Here’s a hyperlink to “Qube, The Film”, a video made with Qube, my digital Rubik’s dice simulator. Mathematically, all the motion is pushed by the 3-by-3 matrices on show.
Qube makes use of a 3x3x3 array of an identical cubelets . The coordinates [x,y,z] of the facilities of the cubelets are all the attainable combos of -2, 0 and +2. The vertices of a person cubelet type a 8-by-3 matrix. The width of a cubelet is managed by multiplying its vertices by a 3-by-3 diagonal scaling matrix.
Our video begins with the width close to zero, so the cubelets seem like factors on the facilities. The width is elevated to a price the place the cubelets nearly contact one another. The small hole is maintained for its visible impact.
The first operations with a Rubik’s dice are the simultaneous rotation of all the cubelets contained in one of many six faces. For instance, this video reveals a slow-motion rotation of the “Left” face in regards to the x-axis.
Additionally it is attainable to rotate the three inside “slices”, in addition to the whole dice about one of many coordinate axes.
The sort of a cubelet is the variety of nonzeros within the coordinates of its middle.
sort = nnz([x,y,z])
The cubelet within the middle of the puzzle has sort = 0 and the six cubelets with sort = 1 are situated on the middle of every face. We use 0:1 to indicate the set of those seven cubelets. In an actual, bodily Rubick’s Dice, 0:1 is a single strong central core that holds the whole puzzle collectively.
There are twelve sort = 2 cubelets situated on the perimeters of every face and eight sort = 3 cubelets on the corners of the puzzle. So, 0:3 denotes the whole dice and 2:3 is the corners and edges with out the central core. Qube, The Film reveals the kinds on this order:
0:3, 0:2, 0:1, 0, 1, 2, 3, 2:3
Cautious readers of this weblog ought to acknowledge 2:3 as stage one of many Menger sponge fractal.
Revealed with MATLAB® R2022a