Physical 4-dimensional Puzzle!?!

Physical 4-dimensional Puzzle!?!

This is a new and very special puzzle just shared with the world from one of my creative heroes, Melinda Green. It's a bit magical that it can even exist, and a number of mathematical properties had to be just so for it to be possible.

The program MagicCube4D has long allowed playing the 4-dimensional analogue of a 2x2x2 Rubik's "Pocket" cube on the computer. With this combinatorially equivalent physical puzzle, you can play the 2x2x2x2 in your hands. It goes to show how keeping hope alive for something over a long period of time, and considering the possibility of it in many different ways, can pay off.

The way I like to think of the puzzle is to consider the 4-dimensional hypercube projected to the 3-sphere. Then regard the 3-sphere topologically as two balls glued along their boundary. Both of the 2x2x2 blocks in the physical puzzle are one of those balls, representing a hemisphere of the 3-sphere and of the higher dimensional puzzle.

The 2^4 puzzle cuts a spherical hypercube along equators of the 3-sphere, defining a dual 16-cell. Each of the 16 little cubes in Melinda's puzzle acts like a cell of the 16-cell (a tetrahedron). Note how each little cube has 4 colors, one for each face of a tetrahedron.

Here's an image of what I'm describing a dimension down:
https://photos.app.goo.gl/bnBF6RZbWQtKVk8t1

But it works better a dimension up! And this is because the tetrahedral group is a subgroup of the octahedral group (whereas in the picture above, the cyclic group of order 3 is not a subgroup of the cyclic group of order 4, so we can't represent triangles as squares in a nice way). Group structure is why those "tetrahedra" can be represented by cubes in the physical puzzle.

The math is fun, but I must say working with the puzzle is quite satisfying too. It's got a je ne sais quoi of simplicity reminiscent of the original Rubik's cube... tons of depth in a simple, colorful package.

Melinda's homepage for the puzzle lives here: http://superliminal.com/cube/2x2x2x2



Originally shared by Melinda Green
https://www.youtube.com/watch?v=_D4m1Kit3TI&feature=autoshare

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