MASS COLLOQUIUM: ABOUT THE NUMBERS 12 AND 24NOTES OF A TALK GIVEN BY ROGER HOWE

LIUQUAN WANG AND ZHEYI XU

Abstract. The numbers 12 and 24 come up often in contexts that involve symmetry. Forexample, a cube has 12 edges, and a dodecahedron has 12 faces. This talk will discuss theextent to which various appearances of 12 and 24 should be considered the same. We willargue that some appearances should definitely be considered the same, and speculate aboutothers.

1. Relations of five 12s in features of polyhedrons

Have a look at the polyhedrons in the following figure.

Figure 1. Tetrahedron, Cube/Hexahedron, Octahedron, Dodecahedron and Icosahedron

These are the Platonic solids, i.e. regular, convex polyhedrons with congruent faces ofregular polygons and the same number of faces meeting at each vertex [1]. The features ofthe Platonic solids are shown in Table 1.

In this table, there are five 12s. Is this just a coincidence or are they related? We willillustrate some relations between them using dualities.

Date: October 10, 2013. The talk was given by Professor Roger Howe from Yale University as a colloquiumfor MASS Program of Pennsylvania State University. We thank Professor Howe for providing the slides forinformation and images used in this paper.

1

2 LIUQUAN WANG AND ZHEYI XU

Name: Tetrahedron Cube Octahedron Dodecahedron IsocahedronFaces: 4(triangles) 6(squares) 8(triangles) 12(pentagons) 20(triangles)Edges: 6 12 12 30 30

Vertices: 4 8 6 20 12Symmetries: 24 48 48 120 120Orientation 12 24 24 60 60-preserving:

Table 1. Features of the Platonic solids

1.1. Cube-Octahedron Duality. Let us embed a cube into an octahedron as shown inFigure 2(a).

Figure 2. (a) Embedding a cube into an octahedron; (b) Embedding anoctahedron into a cube

We can see that there’s a one-to-one correspondence between

(1) the faces of the cube and the vertices of the octahedron;(2) the edges of the cube and the edges of the octahedron;(3) the vertices of the cube and the faces of the octahedron.

This means that the cube is in a sense dual to the octahedron and shows why the numberof edges of the cube and the the number of edges of the octahedron are both 12. Also, ifwe embed the octahedron into the cube as shown in Figure 2(b), the same correspondencearises.

1.2. Dodecahedron-Icosahedron Duality. We can embed a dodecahedron into an icosa-hedron in a natural way as shown in Figure 3.

Figure 3. Embedding a dodecahedron into an icosahedron

MASS COLLOQUIUM: ABOUT THE NUMBERS 12 AND 24 3

It’s obvious that a one-to-one correspondence arises between

(1) the faces of the dodecahedron and the vertices of the icosahedron;(2) the edges of the dodecahedron and the edges of the icosahedron;(3) the vertices of the dodecahedron and the faces of the icosahedron.

This is another natural duality between polyhedrons, which explains how the 12 vertices ofthe icosahedron are related to the 12 faces of the dodecahedron.

There’s an interesting artwork involving a dodecahedron. The Dutch artist M.C.Eschermade a mixed media work named Gravitation in 1952 [2], see Figure 4.

Figure 4. Gravitation by M. C. Escher

It depicts a nonconvex regular polyhedron known as the smallest stellated dodecahedron.It’s made up by extending each face of a dodecahedron to form a stellated dodecahedron,and then leaving a trapezoidal doorway. There are twelve turtles without shells in the star-shaped house. The turtles are in six colored pairs (red, orange, yellow, magenta, green andindigo) with each turtle directly opposite its counterpart.

If we regard the turtles with different colors as the same, then exactly one fifth of symme-tries (resp. orientation-preserving symmetries) of the dodecahedron are also the symmetries(resp. orientation-preserving symmetries) of the stellated dodecahedron.

1.3. The number 12 in the cube, tetrahedron and dodecahedron. There are alsosome dualities between a cube and a tetrahedron.

Figure 5. Embedding a tetrahedron into a cube

4 LIUQUAN WANG AND ZHEYI XU

Once we embed a tetrahedron into a cube as shown Figure 5, we can see that there aresome correspondences between them. Each vertex of the tetrahedron is matched with aface of the cube, and also with the four edges in that face. Each face of the tetrahedronis matched with three edges of the cube in one corner. But these are not what we reallyinterested in. We shall explain how the 12 orientation-preserving symmetries of tetrahedronappear.

From our embedding, we easily found that every symmetry of the tetrahedron comes froma symmetry of the cube. Since there are two choice of position of the tetrahedron aftersymmetry (resp. orientation-preserving symmetry) of the cube, we know the number ofsymmetries (resp. orientation-preserving symmetries) of the tetrahedron is exactly half of thenumber of symmetries (resp. orientation-preserving symmetries) of the cube. Because thereare 24 orientation-preserving symmetries of the cube, there are 12 orientation-preservingsymmetries (i.e. rotations) of the tetrahedron. They permute the edges of the cube simplytransitively.

We can also give an explanation of the relation between the 12 edges of a cube and the12 faces of a dodecahedron. In fact, this comes from the construction of a dodecahedronstarting with a cube. We describe the process here.

(1) Take an edge of the cube.(2) Choose a plane containing the edge, and not cutting the cube.(3) The cube is contained in the half-space on one side of the plane.(4) Rotate space by all rotations of the tetrahedron.(5) The intersection of the 12 half-spaces is a dodecahedron with pentagonal sides.

