$3D$ figure with rotation group isomorphic to $D_3$

Question

When considering plane figures, the dihedral symmetry groups occur with those figures that include orientation-reversing symmetries. It strikes me as mildly funny that these same groups appear just as rotation groups of $3D$ figures. What is an example of a $3D$ figure with rotation group isomorphic to the group of symmetries of an equilateral triangle?

Answer 1

If you take an equilateral triangle as the base of a (right) prism, this will have symmetries corresponding to each symmetry of the original triangle, and they'll all be rotations.

Certainly any rotational symmetry of the triangle a symmetry of the prism as well.

To achieve the former reflections as rotations, imagine our prism is situated with

• a copy $T$ of the original triangle in the $xy$-plane, and a

• vertically-shifted version $T'$ in the plane $z = 1$, and another vertically-shifted copy $T''$ in the plane $z = -1$.

• Let's say $A, B$, and $C$ are the vertices of $T$, with appropriate "primes" for corresponding vertices of $T'$ and $T''$.

To get the rotation of the prism that corresponds to the reflection swapping vertices $A$ and $B$ of $T$, use the line segment connecting $C$ and the midpoint of edge $\overline{AB}$ as the axis of rotation. In cycle notation, this is the permutation $(A'B'')(B'A'')(C'C'')$ of the vertices of the prism.

To see the isomorphism with the original symmetry group of the triangle, note that these rotational symmetries of the prism permute the vertices of $T$.