What is the symmetry group of an icosahedron? What’s the stabiliser of an edge?
How many edges are there? How do you know the symmetry group of the icosahedron is the same as the symmetry group of the dodecahedron?
Do you know the classification of higher-dimensional polyhedra?
Solution
1. Symmetry group and edge count of the icosahedron:
1.1. A regular icosahedron has \(V = 12\) vertices, \(F = 20\) triangular faces, and \(E = 30\) edges (since \(3F = 2E \implies 3(20) = 2E \implies E = 30\)).
Euler characteristic \(V - E + F = 12 - 30 + 20 = 2\).
1.2. The group of orientation-preserving (rotational) symmetries is isomorphic to the alternating group:\[ I \cong A_5 \quad (\text{order } 60). \]The full symmetry group, including improper isometries (with central inversion \(-I_3\)), is:\[ I_h \cong A_5 \times \mathbb{Z}_2 \quad (\text{order } 120). \]
action on the 5 inscribed compound cubes permutes them as \(A_5\), and \(-I_3\) commutes with all rotations.
2. Stabilizer of an edge:
2.1. In the rotational symmetry group \(I\), the action on the set of 30 edges is transitive.By the Orbit-Stabilizer Theorem:\[ |\operatorname{Stab}_I(e)| = \frac{|I|}{|\operatorname{Orb}(e)|} = \frac{60}{30} = 2. \]The non-trivial element in \(\operatorname{Stab}_I(e) \cong \mathbb{Z}_2\) is a rotation by angle \(\pi\) (half-turn) about the axis passing through the center of the icosahedron and the midpoint of the edge \(e\).
Orbit-Stabilizer Theorem.
2.2. In the full symmetry group \(I_h\), \(|\operatorname{Stab}_{I_h}(e)| = \frac{120}{30} = 4 \cong V_4\) (the Klein 4-group, generated by the half-turn, reflection across the edge’s perpendicular bisector plane, and reflection across the plane through the edge and center).
Orbit-Stabilizer Theorem in \(I_h\).
3. Duality between the icosahedron and the dodecahedron:
3.1. The regular icosahedron and the regular dodecahedron are geometric duals of each other:Connecting the centers of the 20 triangular faces of an icosahedron produces the 20 vertices of a regular dodecahedron.
face-vertex dual incidence relations.
3.2. Any isometry preserving the icosahedron permutes its faces, and therefore maps the set of face centers to itself.Thus any symmetry of the icosahedron induces a symmetry of the dual dodecahedron, yielding a canonical isomorphism of symmetry groups \(\operatorname{Isom}(\text{Icosahedron}) \cong \operatorname{Isom}(\text{Dodecahedron})\).
functoriality of dual polytope construction.
4. Classification of regular convex polytopes in higher dimensions:
4.1. In dimension \(d = 4\): There are exactly 6 regular convex 4-polytopes:- 4-simplex (5-cell, self-dual),- Hypercube / Tesseract (8-cell),- 16-cell (cross-polytope, dual to 8-cell),- 24-cell (self-dual),- 120-cell (hyperdodecahedron),- 600-cell (hypericosahedron, dual to 120-cell).
classification of 4D Coxeter-Dynkin diagrams (\(A_4, B_4, F_4, H_4\)).
4.2. In dimensions \(d \ge 5\): There are only 3 regular convex polytopes (the classical families):- \(d\)-simplex (\(A_d\)),- \(d\)-cube (\(B_d\)),- \(d\)-orthoplex / cross-polytope (\(B_d\)).
Schläfli symbol classification and non-existence of exceptional regular polytopes for \(d \ge 5\).
5. Conclusion:The icosahedron has 30 edges, rotational group \(A_5\) (stabilizer \(\mathbb{Z}_2\)), full symmetry group \(A_5 \times \mathbb{Z}_2\) (stabilizer \(V_4\)), shares symmetries with the dual dodecahedron, and fits into the finite classification of regular polytopes. Q.E.D.
1 through 4.