What was your introduction to polyhedra?

For general discussion of polyhedra, not necessarily Stella-specific.
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oxenholme
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What was your introduction to polyhedra?

Post by oxenholme » Sat Jan 19, 2008 6:58 am

Around 1961 form master "Bert" Robinson at Bradford Grammar School had some beautifully constructed polyhedra on display in the classroom - the four I remember were Great Dodecahedron, Great Dodecadodecahedron, Great Icosidodecahedron and Icosahedron.

His source was Mathematical Models by H Martyn Cundy and A P Rollett.

I soon acquired a copy of the paper on Uniform Polyhedra by Coxeter, Miller and Longuet-Higgins. I'd made Miller's Monster before I left school.

The snub polyhedra had me defeated, but I was lucky enough to get in touch with George Olshevsky from Toronto University. It wasn't too long before I'd finished the Small (!!!) Inverted Retrosnub Icosicosidodecahedron.

By chance I found out that H Martyn Cundy lived in the same town - he gave me some papers, and before too long I embarked on making a number of symmetrical compounds. I worked them out from scratch - no computers, no mathematics.

And then came Stella.

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Brendan
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Post by Brendan » Sat Jan 19, 2008 11:15 am

I already knew about the Platonic solids from general references, then I found a hardcover copy of Alan Holden's Shapes, Space and Symmetry in the library. I borrowed that one several times and at university. I still find new things in it each time I look at it. :shock: Then I found the Wooden Book about Platonic and Archimedian solids. That continued my interest in polyhedra so I ordered the new edition of the SSS book and got Peter Cromwell's Polyhedra book at another bookstore. After that, I ordered Wenninger's Polyhedron Models and Duel Models.

So far, I have made paper models of the Platonic, Archimedian and Kepler-Poinsot solids with nets I found online. But those got destroyed when I was away at university and I found only some of them stuffed into a box of stuff. Now even those are gone. :(

Now that I have Stella4D that allows models to be scaled, I am also thinking about building some kind of shelving system for the models and got a book about shelves from the library.

Squeaky
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My first venture

Post by Squeaky » Sat Jan 19, 2008 11:24 am

Hi Oxenholme,

Not too distant from North Derbyshire.

I love the look of your models.

I really stumbled into polyhedra and since that time I have become very interested in the subject. My wife tells me that I do not have 'normal' hobbies but obsessions.

I do a lot of photography and took a series of shots at a place called Padley Gorge. These I made into a panorama and decided to mount them on some sort of sphere.

OMG did I have problems with the geometry. I have been an engineer making, at one period, prototypes of some quite complicated stuff and just launched into the 'sphere' project with nothing more than a bit of basic geometry.

Image


Everything was O.K. till I tried to fit the triangles ... Eventually I made a cardboard model, of the whole thing, and worked out the mitre angles with a 'knife and fork'.

I made the model, as shown above, and made it revolve with the aid of a microwave turntable motor.

Then I found a copy of 'Geodesic Domes' by Borin Van Loon .... and I was hooked.

I still rather struggled with mitre angles till I stumbled across Stella.


Squeaky

Squeaky
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What is it?

Post by Squeaky » Sat Jan 19, 2008 4:46 pm

Hi Oxenholme,

What is the name of this particular model?

Image

I once attempted to make five interlocked tetrahedrons and most of it ended up in the scrap box.

This model of yours look suspiciously like the one I failed at ...

Where did you find the net?

Squeaky

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oxenholme
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Post by oxenholme » Sat Jan 19, 2008 5:19 pm

Five Tetrahedra is second from the right on the top shelf...

The compound that you've picked out is Four Tetrahedra - each is rotated 30° by vertex or opposite face from a hypothetical base tetrahedron.

I drew the polyhedron on 14th July 1997 from scratch and then worked out the net, again from scratch.

On the same day I drew also the compound of Eight Tetrahedra obtained by combining left and right enantiomorphs but I never took it any further.

Squeaky
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Post by Squeaky » Sat Jan 19, 2008 7:02 pm

Hi Oxenholme,

I wonder .... do you still have copies of your drawings for the net for the four tetrahedra?

Would you share them with the rest of the forum?

I would love to make this model some time in the future.

