While it is clear you can't have orientation parity in a circle, you could technically build a fudged "Super Constellation Six" by including the 8 small green and 13 tiny blue pieces surrounding the Triskaidecagon (13-gon) center:

I imagine that solving this design would create a nightmare for position-matching all those many tiny parts, not to mention the 13-fold orientation of the centers! However, to build this, you would need an extremely precise tolerance on all the parts, as well as a very tight-fitting puzzle to prevent loss of the inner pieces or accidental rotation of the centers. But by that point, would you even still be able to turn it?

Since we know that the original Tuttminx doesn't exactly have rotational symmetry when doing jumbling moves, I decided to do it on a sphere so it didn't look so fractured. To get better looking results, I increased the resolution of the sphere approximation and changed it to render only the cuts, not the squares that make up the "sphere."
Here's just 2 faces getting turned. (Pent' Hex) x 68:
And to make sure extra faces don't do something different, here it is with 2 pentagons and 2 hexagons. (P1 P2 H1' H2')x38:
Some of the edge pieces don't look like they got chopped up enough so I tried again with different rotation directions. (P1 P2' H1' H2')x38: Looks like this one missed them too. Oh well.

Anyways... Interesting new pieces. Every time I see these fractal pattern of pieces, I'm completely baffled on how to solve them. I'm not talking about just those tiny dusty pieces. I don't even see an immediate solution for the square and long rectangle ones.


As a note, all my cuts are equidistant from the core. I'm not sure if a true Tuttminx has that property or not.

I certainly didn't expect these pieces....
And I think I now see why... Take a look at this image.

I think your pentagonal cut is too deep and the iterative process is trying to correct for this.... and in effect almost making the master version of the puzzle. I think the pentagonal cut should be where the red circle is. Either that or the hexagonal cut should be deeper and you can see that circle starting to form too. This shape is a Truncated Icosahedron and these equations can be used to get you the relative depth of the hexagonal cuts and the pentagonal cuts:

r(pentagonal) =

r(hexagonal) =

Could you use these cut depths on a sphere of radius slightly greater then:

r =

and see what happens? I suspect if you used a sphere exactly equal to or less then r then you'd end up with a 'doctrinaire' puzzle without the need for fudging as there would be no corner pieces.

Carl

That should have been "definition of fudging..." not jumbling. Typo on my part.

Carl

Sound fair. I made the hexagons deeper. I just played it by eye until the rectangles looked like they disappeared. Of course they didn't really disappear because the two radii are different. Here's the approximate relative radii:
Sphere = 1
Pentagons = 0.89
Hexagon = 0.85Yep. The weird pieces seem to have disappeared. I kind of miss them now

I still think the difference in radius would cause the edges of the edges to never ever line up perfectly. Time to run the simulation and find out.
Here's the approximate relative radii:
Sphere = 1
Pentagons = 0.939
Hexagon = 0.915The difference in radii doesn't result in any funny jumbling and creates a stable little sliver of a crescent. This should be the same rectangle type piece we saw in my first renderings. This time I have a vauge glimmer of an idea of how to solve it.
It's really hard to tell which cuts are active because the arcs are so close (so close you probably couldn't build this puzzle), but I'm pretty sure I have it figured out. At each "edge position" on a face, it will always move 1 big edge + 1 thin crescent edge regardless if you are turning a pentagon or a hexagon. The difference is the order that they are in. The pentagon moves them with the big edge on the inside and crescent on the outside. The hexagons are the exact opposite with the crescents on the inside and big edges on the outside.

Would it still be fudging to remove the thin crescent piece because there isn't an infinite number of them?

Never say never. I wonder is there is a shape mod that would help exaggerate the size of this piece. Now that I know its there I'd LOVE to have a puzzle that allowed you to play with it.
I'm tempted to say no. Would you consider a Void Cube fudged? It has a finite number of pieces removed. How about Gerardo's Cube? And I'm sure there are many others that fall in this catagory. The only reason I see that it might be considered fudged is that in this case you could remove those pieces and it NOT be obvious that they have been removed.

Carl

Got it. Here is an idea that will allow you to play with that piece PLUS the others you didn't see how to solve. And the answer lies in your first set of pictures. Start with the Futtminx and cut each corner into 6 pieces and each pent/hex edge into 2 pieces and each hex/hex edge into 3 pieces. The pent face center becomes much bigger. Look at this:



Oskar want to take a shot at this?

Carl

Ok... my PC finally finished.

I notice his image is brighter then mine. bmenrigh did you do any post processing to the image? I'm not sure why the two are different at the moment as you used the same code I did.

Carl

Ray tracing is a sacred mathematical covenant, any post processing of the final image is blasphemy. I didn't do any post processing. I did compile POV-Ray 3.7.RC3 so that I could make use of my 8 CPUs. It wouldn't accept your file as provided so I had to add a few directives. One of them was "assumed_gamma 1.0" in global_settings { [...] }. That probably accounts for the brightness difference.

I tried to be minimally invasive in my modifications since I have never used POV-Ray before and only barely understand the structure of your code. My goal was just to get it running on 3.7. You can see the source to what I rendered here.

It's a hobby to me. So I'm much less serious about it. I was just curious...

That I am sure is it.

Feel free to be as invasive with my code as you like. Its a great way to play with and learn POV-Ray. I take the code of others and try to understand what they are doing and when it clicks (it doesn't always) I learn something new.

Carl