How to count the number of symmetries of a 3-d object?

Question

Suppose, I have 3-d shape with a finite number of sides, what would be the general procedure for finding how many symmetries it has? For example, suppose I have a cube, If I rotate the cube or even flip it, the cube is same. So how do I find how many 'actions' that I can do on the cube and still get an equivalent figure?

I have written some attempts that I've done underneath:

1. Manual counting in case for a cube

Now this is an attempt to do this by brute force, the arrows denote the way I am rotating the cube and the transformation underneath is the opposite transforms which would undo that. I get eight but I don't think this is the right answer neither is it generalizable

2. Something to do with integer solutions?

I was recently reading a book called "Strange Curves, Counting Rabbits, & Other Mathematical Explorations Book by Keith Martin Ball" in it he species the cube using points.

So, I'm thinking the number of symeteries is related to the integer solutions of

$$ x,y,z \leq 3 $$ with $ x,y,z \geq 0 $$

Research attempts:

I saw this "poly enumeration theorem" but I can't grasp it because I don't understand group theory: https://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem

and, I saw this stack:

What are the symmetries of the tetrahedron? , but even after learning some group theory, I can not understand it

Answer 1

We have rotations around an axis, reflections in a plane, and combinations of rotations and reflections (rotoreflections).

Consider how we can put a rotation axis through the center of the cube, and through which angles we can then rotate to find the same cube.

We have the following rotations:

• Axis through the center of a face (3 options) after which we can rotate by 90, 180, or 270 degrees. This gives us 9 rotations.
• Axis through the center of an edge (6 options) after which we can only rotate by 180 degrees. It's another 6 rotations.
• Axis through a corner (4 options) after which we can rotate by 120, or 240 degrees. It gives another 8 rotations.

So we have a total of 23 rotation symmetries.

We have the following reflections:

• Plane parallel to a face giving 3 reflections.
• Plane that contains an edge for 6 more reflections.
• In theory we could also have a reflection plane through a corner without containing an edge, but that does not work for a cube.

So we have 9 reflection symmetries.

That leaves the roto reflections that are a bit harder to enumerate. Typically we can encode each symmetry as a set of 8 numbers. Each number identifies a corner, which we will number 1-8. Now we have to combine each rotation with each reflection to see if that yields a new symmetry. I won't do that here. Let it suffice that there are 15 unique rotoreflections.

To summarize, the symmetries of the cube are:

1. Identity (1).
2. Rotations (23).
3. Reflections (9).
4. Rotoreflections (15).

Oh, and it's not a coincidence that identity plus rotations number 24, which is the same number as the number of reflections and rotoreflections. It's how we can verify that we did not miss any.