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### Re: Six pentagrammatic prisms

Posted: **Mon Jan 21, 2008 9:10 am**

by **oxenholme**

Dinogeorge wrote: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.

Same set of 60 corners, now two prisms per each:

Do you notice a certain similarity to Miller's Monster?

Posted: **Mon Jan 21, 2008 9:14 am**

by **oxenholme**

Great!

Can you show me how to do this with Stella? It would be interesting to see what it looks like with (say) a 45° twist...

### Re: Six pentagrammatic prisms

Posted: **Tue Feb 12, 2008 9:04 am**

by **Dinogeorge**

oxenholme wrote: Do you notice a certain similarity to Miller's Monster?

Sure. The compound of twelve pentagrammatic prisms is a faceting of the rhombicosidodecahedron, and Miller's monster is a faceting of a quasiuniform rhombicosidodecahedron in which the rhombi-squares are made into just a little off-square rectangles. Put in the coplanar pentagram pairs, and they're quite close lookalikes.

Posted: **Fri Feb 15, 2008 11:23 am**

by **marcelteun**

At school I was already interested in geometry, but my introduction to polyhedra was when I visited the Science Museum in London in 1988. There they had a collection of all uniform polyhedra. I had no idea what these models actual represented, but was fascinated and hooked immediately.

This was a time when internet didn't exist (for me) and the library didn't have any books on polyhedra and for a long time I felt pretty lonely in having such a peculiar hobby. Later I came in contact with a teacher at the University who had the same interest and he suggested me to buy Magnus' book Polyhedron models.

I see myself as a polyhedron model builder, but preferably I wish to make new models, i.e. it is not my goal to build all uniform polyhedra (though sometimes I just build one just for fun.) I occupied myself a long time with cube compounds, until I found out that H. Verheyen wrote a book about this.

Posted: **Wed Feb 27, 2008 7:01 pm**

by **Nordehylop**

I got into polyhedra last fall via a Wikipedia search for "dodecahedron". I was amazed that I hadn't ever heard of any of these things before, and so that night I built one out of poster board. After that I was hooked. I've made around 30 models and incurred $6.20 of library fines for Wenninger's "Polyhedron Models". I live my life in constant fear that one of the librarians will find out about all those tiny holes in the corners of the nets...

oxenholme, I love your models. Are they all compounds?

Posted: **Mon Mar 03, 2008 6:17 am**

by **oxenholme**

Nordehylop wrote:oxenholme, I love your models. Are they all compounds?

Those in that pic are all compounds, but I have many more around the house of all types. I'll try to remember to take my camera home with me and post pics of a few more.

Posted: **Sun Apr 13, 2008 5:12 pm**

by **robertw**

oxenholme wrote:
Great!

Can you show me how to do this with Stella? It would be interesting to see what it looks like with (say) a 45° twist...

Sorry for very belated reply! As I mentioned, it's tricky, and I even ran into a bug along the way, which I have fixed in my version so will appear in the next release.

The method doesn't easily allow for an arbitrary choice of angle.

The idea is to create a compound of 2 tetrahedra, rotated from each other by some angle around a vertex. Then augment this compound to all the faces of a tetrahedron and the desired polyhedron should pop out.

Here's what I did:

- Start with a N-gonal prism, where N is a multiple of 3 (so we can facet a triangle into it) and which includes vertices at the desired angle apart. For example, 45 degrees between vertices gives an octagon (360 / 45 = , so N must be divisible by 3 and 8, hence N = 24.
- Drop to 3-fold pyramidal subsymmetry.
- Enable "Maintain reflexibility" to make faceting quicker.
- Facet to create two triangular prisms, rotated 45 degrees from each other (count around 8 steps between vertices). Should look like this:

(EDIT: Oops, image showed a 30 degree turn, fixed to 45 degree now)
- Create the faceted model and put in a memory slot.
- Augment a pyramid (tetrahedron) onto two coplanar triangles, then excavate the prism from the base (using what was stored in memory). Make sure you hit the Up arrow until the augmentation preview aligns with the base.
- You now have a compound of 2 tetrahedra. Put this in memory.
- Load a tetrahedron and excavate the model in memory from all of its sides. This is where the bug I found occurs. It won't actually let you do it! It will either give an error or hang depending on whether "Keep coincident faces" is ticked. Will be fixed in next version.
- A copy of the original tetrahedron remains, so select one of its faces and use Ctrl+D to delete it (or "Poly->Delete One Part of Compound").
- If all one colour, use Shift+A to colour it as a compound ("Color->Color as a Compound").

The result at 45 degrees looks very close to the standard compound of 5 tetrahedra with one missing, but it is not quite the same. In fact that would happen at 44.4775 degrees!

Rob.