Math CentralQuandaries & Queries


Question from Vincent, a student:

I want to ask is there a formula to find out the number of axes of rotation & plane of reflection of a 3D figure, like pentagon, pyramid?

Thank you for answering!


It is not clear what kind of formula you expect. The only method to determine the symmetries of a polyhedron that I know is to look at the figure. However in the three cases of REGULAR POLYHEDRA, if the number of edges is E then the number of planes of reflection is
(3/2)*(SQRT[4E+1] - 1).
See REGULAR POLYTOPES by H.S.M. Coxeter, page 68.

There are few possible types of finite symmetry groups in three dimensions. The complete list of 14 types can be found in the appendix of INTRODUCTION TO GEOMETRY by H.S.M. Coxeter, page 413, with their description in section 15.5.
For any polyhedron, an axis of rotation must pass through either (a) the centre of a face, or (b) the midpoint of an edge, or (c) a vertex. Any mirror (or plane of reflection, if you prefer) must intersect the polyhedron in either (a) a line of symmetry of a face (the mirror is perpendicular to the face and intersects it in a line that joins the centre of the face either to a vertex or to the midpoint of an edge), or (b) an edge (in which case it bisects the angle between the two faces that come together at that edge).

Here are the figures with the most symmetry: The first three come from the five Platonic solids.
7 axes (four join a vertex to the centre of the opposite face, three join midpoints of opposite sides).
6 mirrors (each joins an edge to the midpoint of the opposite edge).

13 axes (in the cube, 3 axes join the centres of opposite faces, 6 join midpoints of opposite edges, and 4 join pairs of opposite vertices; the octahedron is the dual figure, so just interchange the roles of faces and vertices).

9 mirrors (for the cube, 6 contain pairs of opposite edges and 3 pass through midpoints of four parallel edges).

31 axes (for the dodecahedron, 6 join the opposite faces, 10 join opposite vertices, and 15 join midpoints of opposite edges).

15 mirrors (each contains two edges of the dodecahedron and the centres of four faces).

Then there are PYRAMIDs
with (at most) 1 axis joining the centre of the face at the bottom to the vertex directly above it,
and as many mirrors containing that axis as there are lines of symmetry of the base figure.

(with congruent polygons at the bottom and top, and sides that are rectangles (for the prisms) or triangles (for the antiprisms).

These have 1 vertical axis and mirrors that contain that axis and meet the top and bottom polygons in lines of symmetry. There might also be a horizontal mirror between the top and bottom polygons, and any number of axes in that plane, depending on the symmetry and relative position of the top and bottom polygons.



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