Heuristic or function for getting the number of possible unique 3D orientations of a cuboid I'm looking into the number of possible orientations for different 3d shapes.
Is there a function that can give the number of unique orientations of a 3d object (e.g. a cuboid shape) given a rotation unit of e.g. 45 degrees in all axes?
any ideas?
 A: If you were considering a finite group of symmetries of 3d space $G\subset O(3)$, then the simple answer would be $$\frac{|G|}{|G\cap H|},$$ where $H$ is the symmetry group of the object.
However I do not think that is your question: There are infinitely many combinations of rotations of $45$ degrees through the co-ordinate axes, hence infinitely many resulting orientations of a cuboid.
Instead I think you are referring to actual rotations of multiples of $45$ degrees through  a co-ordinate axis (not combinations of them), of which there are $22$.  However once you disregard arbitrary combinations, your set of symmetries is no longer a group, so you cannot use the simple formula above.  I do not know of any general method for this situation, but we could look at a few cases.
$1)$ The (rectangular) cuboid parallel with co-ordinate axes, with distinct height, width and length.
The initial resting position of the cuboid gives the same orientation as a $180$ degree rotation through any of the $3$ co-ordinate axes.  Thus $4$ of your $22$ rotations result in this orientation.  The remaining $18$ rotations divide into pairs that result in the same orientation.  Thus you get $1+9=10$ distinct orientations.  Note that the unequal sizes of the sets of rotations that give the same orientation ($4$ and $2$) is what makes this harder than the group case, where they are all the same size and we can just divide the total symmetries by a fixed number.
$2)$ The (rectangular) cuboid parallel with co-ordinate axes, with equal length and width and distinct height.
Now the resting position gives the same orientation as a $180$ degree rotation through any of the $3$ co-ordinate axes, but also the $2$ $90$ degree rotations through the height axis.  Also a $45$ rotation through the height axis gives the same orientation as a $135$ or $225$ or $315$ degree rotation.  Thus your rotations break up into $1$ orientation corresponding to $6$ rotations, $1$ orientation corresponding to $4$ rotations and $6$ orientations corresponding to $2$ rotations each: $22=1\times 6+1\times 4+ 6\times 2$.  Thus you get $8$ orientations.
$2)$ The cube parallel with co-ordinate axes.
Now the resting position gives the same orientation as a $180$ degree rotation through any of the $3$ co-ordinate axes, but also any of the $2$ $90$ degree rotations through each of the $3$ axes.  Thus this orientation corresponds to $1+3+2\times 3=10$ rotations.  The remaining $12$ rotations break up into $3$ orientations, corresponding to $4$ rotations each, so you get $22=1\times 10+3\times 4$.  Thus you have $4$ orientations.
