How to calculate all 24 rotations of 3d array?

Question

I have a 3d numpy array describing a polycube (imagine a 3d tetris piece). How can I calculate all 24 rotations?

Numpy's array manipulation routines includes a rot90 method, which gives 4 of the 24, but I'm clueless how to calculate the rest. My only idea is to convert the 3d array to a 2d matrix of co-ordinates, multiply by a rotation matrix, and convert back. But I'd rather work directly with the 3d array.

Example 2x2x2 array:

>>> from numpy import array >>> polycube array([[[1, 0],         [1, 0]],         [[1, 1],         [0, 0]]]) 

Example 3x3x3 array:

array([[[1, 1, 0],         [1, 1, 0],         [0, 0, 0]],         [[0, 0, 0],         [1, 0, 0],         [1, 0, 0]],         [[0, 0, 0],         [0, 0, 0],         [0, 0, 0]]]) 

Edit: I only want the 24 orientation-preserving isometries, not all 48 rotations and reflections (though it would be interesting to know how to make them too). If it helps to test, I believe the 3x3x3 example has no rotational symmetry and is chiral (so the 48 are distinct).

Motivation: I'm writing a solver for a Soma cube-style puzzle.

Answer 1

Update: Simplified after Numpy 1.12.0 added an axes argument to the rot90 function

Here's how I made all 24 rotations:

from numpy import rot90, array  def rotations24(polycube):     """List all 24 rotations of the given 3d array"""     def rotations4(polycube, axes):         """List the four rotations of the given 3d array in the plane spanned by the given axes."""         for i in range(4):              yield rot90(polycube, i, axes)      # imagine shape is pointing in axis 0 (up)      # 4 rotations about axis 0     yield from rotations4(polycube, (1,2))      # rotate 180 about axis 1, now shape is pointing down in axis 0     # 4 rotations about axis 0     yield from rotations4(rot90(polycube, 2, axes=(0,2)), (1,2))      # rotate 90 or 270 about axis 1, now shape is pointing in axis 2     # 8 rotations about axis 2     yield from rotations4(rot90(polycube, axes=(0,2)), (0,1))     yield from rotations4(rot90(polycube, -1, axes=(0,2)), (0,1))      # rotate about axis 2, now shape is pointing in axis 1     # 8 rotations about axis 1     yield from rotations4(rot90(polycube, axes=(0,1)), (0,2))     yield from rotations4(rot90(polycube, -1, axes=(0,1)), (0,2)) 

Test that all 24 rotations are indeed distinct:

polycube = array([[[1, 1, 0],         [1, 1, 0],         [0, 0, 0]],         [[0, 0, 0],         [1, 0, 0],         [1, 0, 0]],         [[0, 0, 0],         [0, 0, 0],         [0, 0, 0]]])  assert len(set(str(x) for x in rotations24(polycube))) == 24