Projections of the 600-cell

Click to enlarge:

Introduction.

The 600-cell, as one may discover, is a singular example of a regular polytope. As its name hints, this is a particular arrangement of 600 cells, each congruent to a regular tetrahedron. The Schläfli symbol for the 600-cell is {3,3,5}. The first two 3's in this symbol designate that the cells are tetrahedra {3,3}, and the 5 designates that exactly 5 tetrahedra surround each of the 720 edges. Whatever your definition of regular, one should notice that this figure is regular in the extreme, for its symmetry group has only 5 orbits, the vertices, the edges, the triangular faces, the cells, and the model itself.

It is impossible to make such an arrangement of 600 regular tetrahedra in 3-dimensional space, but 4-dimensions suffice. For this reason, among others, it is an example of a regular "polychoron". The model pictured above should be considered as the image of an orthogonal projection from R4 to R3. Of course, you are not looking at the model, but instead at a few two-dimensional renditions of the model. The usual way of taking a photograph, however, is not an orthogonal or even a linear projection. Perhaps, instead, the best way to describe it is as a "perspective" projection, or projection "to a point". In any case, what you see before you represent projections of a projection.

As it turns out, this particular model also represents some non-convex regular star polychora. The reason is that this model only shows the edges and vertices of the 600-cell, not the faces or the cells. These 720 edges and 120 vertices coincide with those of 3 other regular polychora. This phenomenon may be more familiar in the 3-dimensional case, where, for example, the edges and vertices of the regular icosahedron {3,5} coincide with those of the great dodecahedron {5,5/2}. The Schläfli symbols for all of these polychora are

{3,3,5}, {5,5/2,5}, {3,5,5/2}, and {5,3,5/2}.

The model was constructed using drinking straws, chenile stems (a.k.a. pipe-cleaners), and model glue.

The Binary Icosahedral Group.

The vertices of the 600-cell, together with an operation defined below, constitute a group isomorphic to the binary icosahedral group.

The icosahedral group H3 is the set of all rigid motions preserving an icosahedron. It is isomorphic to the direct product of the alternating group A(5) and the two-element group C(2). The subgroup isomorphic to A(5) acts as a set of rotations, and is denoted by H(3,+). If the icosahedron is situated so that the center of the icosahedron and the origin coincide, then the subgroup H(3,+) occurs as a subgroup of the orthogonal group SO(3), the rotations in 3-space fixing the origin. Associated to the orthogonal group SO(3) is a simply connected group Spin(3) and a surjective homomorphism

p : Spin(3) ---> SO(3).

The binary icosahedral group is the preimage of H(3,+) lying inside Spin(3). Since the index of the covering map p is 2 and the order of the alternating group A(5)=H(3,+) is 60, the order of the binary icosahedral group is (2)(60)=120.

The binary icosahedral group has an elegant presentation in terms of generators and relations. This is closely related to a presentation for the rotation group of the icosahedron. First, the group of the icosahedron can be given as a variant of Hamilton's icosian calculus:

H(3,+) = <a,b,c | a2 = b3 = c5 = abc = 1>.

One obtains the binary icosahedral group simply by omitting the "= 1" in the relations:
I = < a,b,c | a2 = b3 = c5 = abc>.

To prove this, one can establish that the central element z=a2=b3=c5=abc must have order 2. Since there is a homomorphism from I onto H(3,+), the group I has order 120.

Hamilton's Quaternions.

Hamilton's quaternions are a generalization of the complex numbers. As a real vector space, the space of quaternions is isomorphic to 4-dimensional space R^4. Thus, every
quaternion is written

q = w + ix + jy + kz,

where w, x, y, and z are real numbers. One can multiply quaternions by obeying all necessary distributive laws along with Hamilton's equation
i2 = j2 = k2 = ijk = -1.

Given a quaternion q=w+ix+jy+kz, there is also a conjugate quaternion
conj(q) = w - (ix + jy + kz).,

analogous to conjugation of complex numbers. This allows one to define the length of a quaternion because multiplying
q conj(q) = w2 + x2 + y2 + z2

yields a non-negative real number. The length of q is defined as
|q| = sqrt[q conj(q)].

This length function is a sort of norm because it is preserved under quaternion multiplication,
|pq| = |p| |q|.

One can show that the set of quaternions with unit length is a group. This group is conveniently denoted S3 because it corresponds to the vectors in 4-space with unit length.

The binary icosahedral group arises as a particular finite subgroup of S3. Let a and b be the roots of the quadratic equation

x2 = x + 1.

chosen such that a < b. Notice that these are the constants which are used in the closed formula
F(n) = (bn-an)/(b-a)

for the nth Fibonacci number. Then the binary icosahedral group is the smallest subgroup of S3 containing both of the quaternions p=(b+i+ja)/2 and q=(b+j+ka)/2.

