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 me

Answer 1

First let's determine the 48 isometries, as matrices that permute the basis vectors, possibly flipping some signs:

def isometries():
    basis = numpy.eye(3)
    for ix in range(3):
      for iy in range(3):
        if iy == ix:
          continue
        for iz in range(3):
          if iz == ix or iz == iy:
            continue

          for sx in range(-1, 2, 2): # -1, 1
            for sy in range(-1, 2, 2):
              for sz in range(-1, 2, 2):
                yield numpy.array([sx * basis[ix], sy * basis[iy], sz * basis[iz]])

Then let's filter for the rotations, which are the isometries of determinant 1:

def rotations():
    return filter(lambda x: numpy.linalg.det(x) > 0, isometries())

Finally, we can apply such a rotation to a polycube by shifting the array indices such that the inner point is the origin, and multiplying the index vector by the rotation matrix:

def rotate(rotation, shape):
    shift = numpy.array([1,1,1])
    res = numpy.zeros(27).reshape(3,3,3)
    it = numpy.nditer(shape, flags=['multi_index'])
    while not it.finished:
        v = numpy.array(it.multi_index) - shift
        w = rotation.dot(v)
        dest = tuple(w + shift)
        res[dest] = it[0]
        it.iternext()
    return res

For example:

>>> rotate(list(rotations())[5], polycube)
array([[[ 0.,  0.,  0.],
        [ 0.,  0.,  0.],
        [ 0.,  0.,  0.]],

       [[ 0.,  1.,  1.],
        [ 0.,  0.,  0.],
        [ 0.,  0.,  0.]],

       [[ 1.,  1.,  0.],
        [ 1.,  1.,  0.],
        [ 0.,  0.,  0.]]])