Compound of 2 Icosahedra

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jos smits
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Joined: Fri Oct 24, 2014 2:07 pm

Compound of 2 Icosahedra

Post by jos smits » Fri Oct 31, 2014 3:32 pm

Just bought Great Stella4D. Happy to join the club of symmetrists. Having built quite a few models over the past years I am ready to tackle the compounds of 2 icosahedra as shown here:

http://mathworld.wolfram.com/Icosahedron2-Compound.html

With only trigonometry I find it rather difficult and Great Stella only presents me with the first model (from left to right). Who can make the two missing beauties with GreatStella or teach me how to do it ?
Many thanks in advance.

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oxenholme
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Post by oxenholme » Tue Nov 04, 2014 4:03 pm

Just to observe that the third one can be worked out easily without trig as it consists of 4 pentagonal pyramids and two pentagonal antiprisms with the pentagons paired so that their vertices coincide with alternate vertices of their circumscribing decagons.

I would tackle the second one by drawing two icosahedra looking face on with the vertices of the two faces coinciding with alternate vertices of their circumscribing hexagon.

Both quite enjoyable exercises!

I made the first one some time ago - and the corresponding compound of two great dodecahedra.

My copy of Stella died about 5 computers back, so once again I have to work things out from first principles!!!

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robertw
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Post by robertw » Fri Nov 07, 2014 9:38 am

Sorry for the late reply! Somehow I missed this when first posted.

First compound:
  • Start with any polyhedron that has a hexagram, eg a hexagrammic prism (really a compound of two triangular prisms). Create this by opening the built-in polyhedron list (with "File->Polyhedron List", the matching toolbar button, or hitting Ctrl+N), type in 6/2.4.4 and hit enter.
  • We will augment an icosahedron onto each of the coplanar triangles on one side of the prism. First, drop to a lower symmetry so that it only adds icosahedra on one side not both. Find the rotational symmetry drop-down list in the toolbars. It should be showing "6-fold Dihedral". Change this to "6-fold Pyramidal".
  • Select one of the coplanar triangles in either hexagram (Shift+left click, or double left-click on it).
  • Prepare to augment the polyhedron using "Poly->Augment Polyhedron", the matching toolbar button, or hit the "a" key.
  • In the box that appears:
    • Type "i" in the "Symbol" box (this is enough for it to presume you mean the icosahedron). It will select "Augment using...Built-in model" for you.
    • You can leave other settings as their defaults.
    • Hit OK (or Enter)
    • For some reason I get a warning about producing a model with lower symmetry. I don't think it's supposed to do that, but just hit OK.
    • You are now looking at a preview of the augmentation. Hit OK to accept it.
  • The compound of 2 icosahedra should now appear, but with the hexagonal prism still attached to it.
  • Select one of the coplanar triangles on the other side of the hexagonal prism.
  • Augment again. This time choose these options:
    • Augment using...Prism.
    • Select "Excavate (inwards)".
    • Hit OK, and then OK again to accept the preview.
  • Voila! It is probably in a single colour now. Use "Color->Color as a Compound" or hit Shift+A to colour each icosahedron differently.
Second compound:
  • As observed above, this one has two antiprisms in the middle with pyramids on top, so...
  • Start with a compound of two pentagonal prisms. Do this by opening the polyhedron list and typing 10/2.3.3.3 and hit OK. 10/2 is a compound of two pentagons.
  • Select one of the coplanar pentagons and augment it using pyramids. Remember to switch from excavate back to augment. If you forget, or get any options wrong, just hit "a" again to return to the dialog box.
  • Use "Color->Color as a Compound" or hit Shift+A again to colour each icosahedron differently.
Let me know how you go!

jos smits
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Post by jos smits » Tue Nov 11, 2014 9:24 am

Hallo Robert !

Many thanks for your very precise description. Within 10 minutes I had the models rolling over the screen, giving me a thrill as never allover because it was the first time I 'created' something in Stella. Now there is a lot of learning to do but with f.e. your fantastic models of prisms and antiprisms on Wikipedia this can only be a pleasure.
Questions arise from answers. Having seen the clusters of Rh. Triacontahedra on R. Austin's Facebook posted on Nov. 8 I tried to make the same but could not even augment one Rh.Tr. with another one saved as a file. (Message: doesn't have any matching 4-sided faces.)....... ??

Are you familiar with the work of Sandor Kabai ? He did a lot with Oblate Rhombohedron and Prolate Rhombohedron (golden rhombs) as building blocks for Rh. Triacontah., Rh.Icosahedron, Rh. Dodecahedron of second kind and many many other things. I find his work fascinating and just incase you are looking for some new models to complement Stella Library I would certainly recommend these jewels because you can build so many things with them.

http://www.kabai.hu/

Here is a picture of your Stellation of small stellated truncated dodecahedron. I made it about 10 years ago in Greece without Stella but with your personal help. Remember ? Great to see you are using it as your logo. https://www.dropbox.com/s/ew8uc7178q93j ... b.jpg?dl=0

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robertw
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Post by robertw » Tue Nov 11, 2014 12:33 pm

Just guessing, but maybe augmenting from a file isn't working because you saved the icosidodecahedron, with RTC as the dual, rather than saving the RTC as the main model.

I suggest not augmenting from a file. Instead, stick the RTC in a memory slot (eg "Edit->Put Model In Memory->Memory 1" or type "m1") and augmenting from there. As long as the view showing the RTC (ie the dual view) is selected, that's the model it will put in memory.

jos smits
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Joined: Fri Oct 24, 2014 2:07 pm

Post by jos smits » Tue Nov 11, 2014 5:40 pm

Many thanks once again Robert. This way it is working.

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