Remark 1.1. For exactly one position of the original plane, the dodecahedron will be regular.In the extreme case, when the planes are equally inclined to both adjacent sides, the

pentagons degenerate to rhombuses. The result is the rhombic dodecahedron, see Figure 6.

Figure 6. Rhombic Dodecahedron

2. occurrence of the numbers 12 and 24 in the theory of finite simplegroups.

In this section, we will point out some occurrences of the numbers 12 and 24 in the theoryof finite simple groups. For more information on groups, readers can refer to the book [3].

2.1. Sporadic Groups. Let’s recall some definitions first.

Definition 2.1. A simple group is a group G that does not have any normal subgroupsexcept for the subgroup consisting only of the identity element, and G itself.

MASS COLLOQUIUM: ABOUT THE NUMBERS 12 AND 24 5

When mathematicians tried to classify all simple groups, they found 18 countable infinitefamilies and 26 exceptional groups that do not belong to any family. Those groups arecalled sporadic groups. And Mathieu groups M11,M12,M22,M23,M24, multiply transitivepermutation groups on 11, 12, 22, 23 or 24 objects, were the first sporadic simple groupsdiscovered.

Definition 2.2. A permutation group G acting on n points is k−transitive if, given two setsof points a1, a2, · · · , ak and b1, b2, · · · , bk with the property that all the ai are distinct and allthe bi are distinct, there is a group element g in G which maps ai to bi for each 1 ≤ i ≤ k.

The data in Table 2 shows the orders of Mathieu groups.

Group Order Order(product) Factorized orderM24 244823040 3 · 16 · 20 · 21 · 22 · 23 · 24 210 · 33 · 5 · 7 · 11 · 23M23 10200960 3 · 16 · 20 · 21 · 22 · 23 27 · 32 · 5 · 7 · 11 · 23M22 443520 3 · 16 · 20 · 21 · 22 27 · 32 · 5 · 7 · 11M12 95040 8 · 9 · 10 · 11 · 12 26 · 33 · 5 · 11M11 7920 8 · 9 · 10 · 11 24 · 32 · 5 · 11

Table 2. Features of Mathieu Groups

We mention some interesting properties here.

Proposition 2.3. (Mathieu Groups)(1) M22 ⊂M23 ⊂M24;(2) M11 ⊂M12 ;(3) M24 is 5-transitive on set of 12 letters;(4) M24 is the symmetry group of Golay code, which is a code using bit sequence of length24.

In order to understand the idea of code better, we need to have the basic idea of lattices.

Definition 2.4. A lattice in Rn is a discrete subgroup of Rn which spans the real vectorspace Rn.

A useful fact is that every lattice in Rn can be generated from a basis for the vector spaceby forming all linear combinations with integer coefficients.

The lattice in R2 can be represented as the set L = {ma +nb : m,n ∈ Z}. Some differenttypes of lattice in R2, i.e. planar lattice, are shown in Figure 7 and Figure 8.

We focus on some lattices which have special properties.

Definition 2.5. An even unimodular lattice is a lattice s.t. all vectors have even squarelength.

Such lattices exist only in 8d-dimensional spaces, where d is a integer. When the dimensionis 8, the lattice is called E8 lattice. When the dimension is 24, it is called Leech Lattice.

Proposition 2.6. (Leech lattice)(1) It’s a 24-dimensional even unimodular lattice;(2) Its minimal square length is 4;(3) The symmetries of Leech lattice, up to ±1, form the group Co1,(the largest of Conwaygroups, which contains Mathieu groups).

6 LIUQUAN WANG AND ZHEYI XU

Figure 7. Rectangular lattice (a · b = 0) and Square lattice (a · b = 0 and|a| = |b|)

Figure 8. Fat Rhombic lattice (|a| = |b|), Hexagonal lattice (|a| = |b|,∠(a,b) =60

◦) and Thin Rhombic lattice (|a− b| = |a|)

2.2. Codes. Now we discuss an application of these mathematical ideas.

Definition 2.7. A binary linear code based on X is a set of subsets of X that is closedunder the operation of addition. The addition is defined as A + B = (A − B) ∪ (B − A),which is called the symmetric difference of A and B.

The addition on the vertices of a cube is shown by the following figures.

Figure 9. Addition of the vertices on a cube

The extended Hamming code is a kind of error-correcting code based on 8-bit words. So,we can view Hamming code as a group on some of the 8 vertices of a cube, using the additionabove.

MASS COLLOQUIUM: ABOUT THE NUMBERS 12 AND 24 7

A Hamming code contains 16 elements (words), and all 14 code words except for the emptyset and the whole set have size 4. There are three types of Hamming codes depending onthe different selection of the 14 words.

Figure 10. Three types of Hamming codes

We can figure out that E8 is constructed from Hamming Code (see Section 1.3 of [3]):since all code word lengths are multiples of 4, the lattice is even, and since Hamming Codehas dimension 4, the lattice is self-dual.

The same method can also be used in construction of Leech Lattice from extended Golaycode (see Section 2.8 of [3]). It is the unique 12 dimensional code with minimal length 8 andit has 759 code words of length 8, which are also called octads.

References

[1] Platonic solid, http://en.wikipedia.org/wiki/Platonic_solid[2] Gravitation (M. C. Escher), http://en.wikipedia.org/wiki/Gravitation_(M._C._Escher)[3] Lattices and codes, a course partially based on Lectures by F.Hirzebruch. Wolfgang Ebeling, Braun-

schweig; Wiesbaden: Vieweg, 1994