Miller's Monster

[oxenholme]

I made it out of white cartridge paper back in the 60s.

I'm thinking of making it again, but this time in colour. Presumably six colours for the coplanar pairs of pentagrams, ten colours for the coplanar pairs of triangles, but how many colours will I need for the coplanar pairs of squares?

[guy]

According to Wenninger, there are 30 pairs of squares. They are diametral or "hemi" faces, so there is no parallel pair. To colour each pair differently would need 30 colours.

That makes 6+10+30=46 colours in all. Ouch!

Wenninger describes a simpler colour scheme. He uses a simplified set of 5 colours for the icosahedral (triangular) faces, based on the observation that the 20 face planes are those of 5 tetrahedra: one colour for each tetrahedron. He also uses a single colour for all squares. This gives 6+5+1=12 colours which is a bit more practical.

Although the pentagons cannot be divided into equal subgroups in the way that the triangles can, I am not sure about the squares. I suspect that they might be divisible in various ways, depending on your preference. One way to find out would be to study the Archimedean duals having 60 faces, pick (the?) one with equivalent face planes, and look for sub-symmetries there.

[022ahnoor]

. Presumably six colours for the coplanar pairs of pentagrams, ten colours for the coplanar pairs of triangles

[Peter Kane]

With Stella, how can you set up the colour arrangement described by Guy ?

[Ulrich]

I tried it in the following way:

Take the compound of 5 tetrahedra and put it into memory 1. Then take Millers Monster, select a pentagram (they all should be red in the auto colour option), you hit c and you slightly shift the red colour a bit lighter or darker to avoid colour conflicts in the following steps. You set the radius of the model to the same like that of the compound in memory 1. Select a triangle and hit shift-h to hide all of them, and you do the same with the coplanar one. Then hit shift-ctrl-t to toggle between shown and hidden faces. Now you see all triangles only. You reduce the symmetry to tetrahedral and you add the model from memory 1. If you select a triangle of a tetrahedron now and you look at that face (view->orientation->look at selected face), you see two coplanar triangles parallel above the selected face. You give them the colour of the selected one, using ctrl-c. You repeat this with all faces of the 5 tetrahedra compound, then you delete the tetrahedra one by one. Now you see an arrangement of 5 groups of 4 pairs of coplanar triangles, each group in tetrahedral orientation. (You best save it now in a memory, so that you can retrieve it, when needed.)
You hit shift-ctrl-h to show all faces, select a pentagram, hit h to hide all of them and shift-ctrl-t to toggle between shown and hidden faces. You see twelve pairs of coplanar pentagrams in a dodecahedral arrangement. You colour them like 6 groups of parallel faces, using ctrl-c.
After showing all faces again, you select a square and give all of them the same colour using shift-c.
Al colours can easily be modified afterwards.

Ulrich

[Peter Kane]

Many thanks Ulrich, I'll give it a try.

[robertw]

There's a much easier way. Stella already has various colour arrangements built in. From the menu, select these:

• Color->Basic Color Scheme->Auto Color (probably already set).
• Color->Special Color Arrangements->Dodecahedral Arrangement 2 (6 colors)
• Color->Special Color Arrangements->Icosahedral Arrangement 1 (5 colors)

[Ulrich]

much easier indeed, cool!

[Peter Kane]

Thanks Rob. I did try that, but I must have screwed it up. I just tried it now and it appears to work.

Pete K

[elijahmay]

I need help to know what to fold. I am having trouble with Miller's Monster. I have checked the website but I still don't know how to fold some nets on the polyhedron.

[robertw]

My first suggestion is start simple. If you haven't made many simpler polyhedra first, I'd start there. This one's a huge project to start with.

Once you've done other simpler ones, you'll probably have a better idea how to make this one.

But otherwise, I would need more details to know how to help you with this. When you say "what to fold", I don't know which stage you're really having trouble with. You posted an image, so I'm guessing that's one of the nets causing an issue? Where have you got up to so far? Eg:

• Printed nets and cut out? Be sure to score the edges before cutting out the nets too
• After after cutting, crease the scored lines before gluing them together. You can check in Stella to see whether to use a mountain or valley fold for each edge
• Also see Stella to figure out which tabs should glue to which tabs. You may also want to use "Nets->Show Edge Data->Edge Connection IDs" which will show IDs on the tabs which you can match up to know what glues to what.

[elijahmay]

Thank you. I have printed all the nets and folded the stars. I am having trouble with the bottom section of the picture where it shows the earlier stage. I just don't know how to get from the net out of the printer to the earlier piece on that picture. I also am having trouble with how to get to the foldable nets on great Stella and not the single nets it shows me. Thank you so much!

[robertw]

Did you click on it for the larger image? May still not be clear, but there's a small red triangle on each side connecting the orange parts together.

[elijahmay]

would Stella help me with how to get to that stage with the red triangles?

[robertw]

Stella would print all the parts out, including the red triangles, but it can't tell you what order to put things together. You can use "Nets->Show Edge Data->Edge Connection IDs" to print IDs on tabs that you can match up though.

Or just look for the triangles with matching edge lengths to connect those orange parts.