What do 2x2x2's; Shallow, Normal, and Deep Mixup Cubes; and Shallow Helicopter Cubes have in common?

This POV-Ray code for one:

http://wwwmwww.com/Puzzle/MixUp/Mixup0.povI wrote this to visualize what happens as you vary the depth of the cut on a Mixup Cube. In this code you can change the one line where I define the depth and produce all the below animations and images. It may be more accurate to think of the depth value as the ratio of the width of the puzzle (its edge length as its a cube) to the width of the slice layer.

#declare depth = 0;

Means there is simply no slice layer so we have your basic 2x2x2.

#declare depth = 1/(3+sqrt(2));

This value, about 0.2265, produces a deeper cut Mixup Cube. Here the

T-Centers are the same size as the Face Centers.

The puzzle is doctrinaire and all 45 degree slice layer turns are allowed. Faces are limited to 90 degree turns.

#declare depth = 1/3;

This value, about 0.3333, keeps the face cuts proportional, as they are on a normal 3x3x3. The Face Center square is the same size as the square on the Corners.

The puzzle is doctrinaire and all 45 degree slice layer turns are allowed. Faces are limited to 90 degree turns. It has all the same surface pieces as the one above so it could be considered the same puzzle.

#declare depth = (sqrt(2)/2)/(1+sqrt(2)/2);

This value, about 0.4142, is the normal Mixup Cube.

The puzzle is doctrinaire and all 45 degree slice layer turns are allowed. Faces are limited to 90 degree turns. The T-Centers have either be removed or hidden below the surface of the puzzle.

#declare depth = 1/(2*sqrt(2)-1);

This value, about 0.5469, is a shallow cut Mixup Cube.

The puzzle is NOT doctrinaire and jumbles as some 45 degree slice turns are blocked after the first one. The puzzle can be turned into a doctrinaire puzzle with some extra cuts and fudging as PuzzleMaster6262 has done

here. But is that really needed to make this a puzzle you can fully scramble? I think, but haven't proven, that this could make a very interesting puzzle as presented in the animation below.

The Face Center square is the same size as the square on the Corners.

Note: After the first turn of this animation the second slice turn is limited to 90 degrees.

#declare depth = 3/(1+3*sqrt(2));

This value, about 0.5722, is another shallow cut Mixup Cube. I'll call it the Caged 3x3x3.

The puzzle is NOT doctrinaire and jumbles and has all the same surface pieces as the one above so it could be considered the same puzzle.

Here the Face Centers, the T-Centers, and the X-Centers are all the same size which makes it look like the face of a 3x3x3 is caged in the center of each face.

Note: After the first turn of this animation the second slice turn is limited to 90 degrees.

#declare depth = 1/sqrt(2);

This value, about 0.7071, is another shallow cut Mixup Cube. I'll call it the Caged 2x2x2.

The puzzle is NOT doctrinaire and jumbles but has lost the Face Centers and T-Centers seen on the above two puzzles.

Note: After the first turn of this animation the second slice turn is limited to 90 degrees.

#declare depth = 3/(1+2*sqrt(2));

This value, about 0.7836, is yet another shallow cut Mixup Cube and it is different from the above 3.

The puzzle is NOT doctrinaire and jumbles and it has Face Centers and T-Centers again but these AREN'T the same pieces that were present on those above.

The Face Center square is the same size as the square on the Corners.

Note: After the first turn of this animation the second slice turn is limited to 90 degrees.

So at this point we have covered all the interesting values with depth set to a value less then one. Are we done? If not what does it mean to have a slice layer that is wider then the edge length of the puzzle? I'll slow you...

#declare depth = 1+cut;

This value, just greater then 1, produces a shallow Helicopter Cube.

Here is the animation produced with this code with just that one line changed.

Note: After the first turn of this animation the second slice turn is NOW limited to 180 degrees.

So we can say:

depth < (sqrt(2)/2)/(1+sqrt(2)/2) ~ 0.4142 we have a doctrinaire twisty puzzle. All 45 degree slice turns are allowed.

(sqrt(2)/2)/(1+sqrt(2)/2) < depth < 1 we have a shallow Mixup Cube. Some slice turns are limted to 90 degrees and the puzzle is no longer doctrinaire.

1 < depth < sqrt(2) we have a shallow Helicopter Cube. If we don't consider the jumbling moves present on a typical Helicopter cube we again have a doctrinaire puzzle but now some of the slice turns are limited to 180 degrees.

sqrt(2) < depth and we just have a 1x1x1 cubie. All the cuts are outside the surface of the puzzle.

I had never looked at the Helicopter Cubes and the Mixup Cubes as being related by a single parameter before. I hope others find this interesting.

Enjoy,

Carl