drawing. They fit in the formulae in the appendix...

I used spreadsheets to do the calculations. Sadly my computer died a week ago so I will have to redo them...

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Peter Kane
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Post by Peter Kane » Sun Nov 29, 2009 1:40 pm

oxenholme wrote:
The parameter values are shown by each line drawing. They fit in the formulae in the appendix...

I used spreadsheets to do the calculations. Sadly my computer died a week ago so I will have to redo them...
Thanks for the link, but I was unable to access the post, even after registering. However, following your hint, I'll take another look at the paper.

Lovely result, by the way and in only two hours for that last piece! It takes me that long to get my paraphernalia together. I often stick a piece of card UNDER the tabs on the last opening so that I have something to push against when I glue the final piece in place. You probably do something similar. If you use my suggestion, I’m sure it will take you more than two hours next time...
Cheers,
Pete K

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oxenholme
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Post by oxenholme » Mon Apr 05, 2010 8:23 pm

I finished the final Isohedron today...

Image

The one on the left has octahedral symmetry. The one on the right has cubic symmetry

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