As a note, all my cuts are equidistant from the core. I'm not sure if a true Tuttminx has that property or not.
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.
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 = 0.85
PPHH-38turns-89v85.png [ 320.32 KiB | Viewed 1474 times ]
Yep. The weird pieces seem to have disappeared. I kind of miss them now
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:
Could you use these cut depths on a sphere of radius slightly greater then:
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.
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 = 0.915
PPHH-38-shallow.png [ 400.06 KiB | Viewed 1474 times ]
The 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?