Squeaky

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Jabe
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Re: My first venture

Post by Jabe » Sun Jan 20, 2008 7:01 am

Squeaky wrote: Image
Everything was O.K. till I tried to fit the triangles ... Eventually I made a cardboard model, of the whole thing, and worked out the mitre angles with a 'knife and fork'.
I've once used the same polyhedron to make a panoramic view, but used a computer generated environment for the pictures - I also found the triangles a bit tricky to get the pictures in the right viewing angle.
May the Fourth (dimension) be with you.

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robertw
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Post by robertw » Sun Jan 20, 2008 1:52 pm

Squeaky wrote:I wonder .... do you still have copies of your drawings for the net for the four tetrahedra?

Would you share them with the rest of the forum?

I would love to make this model some time in the future.
I'm sure the model can be made with Stella. I'll have a think about it and get back to you.

Rob.

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Dinogeorge
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Re: What was your introduction to polyhedra?

Post by Dinogeorge » Mon Jan 21, 2008 1:37 am

oxenholme wrote:The snub polyhedra had me defeated, but I was lucky enough to get in touch with George Olshevsky from Toronto University. It wasn't too long before I'd finished the Small (!!!) Inverted Retrosnub Icosicosidodecahedron.
Yike! A blast from the past! Did you use my computer-generated nets for the yog-sothoth (my personal name for that figure)? Bruce Chilton used them and built a model about three feet across, which we displayed at the Shaping Space conference at Smith College in 1984. Here's a photo I swiped from George Hart's website of Coxeter admiring the model at the conference. Alas, Bruce had to dispose of it a few years later. It was taking up too much space.

Image

Other of Bruce's models surround the big yog-sothoth, including a huitzilopochtli (vertex figure of the great prismosaurus, toward the back in light turquoise), a number of shadows of 4D star-polychora, and some facetings of Miller's monster. I'm not entirely sure, but I think that's George Hart himself in the picture next to Coxeter.

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oxenholme
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Re: What was your introduction to polyhedra?

Post by oxenholme » Mon Jan 21, 2008 5:45 am

Dinogeorge wrote:Yike! A blast from the past! Did you use my computer-generated nets for the yog-sothoth?
I wondered whether it might be you!!!

Yes, I used them, and I still have them. My yog-sothoth is approx 23.25 inches circumspherically.

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oxenholme
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Post by oxenholme » Mon Jan 21, 2008 5:50 am

Squeaky wrote:I wonder .... do you still have copies of your drawings for the net for the four tetrahedra?
I do not throw things away, but I have yet to locate them. They will turn up eventually!

Meanwhile, I wonder whether compounds obtained this way can be generated by Stella?

While I was looking for my drawings I came across one of a compound of Five Pentagramic Prisms with dodecahedral symmetry. One of these days I will get around to making one...

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robertw
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Post by robertw » Mon Jan 21, 2008 6:42 am

It can indeed be done in Stella, although it was a little tricky.

Click here to download the .stel file

Once in Stella you can print out the nets :D

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Dinogeorge
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Six pentagrammatic prisms

Post by Dinogeorge » Mon Jan 21, 2008 7:18 am

oxenholme wrote:While I was looking for my drawings I came across one of a compound of Five Pentagramic Prisms with dodecahedral symmetry. One of these days I will get around to making one...
Are you sure you don't mean six pentagrammatic prisms? Five of those prisms have only 50 corners, a number that doesn't evenly divide 120, the order of the icosahedral symmetry group.

Here's the compound of six with dodecahedral symmetry. It has the 60 corners of a (small) rhombicosidodecahedron:

Image

It makes for a very pretty model. By the way, good to hear from you after some 30 years! :D

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Dinogeorge
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Re: Six pentagrammatic prisms

Post by Dinogeorge » Mon Jan 21, 2008 7:48 am

Dinogeorge wrote:Here's the compound of six with dodecahedral symmetry.
And I couldn't resist making the compound of twelve by merging the six with its mirror image. :o Same set of 60 corners, now two prisms per each:

Image

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oxenholme
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Re: Six pentagrammatic prisms

Post by oxenholme » Mon Jan 21, 2008 9:08 am

Dinogeorge wrote:Are you sure you don't mean six pentagrammatic prisms? Five of those prisms have only 50 corners, a number that doesn't evenly divide 120, the order of the icosahedral symmetry group.

:D
Yes! I meant six...

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