Compound of Two Deltahedra

For general discussion of polyhedra, not necessarily Stella-specific.
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oxenholme
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Compound of Two Deltahedra

Post by oxenholme » Mon Aug 17, 2009 12:22 pm

Due to a never-ending tale of woe with broken computers I haven't got access to Stella at the moment...

You know the "deltahedron" - same edges as the regular dodecahedron, pentagonal dimples each comprising 5 equilateral triangles, the polyhedron is actually a stellation of the icosahedron?

If you make a compound of two of them with octahedral symmetry so that eight of the vertices are twinned, and six pairs of edges intersect each other perpendicularly at their midpoints...

I am correct in thinking that eight of the triple-triangle faces of the one will be coplanar with eight of those from the other?

I'm thinking that it ought to be a fairly attractive compound, and probably quite simple to make from 24 pieces of card - twelve each of the two colours. I fancy it being my next model...

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Post by robertw » Tue Aug 18, 2009 1:15 pm

Your thoughts are right. Interestingly, although some faces occur in coplanar sets of six, none of them overlap, although they seem to share a vertex.

Image

Only 144 facelets, so shouldn't be too hard to make one.

Rob.

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Post by oxenholme » Sat Aug 22, 2009 11:06 am

This compound has an unusual characteristic - it can be made from 24 pieces of card, each folded to make a vertex as the sum of angles at said vertex is 360 degrees. There is no need to cut and join. Obviously the twinned vertices are different.

The only other polyhedron that I've encountered with a similar characteristic is the compound of a similar deltahedron with a great dodecahedron - the edges of which form the diagonals of a circumscribing rhombic triacontahedron. This compound can be made from 20 folded regular hexagons and twelve folded regular decagons. It is quite attractive.

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Post by robertw » Sat Aug 22, 2009 3:53 pm

Yes, that compound you mention is interesting. It has two different types of starry vertices, and both can be folded from flat paper without a cut to the vertex. I keep meaning to make one. Here's the .stel file if any Stella users want to have a look:

http://www.software3d.com/StelFiles/Gre ... dIcos.stel

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Post by oxenholme » Thu Aug 27, 2009 11:40 am

The non-convex Archimedean polyhedron sharing the edges, vertices and triangular faces of the cuboctahedron - but having the diametral hexagons in place of the square faces.

I suspect that all the vertices in the compound of five such, dodecahedral symmetry, can be folded without cutting - with some additional facets inbetween to hold it together.

The balsa cement has arrived from the supplier, I've found a new blade for the Stanley knife, and I have 27 colours of 160gsm card to hand, so it's back in production. Though the compound of two deltahedra will of course only need two colours.

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Post by robertw » Thu Aug 27, 2009 2:09 pm

oxenholme wrote:The non-convex Archimedean polyhedron sharing the edges, vertices and triangular faces of the cuboctahedron - but having the diametral hexagons in place of the square faces.

I suspect that all the vertices in the compound of five such, dodecahedral symmetry, can be folded without cutting - with some additional facets inbetween to hold it together.
Well, it's not Archimedean, just uniform, but I know what you mean, the octohemioctahedron. Stella's library already includes compounds of 4 and 5 of them.

But it doesn't quite work the way you suggest because the hexagons lie in coplanar pairs, so unless the paper is cut within a flat section, nets can't be made the way you suggest.

Rob.

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oxenholme
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Post by oxenholme » Sat Aug 29, 2009 5:44 pm

The compound of five octohemioctahedra is a bit of a teaser - all the faces, both triangles and hexagons, are in coplanar pairs. It would be quite a pain to make. The flat central part of each triangle pair will need to be divided into 6 scalene triangles of alternating colours. Each dart shaped face of the inward facing pentagonal spikes formed from the hexagons will need to be divided into 4 triangles of differing colours.

By using an uncut folded single piece of card you remove the problem of clashing tabs on the visible part of the crossed vertices.

I'm approaching halfway on the compound of two deltahedra. I am doing it in two colours, but I suspect it would look a lot better with one of the deltahedra using ten colours as a stellation of the icosahedron, and the other all black. It's finicky.

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Post by oxenholme » Wed Sep 23, 2009 5:46 pm

I finished the compound. It looks lovely. To my dismay there was barely enough room for it in the display cabinet. Polyhedra are wonderful at filling up space.

Image

Image

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Post by indigotwilight » Thu Apr 21, 2011 10:56 pm

I just tried to create this model (as an exercise) on stella4d. Though the compound was relatively straightforward to create, I noticed the net wasn't looking right.
Shouldn't the net be a hexagonal object with this colour scheme?

Here's a screenshot. What am I doing wrong? I set the colour scheme to 'colour as compound'.

Image


Twi

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Post by robertw » Fri Apr 22, 2011 4:34 am

Nets are made from whatever appears in the stellation view, so have a look at the stellation view to see what's happening. You can also try the 3D folding net view to see how the nets fold up.

Now the stellation view, by default, figures out which stellation chunks are required to recreate the original shape, so nets should be as you'd expect. However, there's currently a limitation that in the stellation view all parts lying in the same plane must be the same colour. This is a problem for the model above because some faces in the same plane should be different colours. So the colouring in the stellation view won't match the original model. In turn, nets won't be grouped by colour properly. The nets you found are correct, but they are spreading partly across the red/yellow border.

You can use Cut/Uncut mode to split at the red-yellow border, but it seems Stella still doesn't want to include all the parts around a single-colour vertex. Pity since they should fit exactly. In some other cases Stella can handle this, so not sure what's different here. Oh, hang on, it'll be the colour problem again. Use "Nets->Net/Paper Color Mixing->Allow Mixing of Colors" (or make the whole thing a single colour). Now it creates some amazing nets! First it creates a single net needing only 12 copies. When I scissor on one of the red-yellow borders, it now creates an even larger net only needing 6 copies! One more click and it gives me the desired net (although showing it with red and yellow mixed when they shouldn't be). 24 copies required of this net, half of each colour.

Rob.

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Post by indigotwilight » Fri Apr 22, 2011 3:01 pm

Thanks Rob, that did the trick. :D

I've noticed compounds of some stellations of the icosahedron seem quite prone to this.

The cut is a good workaround, and I decided to print out a copy of the net to build one. I've got a pack of metallic black card I've been dying to build something with. I think this model would look great in black and gold/copper.

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