It's easier to think about the space of positions of an unsymmetrical object (we're talking rotational symmetry here) such as a teapot, to start with. The space of positions of a dodecahedron is a bit smaller, because some positions of the dodecahedron, which would be different if the dodecahedron was less symmetrical (for example, if one or more of its faces were visibly scratched), count as indistinguishable if the dodecahedron is absolutely, ideally regular - if it is a truly Platonic dodecahedron, in other words. The ideal dodecahedron has fewer distinguishable positions than one that is scratched or otherwise imperfect.
Indeed, we can readily see that the space of positions of an ideal dodecahedron has some of features that agree with what we already know of the PDS. Imagine your dodecahedron sitting on your desk, and that you turn it slowly about a vertical axis. All its positions are different, until you have turned it through 72°, at which point its position becomes indistinguishable from its starting position. Your journey through the 'space of positions' has come back to its starting-point.
However, this is not the only way to return to the starting-point. Suppose you turn the dodecahedron slowly about a horizontal axis, parallel to one edge of the top pentagonal face. After you have turned it through 63.43495° (= inverse tan of 2) the centre of the top face will have taken the position that was originally occupied by the centre of a next-to-top face. If you now further rotate the dodecahedron through 36° about the axis which passes through this centre, and through the centre of the opposite face (the one that started off at the bottom), the dodecahedron will again be in a position that is indistinguishable from its starting position.
In fact, what we have achieved by means of two rotations (through 63.43495° and 36°) can actually be achieved by a single rotation of 72°. Just let the axis pass through the centre of another one of the next-to-top faces (a next-door one). Then the topmost face will move directly into the place of the next-to-top one.
There are five possible choices of axis - six, including the vertical one - which gives us six directions in the space of positions which lead back to the starting-point. Or 12 directions, counting clockwise and anticlockwise turns as distinct. The similarity - to put it no stronger - with the PDS (as defined earlier in this blog) is striking. What is not clear, as yet, is whether it makes dense to regard the 12 privileged directions as 'equably distributed' over the 'celestial sphere' of directions at a point within the 'space of positions'. This is something we need to investigate.
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Returning to the space of positions of a teapot, how big is the space? How much of it is there? For a start, how many dimensions does it have?
It will often be helpful to imagine the teapot as fixed within a transparent sphere, with a reference point and arrow (based at that point) marked on the sphere. When the teapot is rotated into a new position, so is the sphere. We can tell what has happened to the sphere by noting 1) where the reference point now is and 2) in which direction the arrow is now pointing. The position of the point is fixed by two coordinates; the direction of the arrow by one more coordinate. Thus three coordinates suffice to fix the position of the sphere, and hence the position of the teapot: its space of positions is three-dimensional.
Having got a bit of a handle on the space of positions, our next move is to turn it into a sort of geometry, by saying what we mean by a line in the space of positions.
Given two positions, A and B, say, of the teapot, one can always get from one position to the other by rotating the teapot through a certain angle, about a certain axis (this is not obvious, but will be shown below). The 'line' joining A and B consists of all the intermediate positions reached by rotating about the same axis, but through different angles. It is clear that the 'line' is not the sort of line that goes on forever. Really it is more like a circle: if you go far enough along it, you come back to your starting-point. But so long as we are speaking of 'geometry', it will be useful to continue to call it a line.
As promised, we will show that one can always get from A to B by means of a rotation. Think in terms of the circumsphere with its reference point and arrow. How do we get the arrow from position A to position B? Suppose the reference point needs to move from point a to point b. The locus of points equidistant from a and b is a certain plane, passing through the mid-point of the line ab, and also through O, the centre of the sphere. (The plane is the perpendicular bisector of ab.) We can get the arrow from a to b by reflecting in this plane. It is of course a plane of symmetry of the sphere, so the whole sphere is mapped to itself.
The trouble is, that although the arrow has been moved to the right place, it will most likely be pointing in the wrong direction. However, this can be corrected by a second reflection, this time in a plane that passes through b, and through O. So, we can get the arrow into the right position by means of two reflections.
But the effect of two reflections, in two planes, is identical with the effect of a rotation, about the line which is the intersection of the two planes, through an angle which is twice the dihedral angle between the planes.
In this case, the line of intersection necessarily passes through O and is therefore a diameter of the sphere, intersecting the sphere in two antipodal points.
Each of the planes intersects the sphere in a great circle. Taking a more 'intrinsic' view of what is going on, we may regard these great circles as 'lines' in the geometry of the sphere, and the reflections as reflections in these 'lines'. The lines meet in two points, at a certain angle (which is the same at both points). The two reflections are equivalent to rotation about either of these points, through twice the angle between the lines.
It may be the case that the second line of reflection is actually the (spherical) line ab, in which case the meet of the two lines is the mid-point of ab, and the angle of intersection is 90°. The equivalent rotation is rotation about the mid-point, through 180°. This is of course the largest possible angle of rotation.
The smallest possible angle of rotation occurs when the second line of reflection (passing through b) is perpendicular to ab. Then the axis of the rotation is the pole of the great circle ab, and the angle is the length of the arc ab (times 360 divided by the circumference of the sphere, assuming you want it in degrees).
The two extreme cases, together with an intermediate case, are illustrated in the following figure. In each case the line in arrow-position-space, joining A to B is also shown.
(To get better resolution, click on the image)
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