In 1775, mathematician Leonhard Euler proved a fairly-intuitive fact about three-dimension objects. In rough terms, Euler proved that for every pair of orientations of an rigid object, there is an axis of rotation that takes you directly from one orientation to the other (and back again).
Suppose you have have a Rubik’s Cube sitting on a table in front of you. You pick it up, toss it from hand to hand a few times, spinning it in all sorts of ways, and then drop it carelessly back in roughly the same place. Euler’s rotation theorem says that all those spins boil down to a single rotation. There will always be a single rotation about an axis that could take your Rubik’s Cube directly from its original orientation to its new orientation.
This general rule applies to the specific case of isometries: given any two isometries, there will always be an axis of rotation that takes us directly from one to the other. Such axes are known as axes of symmetry, and of particular interest to mathematicians.
Case study: the tetrahedron
The tetrahedron has 12 isometries. This is easily ascertained by counting the number of ways in which a Pyraminx can be placed face down with an upright edge facing towards you. There are four faces, so four ways to place it face down. Having chosen a down face, there will be three upright edges from which to choose your front edge. Thus there are a total of 4 × 3 = 12 isometries.
Now choose one of these isometries to be our initial orientation. Euler’s rotation theorem tells us that for each of the other eleven isometries, there will be an axis of symmetry that will take us directly there.
In the examples below, I’ve chosen my starting point to be blue face down, with the yellow-green edge to the front. One axis of symmetry is perfectly obvious: leaving the blue face on the table, I can rotate my Pyraminx about its top vertex to bring a different edge to the front. In fact, I have a choice of two edges that I can bring to the front this way:
The axis of symmetry that we just rotated around passes through the centre of the blue face, and through the opposite vertex. Since the tetrahedron has four identical (and isometric) faces, there must also be axes of symmetry passing through the centres of the other three faces too. This is indeed the case.
Rotating around the green-face axis:
Rotating around the red-face axis:
Rotating around the yellow-face axis:
Each of these three axes allows rotation between three isometries: the starting isometry and two others. Therefore these four axes of rotation account for only the starting isometry and 8 of other 11. There must be some more axes of symmetry.
And indeed there are. The remaining axes of symmetry pass through the centres of opposite edges.
Rotating around the axis that passes through the centres of the blue-green and red-yellow edges:
Rotating around the axis that passes through the centres of the blue-red and green-yellow edges:
Rotating around the axis that passes through the centres of the blue-yellow and green-red edges:
This accounts for all of our isometries, so there can be no more axes of symmetry.
Order of rotational symmetry
Notice how some axes of rotational symmetry allow us to move between three isometries, while others allow only two. In other polyhedra, some axes will support four, five or even more isometries.
The number of isometries that we can cycle through by rotating about an axis is called the order of rotational symmetry. So, for example, we say that an axis that passes through a face centre and the opposite vertex of a tetrahedron has rotational symmetry of order 3. Equivalently, we say that it has 3-fold rotational symmetry. Finally, since an axis with 3-fold rotational symmetry must reach an isometry every 120°, we may also say that it has 120° rotational symmetry.