The Coxeter group H4.

The vertices of the 600-cell serve as the root system for the hecatonicosahedroidal group. The hecatonicosahedroidal group H4, is an unusual Coxeter group, to say the least. Coxeter groups are characterized as being generated by reflections in Euclidean space. The rank of a Coxeter group is an important invariant; it is merely the number of generating reflections of the group. The rank-2 Coxeter groups are the set of all dihedral groups, the groups acting on regular polygons in 2-dimensional space. This is an infinite family. In all other dimensions, there are only finitely many (spherical or affine) Coxeter groups. All but two of these higher-rank Coxeter groups are crystallographic (meaning they arise in the theory of Lie groups and Lie algebras). The only two Coxeter groups which fail to meet the crystallographic condition are H3 and H4. Note that the Coxeter group H3 is a subgroup of H4.

Suppose v is a unit vector in an inner product space (V,< , >). Then one may associate to v a reflection according to the recipe

Rv(u) = u - 2<u,v>v.

One checks that (a) Rv(v)=-v and (b) if u and v are perpendicular, then Rv(u)=u, establishing that Rv does indeed behave like a reflection. To any set {v(i)} of unit vectors is associated a group, namely the smallest group containing the reflections corresponding to the vectors v(i). The reflections corresponding to the elements of the binary icosahedral group, considered as unit vectors in 4-space, generate the hecatonicosahedroidal group H4.

The characters of the binary icosahedral group.

The characters are interesting. Let W be either of the two 2-dimensional characters appearing below. Then decree that the character U be "incident" to V if U is a component of the tensor product of V and W. One can quickly check that this incidence relation is symmetric, and, more significantly, that the incidence graph for these 9 characters coincides with the Coxeter graph for the extended root system for E8.

```+---+------+-----+-----+-----+------+-----+-----+-----+
| 1 |  12  |  20 |  12 |  30 |  12  |  20 |  12 |  1  | <--- Class Size
| 0 | 1/10 | 1/6 | 1/5 | 1/4 | 3/10 | 1/3 | 2/5 | 1/2 | <--- (theta)/(2pi)
+---+------+-----+-----+-----+------+-----+-----+-----+
| 1 |   1  |  1  |  1  |  1  |   1  |  1  |  1  |  1  |
| 3 |   a  |  0  |  b  | -1  |   b  |  0  |  a  |  3  |
| 3 |   b  |  0  |  a  | -1  |   a  |  0  |  b  |  3  |
| 4 |  -1  |  1  | -1  |  0  |  -1  |  1  | -1  |  4  |
| 5 |   0  | -1  |  0  |  1  |   0  | -1  |  0  |  5  |
+---+------+-----+-----+-----+------+-----+-----+-----+
| 2 |   b  |  1  | -a  |  0  |   a  | -1  | -b  | -2  |
| 2 |   a  |  1  | -b  |  0  |   b  | -1  | -a  | -2  |
| 4 |   1  | -1  | -1  |  0  |   1  |  1  | -1  | -4  |
| 6 |  -1  |  0  |  1  |  0  |  -1  |  0  |  1  | -6  |
+---+------+-----+-----+-----+------+-----+-----+-----+
```

Here, a+b=1 and ab=-1.

Also Known As....

There's more to say here. If I ever find the time, I will flesh out some of this. In the meantime, here's a little outline:

A.k.a. SL(2,GF(5)).

A. k. a. the only finite perfect group admitting a fixed-point-free representation.

A. k. a. the fundamental group of Poincare's homology sphere. Also, the only known finite fundamental group of such a homology sphere.

References.

Michael Artin.  Algebra.  Prentice Hall, Englewood Cliffs, New Jersey, 1991.

H. S. M. Coxeter.  Regular Polytopes.  3rd ed.  Dover Publications Inc., New York, 1973.

H. S. M. Coxeter and W. O. J. Moser.  Generators and Relations for Discrete Groups.  3rd ed.  Springer-Verlag, Berlin and New York, 1972.

P. Du Val.  Homographies, Quaternions and Rotations. Oxford University Press, 1964.

George K. Francis.  A Topological Picturebook.  Springer-Verlag, New York, 1987.

Larry Grove.  The Characters of the Hecatonicosahedroidal Group, J. für Reine und Angew. Math. 265, 1974, 160-169.

James E. Humphreys.  Reflection Groups and Coxeter Groups.  Cambridge University Press, Cambridge, 1990.

John Stillwell. Classical Topology and Combinatorial Group Theory.  Springer-Verlag, New York, 1980.

J. A. Wolf.  Spaces of Constant Curvature.  4th ed.  Publish or Perish, 1977.

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