What does "orientation" of a platonic solid really mean?

Question

Is there any rigorous definition of "orientation" of a platonic solid?

Lots of books mention that the whole group of symmetries of platonic solids consists of rotations composed with reflections, but I'm not sure how we know that the reflections produce only two possible orientations. Furthermore, rotations are said to preserve orientation whereas reflections are said to "invert" the orientation.

Firstly, by "reflection" I assume they mean reflection about a plane?

Secondly, how do we know that there are only 2 possible "orientations" of a platonic solid and not any more or any less than that?

Some supporting visuals/diagrams would be great as well!

Answer 1

Rotations of platonic solids are subgroups of rotations in general. So let's talk why rotations in general preserve orientation. If you take three base vectors $\hat\imath$, $\hat\jmath$, $\hat k$, then the mixed product $(\hat\imath\times\hat\jmath, \hat k)=1 > 0$. If a transformation keeps this product positive, than we say that the transformation preserves orientation. For example, if rotation around $(1,1,1)$ by 120° cyclically swaps base vectors, so we end up with product $$(\hat\imath_1\times\hat\jmath_1, \hat k_1)=(\hat\jmath\times\hat k, \hat\imath)=1>0$$ and we say that this rotation preserves orientation. If instead, we take a reflection around plane $xy$, then $$(\hat\imath_1\times\hat\jmath_1, \hat k_1)=(\hat\imath\times\hat\jmath, -\hat k)=-1 < 0$$ and we say that reflection switches orientation.

One can show that distance preserving transformations (rotations and reflections) can make this mixed product either $1$ or $-1$. So there are only two possible orientations.

Answer 2

There are two ways of orienting a polygon, which we could think of as clockwise and counterclockwise.

A polyhedron is said to be oriented if each of its faces has been oriented in a consistent way, so that, if an edge is shared by two faces, then the order of its vertices given by one face is the opposite that given by the other face.

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If the (connected) polyhedron is orientable at all, an orientation of one face determines the orientations of all the other faces. So there are precisely two orientations, because any given face has precisely two orientations. This is true for any orientable polyhedron, not just Platonic solids.

A symmetry of the polyhedron can therefore either preserve or reverse / invert the orientation. If we think of the polyhedron as situated in three-dimensional space, and its symmetries as arising from isometries of space with an invariant point (the "origin"), then this conception coincides with the one described by Vasily Mitch, though there's quite a bit to unpack there.

By "reflection," your sources most likely mean reflection across a plane, although reflection across a point is also orientation-reversing, and is the product of the reflections across three mutually perpendicular planes. The point could be any point, of course, and the plane could be any plane. But if the transformations are symmetries of a Platonic solid, the point will be the center of the solid (the "origin"), and the planes will pass through this point.

The fact that the dodecahedron has two orientations is illustrated very prettily by the fact that it contains two compounds of five tetrahedra, a "right-handed" one and a "left-handed" one. Half its symmetries preserve these two compounds, and half reverse them.

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Each orientation-preserving symmetry of the dodecahedron permutes the set of five tetrahedra in a given compound, and in this way the symmetry group is seen to be isomorphic to the alternating group $A_5$. Reflection across the center commutes with all symmetries, so the total group is seen to be isomorphic to $A_5 \times \mathbb{Z}_2$.

Chapter 3 of Coxeter's Regular Polytopes is highly recommended.