What is a symmetry of a solid?
Imagine a cube standing in a box that fits it exactly. A symmetry is any move of the cube that leaves it filling the same box again, corners on corners and edges on edges.
There are three kinds of symmetry for solids:
- Plane (mirror) symmetry: reflect the solid in a plane through its middle. Every point goes to its mirror image on the other side.
- Axial (rotation) symmetry: turn the solid about a line (an axis) by some angle.
- Central symmetry: every point goes to the point on the opposite side of one fixed point, the centre, at the same distance.
Doing nothing is also counted as a symmetry (the "identity"). The more symmetries a solid has, the more regular it is.
Planes of symmetry (mirror planes)
A plane of symmetry cuts a solid into two halves that are mirror pictures of each other.
- Cube: 3 planes parallel to the faces (through the middle) and 6 planes through two opposite edges. Total 9.
- Octahedron: also 9.
- Regular tetrahedron: each plane holds one edge and the middle of the opposite edge. There are 6 edges, so 6 planes.
- Dodecahedron and icosahedron: 15 planes.
A solid that is not regular has fewer. A square pyramid has only 4 planes, and a plain box (cuboid) has 3.
Axes of rotation
If a solid fits again after turning by 360° ÷ n about a line, that line is an n-fold axis (axis of order n). Turning 360° brings every solid back, so the interesting question is: what is the smallest turn?
- Cube: 3 four-fold axes (through opposite face centres, turn 90°), 4 three-fold axes (through opposite corners, turn 120°), 6 two-fold axes (through the middles of opposite edges, turn 180°).
- Octahedron: the same numbers, but the four-fold axes pass through opposite corners and the three-fold axes through opposite faces.
- Tetrahedron: 4 three-fold axes (corner to the middle of the opposite face) and 3 two-fold axes (middles of opposite edges).
- Dodecahedron and icosahedron: 6 five-fold, 10 three-fold and 15 two-fold axes.
How many different turns fit? An n-fold axis gives n − 1 turns other than "do nothing". Add one for doing nothing.
Tetrahedron: 1 + 4×2 + 3×1 = 12. Cube: 1 + 3×3 + 4×2 + 6×1 = 24. Icosahedron: 1 + 6×4 + 10×2 + 15×1 = 60.
Centre of symmetry
A point O is a centre of symmetry if for every point P of the solid the point P′ on the other side of O, at the same distance, is also on the solid. In other words, O is the midpoint of every segment PP′.
- Cube, octahedron, dodecahedron, icosahedron: yes. Every corner has an opposite corner.
- Regular tetrahedron: no. The opposite of a corner is the middle of the opposite face, which is not a corner.
A solid with a centre has every face parallel to an opposite face. The tetrahedron has no pair of parallel faces.
The five solids side by side
| Solid | Planes | Axes (number × order) | Centre | Turns |
|---|---|---|---|---|
| Tetrahedron | 6 | 4×3, 3×2 | no | 12 |
| Cube | 9 | 3×4, 4×3, 6×2 | yes | 24 |
| Octahedron | 9 | 3×4, 4×3, 6×2 | yes | 24 |
| Dodecahedron | 15 | 6×5, 10×3, 15×2 | yes | 60 |
| Icosahedron | 15 | 6×5, 10×3, 15×2 | yes | 60 |
Why do cube and octahedron match? They are dual: join the centres of the faces of a cube and you get an octahedron, and the other way round. A dual pair shares the same symmetry. The dodecahedron and the icosahedron are another dual pair. The tetrahedron is dual to itself.
Try it. In the 3D, show the planes of the icosahedron, then switch to the dodecahedron. The planes do not move. Next, count the three kinds of axes on the cube and check the numbers in the table.
Key formulas and definitions
- n-fold axis: smallest turn = 360° ÷ n
- Tetrahedron: 6 planes; axes 4×3, 3×2; no centre; 12 turns
- Cube / octahedron: 9 planes; axes 3×4, 4×3, 6×2; centre; 24 turns
- Dodecahedron / icosahedron: 15 planes; axes 6×5, 10×3, 15×2; centre; 60 turns
- Turns that fit = 1 + sum over axes of (n − 1)
- All symmetries (turns and mirror moves) = 2 × number of turns
Worked examples
1. Count the planes of symmetry of a cube.
Three planes are parallel to a pair of faces and pass through the middle. Six more planes each contain two opposite edges (a diagonal slice). 3 + 6 = 9.
2. List the rotation axes of a cube and the smallest turn for each.
3 axes through opposite face centres: 90°. 4 axes through opposite corners: 120°. 6 axes through the middles of opposite edges: 180°. That is 13 axes in all.
3. Does a regular tetrahedron have a centre of symmetry? Explain.
No. Take a corner P. The point at the same distance on the other side of the middle is on the opposite face, not on a corner. So the solid does not map onto itself under that reflection.
4. How many different turns (including doing nothing) fit a regular tetrahedron?
Four 3-fold axes give 2 turns each (120° and 240°): 8. Three 2-fold axes give 1 turn each (180°): 3. Doing nothing: 1. Total 1 + 8 + 3 = 12.
5. A square pyramid has a square base and four equal triangular faces. Find its planes, axes and centre.
Planes: 2 through opposite corners of the base and the apex, and 2 through the middles of opposite base sides and the apex, so 4. Axes: one 4-fold axis through the apex and the middle of the base. Centre: none, as the apex has no opposite point.
6. How many mirror planes does an icosahedron have, and how are they linked to its 2-fold axes?
15 planes. Each plane is perpendicular to exactly one of the 15 two-fold axes (a plane through the centre and cutting that axis at 90°).
Common mistakes
- Saying a cube has 6 planes of symmetry (only the ones parallel to faces). The 6 diagonal planes through opposite edges count too: total 9.
- Counting each axis twice, once for each end. An axis is a whole line through both ends.
- Thinking a rotation of 45° is a symmetry of a cube. After 45° about a face axis the corners do not land on corners.
- Believing every regular solid has a centre of symmetry. The tetrahedron does not.