A soccer ball has 20 faces that are regular hexagons and 12...

A soccer ball has 20 faces that are regular hexagons and 12 faces that are regular pentagons. Use Theorem $7.4$ to explain why a soccer ball cannot have a $60^{\circ}$ rotational symmetry about a line through the centers of two opposite hexagonal faces.

Key Concepts

Symmetry Axes and Fixed Elements

When examining rotations about a specific axis—such as one passing through the centers of two opposite faces—it is essential to consider the elements (like face centers) that remain fixed under the rotation. The configuration and number of these fixed elements must be compatible with the rotation order. If the arrangement of the surrounding faces or vertices does not permit such a fixed-point structure when rotated by the specified angle, then that rotational symmetry cannot exist.

Group Actions and Combinatorial Constraints

The analysis of polyhedral symmetry often involves applying group theoretic ideas, where rotations act on the sets of faces, edges, or vertices. Theorems in this area—like the one referenced—provide combinatorial restrictions on the possible orders of symmetry rotations. These constraints ensure that the orbits of the action (the sets of elements moved among each other under the rotation) align with the overall structure, meaning that certain rotations (such as a 60° rotation) may be ruled out if they do not preserve the combinatorial and geometric configuration of the polyhedron.

Rotational Symmetry

Rotational symmetry in polyhedra refers to the property that the shape looks the same after being rotated by a certain angle about an axis. This concept requires that the entire arrangement of faces, edges, and vertices remain invariant under the rotation. In problems involving polyhedral symmetry, one tests whether a proposed rotation (by, for example, 60°) can map the polyhedron onto itself without altering its combinatorial structure.

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