This level of geometry NEEDS pictures.

I had the exact same thought.

Pick a corner of the underlying megaminx. Glue in place the three face pieces which are closest to it and glue them in place so they can't spin. Then take the three face pieces which share two edges with pieces we just glued in place, and glue them in place as well. Finally, glue all edges which are shared by two of the face pieces we've glued, and all corners which are shared by three of them.

For the crystal layer, take the triangular piece above the corner you first picked and glue it, the three wedges it shares an edge with, and the three triangles on the other sides of those wedges, in place. Glue them to the mexaminx layer pieces which are glued in place as well.

I found myself wondering if this puzzle had more glue in it then plastic by this point. Everytime I try to picture what is glued to what it just sounds like everything is glued to everything else. This picture did help with the last bit though

I worked it out in my head mentally:



As you may well know, the Pyraminx crystal (PC) uses a megaminx core (MM). Just like the rube cube, the MM centers hold in the MM edges, which in turn hold down the MM corners. The MM centers and edges are rounded out like a sphere to give shape underneath the PC. The MM corners are extended to the PC corners. The PC edges are wedged underneath the PC corners. The Star Minx (SM) would likewise use an extended PC edge as the SM edge, which would hold it's other parts in. The SM edge holds the SM star points in place underneath them, which in turn hold the SM center in place. The SM face centers move exactly in the same manner as the pentagons on the Pentultimate (PU), so naturally, we extend the SM faces out and PU centers, but we cannot connect the PU corners without them falling off, as any grooves or notches we cut into the PU centers / corners to hold in place will be incompatible during mid-rotation (only reason this works for the Skewb is due to the even-sidedness of the square and the duality of the triangles). To alleviate this, grooves will be etched into the walls of the PU parts, and interlocking hidden tabs will be embedded into the grooves to lock in the parts and still allow PU corner/corner and center/center to glide together durning rotation. The SM edge pieces will have additional "spokes" that extend into the PU laters where the grooves iteresect, to prevent the interlocking PU tabs from colliding. Finally, for a fully functional Pentultimate, you need to prevent misaligned midsections in the SM region; this is easily attainable by simply locking six of the axes within the core in place, essentially blocking an entire hemisphere of the internal 12-axis.

Working outwards each layer starting from the MM: parts of the PC are trapped beneath MM parts; parts of the SM are trapped beneath PC parts; parts of the PU are trapped beneath SM parts; so that each layer locks in the layer outside itself.

So there you have it, a fully functional Pentultimate without the infamous "knucklehead" mechanism of the original (the twelve fully redundant floating layers were a bit convoluted ), though I dare say the number of moving parts required to achieve this is probably even greater

Cubic analogs:
MM (6 axis) <--> 3x3x3 (3 axis)
PU (6 axis) <--> skewb (4 axis)

Since their cubic counterparts don't even share the same number of axes, thus proves the intermediary stages of the cube version cannot exist

While we are on the subject, who will volunteer to build this beast:

I think I'll call it the "Ultimate Pentagon Ball"

but the recommended "pyramid of glue" would be unnessesary, as simply locking the aforementioned axes would demobilize said parts without need for extra glue.

Oh... those are easy. I see your red, green, blue, and yellow ball and add in some white lines to boot.



Carl

Sorry for not creating visuals, guys, but pentagonal objects are extremely difficult to make when you've only got the orthogonal axis (X, Y, Z) and right angles to work with.

When I said "block one whole hemisphere," I meant that one axis, and all the other five axis surrounding it, would be blocked. Alternately, a triangle of six axes could be used, as Bram proposed,
viewtopic.php?t=4229
but the recommended "pyramid of glue" would be unnessesary, as simply locking the aforementioned axes would demobilize said parts without need for extra glue.

While we are on the subject, who will volunteer to build this beast:
I think I'll call it the "Ultimate Pentagon Ball"

A 4x4 minx/4x4 starminx/pyraminx crystal ball?

While we are on the subject, who will volunteer to build this beast:
I think I'll call it the "Ultimate Pentagon Ball"

Oh... those are easy. I see your red, green, blue, and yellow ball and add in some white lines to boot.



Carl

How many solved states does this puzzle have? Since the centers can move and there are no 3 color corners I'm guessing its possible to solve the puzzle to a state that appears to be the mirror image of the starting state. If that's doable there should be atleast 2 solved states to this puzzle. Are there any other colorings of the dodecahedron possible where each face has the same neighbors but in another order? I don't think so but I'm not sure I can prove that. Maybe this puzzle has more then 2 solved states.

And a challenge...

Assuming this puzzle has two solved states that are the mirror images of each other, what is the minimum number of moves required to get from one to the other?

Carl

Sorry for the late reply. I think a mirror image cannot be reached with the Starminx. If you pick any mirror plane that bisects the puzzle and cuts 4 faces in half, you'll have 2 edges that need to stay exactly as they are, 2 edges that stay in the same position but need to be flipped, and the other 26 edges need to swap places in pairs. 13 pair swaps -- an odd number -- hence impossible. If you tried to solve the Starminx into a mirror image state, the closest you could get is all the centers and corners correct, and all the edges correct but for 2 edges that need to swap places.

Wow!!! Thanks for that. A late reply is much better then no reply. This reads like a proof... but I noticed you started with "I think...". Mind if I ask how sure are you? Also the mention of "corners" puzzled me a bit. I assume you mean the face pieces which point toward the corners. Since they don't have assigned possitions on each faces (like the face centers on a 4x4x4) could two of those be swapped with the last two edges to make a mirror solution possible?

Carl

While we are on the subject, who will volunteer to build this beast:
I think I'll call it the "Ultimate Pentagon Ball"

What program is that?

Bumrang

http://www.jaapsch.net/puzzles/sphere.htm

I am 100% sure.

why doesn't the winner come out and show it off =(

Agreed... but do we even know if the winner is a member here?

Carl

13 pair swaps -- an odd number -- hence impossible. If you tried to solve the Starminx into a mirror image state, the closest you could get is all the centers and corners correct, and all the edges correct but for 2 edges that need to swap places.

But in my opinion it is incumbent upon Aleh to show the internals. He is the constructor and he should have the honour to do this and he has the permission to do this.