Video

This is a shape modification of my Pentultimate (mechanism v2.5), using caps to create a rhombic triacontahedron.

This modification was first suggested to my knowledge by Robert Webb in 2003 here.

One interesting thing about this modification is that it doesn't change shape as many other pentultimate shape modifications do. It's always a rhombic triacontahedron after every turn.

Every piece orientation matters, as opposed to the Pentultimate where only the triangles matter, or the Icosamate, where only the pentagons matter.

At the end of the video I'm attempting to show how well the puzzle can turn by using only one finger with fast turns. It gets a little loud...

Thanks for looking!

EDIT: I have learned that TomZ made this puzzle first, in 2010.

What a beauty!

One of the things that gets me about solving a Pentultimate is that physically a half-turn (36 degrees) puts the dodecahedron into a shape that almost feels right. It seems the rhombic triacontahedron has this property and the way you've stickered your second photo that much more confusing.

Edit: corrected my reference to a rhombic dodecahedron. The puzzle is a rhombic triacontahedron.

Do opposite faces have same colors? I think it's just been twisted half a turn from solved state.

Fun info about this stickering: Each pentagon has each of the five colors, which could be arranged in 4*3*2=24 possible arrangements, and there are 12 faces, so all the even permutations are used. Each triangle has three colors, and there are 5*4*3/3 = 20 possible colorings of those, and there are 20 triangles, so each possible one is included exactly once.

How beautiful! And it turns like a dream, too!

I may be missing something here, but what pentagons? If you mean the vertices with 5-fold symmetry, then how could you say that each one has each of the five colors? There are more than 5 colors... Can you clarify?

Yeah it was half-twisted. What I was trying to say is that doing this on a dodecahedral pentultimate already "feels right" and his choice of stickering makes it look right too.

Seems like Tom van der Zanden was faster:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2634
Anyway: Variants of the Pentultimate are always amazing.

I'm with you if this puzzle were stickered with only 5 colors but it looks like Jason has used 10 or 15.

Robert Webb did suggest a 5-color sticker pattern, but I didn't try it here. I just used 15 colors, with matching opposite faces to get the interesting half turn look.

Andreas, thanks for the info about TomZ's version. I didn't know I was second to get this one made. Apologies for not mentioning that, Tom.

I'm happy that you made it, but sad because I can't afford to buy it, much less the whole family.

Unless I'm mistaken, the only platonic solid for which this is true is, of all things, the octahedron. In any other platonic shape, the Pentultimate has either non-orientable pieces or duplicate pieces.

(I'm guessing the center triangles on the octahedral version wouldn't lie flat after a 120-degree rotation... I could be wrong though.)

I'm pretty sure they do remain flat.

The rhombic triacontahedron is not a platonic solid (although it's closely related).

I'm not sure there is really a strict definition for how a shape-mod of the Pentultimate should look in the other platonic solids. For a dodecahedron the obvious choice (and arguably the one "correct" choice) has each cutting plane parallel to a face and bisecting the puzzle. For an Icosahedron you could also argue that the one correct choice simply transforms the dodecahedron into its dual.

For a tetrahedron, cube, and octahedron I think the choice is much less clear. If you Fisher / Axis / Ghost / whatever a cubic version of the Pentultimate you could probably make every piece unique and show orientation.

Now we have Pentultimate in a Dodecahedron shape, a ball shape &......Rhombic shape!