Symmetry Groups of Polyhedra

Part of what makes polyhedral origami models beautiful is that they are very symmetric objects. You may have observed that some models have prominent features like pentagon-based symmetry, while others do not. There are many ways to orient these symmetric objects and have them look exactly the same (e.g. any 90° rotation of the cube about an axis through opposite face centers will leave the cube unchanged.) On this page, we will classify the types of symmetry that a polyhedron can have by linking its rotations with symmetry groups.

Note: We will only consider rotations (no reflections or inversions) because each origami model and its mirror image are often different objects.

Cyclic Symmetry

This is the simplest type of rotational symmetry. The only rotations that leave the shape unchanged are those about a single axis, by multiples of \(360^{\circ}/n\).

For example, generic right regular pyramids possess this type of symmetry.

The abstract structure can be represented using the cyclic group \(C_n\). The single \(360^{\circ}/n\) rotation \(r\) repeated \(n\) times results in the initial position. The \(n\) rotations \(r,r^2,\ldots,r^n=e\) (the identity) are all the symmetries.

Dihedral Symmetry

This is the next simplest type of symmetry. For \(n\geq 3\), the shape has one \(n\)-fold rotational axis, as well as \(n\) \(2\)-fold axes (think of flipping the object upside down).

For example, generic right regular prisms and antiprisms possess this type of symmetry.

The abstract structure is the dihedral group \(D_n\) with order \(2n\). It is generated by the cyclic group's rotation \(r\) and a flip \(s\) that turns the object upside down. They satisfy \(r^n=s^2=e\). Notice how rotating then flipping the object is equivalent to flipping then rotating in the reverse direction. (In other words, \(rs=sr^{-1}\).)

Tetrahedral Symmetry

This type of symmetry (often written \(T\)) is most clearly exhibited by the regular tetrahedron. There are four \(3\)-fold rotational axes (each passing through a corner and the opposite face center) and three \(2\)-fold rotational axes (one through each pair of opposite edge midpoints).

The symmetry group is isomorphic to the alternating group \(A_4\) with order \(12\). This is because there is a 1-to-1 correspondence between the possible rotations of the tetrahedron and the even permutations of \((1,2,3,4)\): \(3\)-cycles (e.g. \((1\,2\,3)\)) correspond to corner-centered rotations, and double transpositions (e.g. \((1\,2)(3\,4)\)) correspond to edge-centered rotations. You can verify this by labeling the vertices \(1,2,3,4\) and seeing where they go when you rotate the tetrahedron.

Octahedral Symmetry

The cube and its dual, the octahedron, have this type of symmetry (often written \(O\)). On the cube, there are three \(4\)-fold rotational axes (one through each pair of opposite face centers), four \(3\)-fold rotational axes (one through each pair of opposite corners), and six \(2\)-fold rotational axes (one through each pair of opposite edge midpoints).

The symmetry group is isomorphic to the symmetric group \(S_4\) with order \(24\). Each possible rotation of the cube corresponds to a permutation of \((1,2,3,4)\), where \(1,2,3,4\) label the 4 space diagonals of the cube: \(4\)-cycles (e.g. \((1\,2\,3\,4)\)) correspond to \(90^{\circ}\) face rotations, \(3\)-cycles (e.g. \((1\,2\,3)\)) correspond to corner rotations, swaps (e.g. \((1\,2)\)) correspond to edge rotations, and double transpositions (e.g. \((1\,2)(3\,4)\)) correspond to 180° face rotations. You can verify that each of the \(24\) permutations of these 4 space diagonals corresponds to a unique orientation of the cube. (The idea is that on each face, the 4 axes form a unique permutation of \((1,2,3,4)\) up to cyclic shifts. Since the orientation of the cube is uniquely determined by the top face and how the top face is rotated, the bijection follows.)

Note that \(A_4\) is a subgroup of \(S_4\). Geometrically, the set of symmetries of a regular tetrahedron is a subset of the set of symmetries of a cube. This is because a regular tetrahedron can be inscribed in a cube using 4 of the cube's vertices, and each rotation of this inscribed tetrahedron leaves the cube unchanged as well. Also, \(A_4\) has index \(2\) in \(S_4\), so the other \(12\) cube rotations swap the two inscribed tetrahedra. This is displayed in my Icecube origami model.

Icosahedral Symmetry

The dodecahedron and its dual, the icosahedron, have this kind of symmetry (often written \(I\)). On the dodecahedron, there are six \(5\)-fold rotational axes (one through each pair of opposite face centers), ten \(3\)-fold rotational axes (one through each pair of opposite corners), and fifteen \(2\)-fold rotational axes (one through each pair of opposite edge midpoints).

The symmetry group is isomorphic to the alternating group \(A_5\) with order \(60\). Each possible rotation of the dodecahedron corresponds to an even permutation of \((1,2,3,4,5)\): \(5\)-cycles (e.g. \((1\,2\,3\,4\,5)\)) correspond to face rotations, \(3\)-cycles (e.g. \((1\,2\,3)\)) correspond to corner rotations, and double transpositions (e.g. \((1\,2)(3\,4)\)) correspond to edge rotations. The objects corresponding to \(1,2,3,4,5\) on the dodecahedron form a well-known origami model—the 5 intersecting tetrahedra! You can verify that each of the \(60\) even permutations of these 5 tetrahedra corresponds to a unique orientation of the dodecahedron.

Note that \(A_4\) is a subgroup of \(A_5\), which can be demonstrated geometrically by inscribing a regular tetrahedron with 4 vertices of the dodecahedron. Each rotation of this inscribed tetrahedron leaves the dodecahedron unchanged, too. In addition, \(A_4\) has index \(5\) in \(A_5\), so the other \(48\) dodecahedron rotations swap out this inscribed tetrahedron for one of the other 4.

However, \(S_4\) is not a subgroup of \(A_5\). Geometrically, although it is possible to inscribe a cube inside a dodecahedron, rotating the cube 90° will change how the dodecahedron looks because regular dodecahedra lack \(4\)-fold rotational axes. This corresponds to the algebraic fact that \(A_5\) has no element of order \(4\), but \(S_4\) does.

Further Ideas

It turns out that the rotation group of any polyhedron is one of these five types: two infinite families (\(C_n,D_n\)) and three exceptional groups (\(T,O,I\)). Although I haven't made any 4-dimensional origami models yet, it is interesting to consider the rotational symmetry groups of regular 4D polytopes. For example, try finding the symmetry groups of the 4-simplex (4D equivalent of the 3D tetrahedron) and the hypercube (4D equivalent of the 3D cube).