**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 **R**^{4} to **R**^{3}. 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

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 H_{3} 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

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:

One obtains the binary icosahedral group simply by omitting the "= 1" in the relations:

To prove this, one can establish that the central element z=a

**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

where w, x, y, and z are real numbers. One can multiply quaternions by obeying all necessary distributive laws along with Hamilton's equation

Given a quaternion q=w+ix+jy+kz, there is also a conjugate quaternion

analogous to conjugation of complex numbers. This allows one to define the length of a quaternion because multiplying

yields a non-negative real number. The length of q is defined as

This length function is a sort of norm because it is preserved under quaternion multiplication,

One can show that the set of quaternions with unit length is a group. This group is conveniently denoted S

The binary icosahedral group arises as a particular finite subgroup of S^{3}. Let a
and b be the roots of the quadratic equation

chosen such that a < b. Notice that these are the constants which are used in the closed formula

for the nth Fibonacci number. Then the binary icosahedral group is the smallest subgroup of S

**The Coxeter group H _{4}.**

The vertices of the 600-cell serve as the root system for the
hecatonicosahedroidal group. The hecatonicosahedroidal group H_{4}, 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 H_{3} and H_{4}. Note that the Coxeter group H_{3} is a subgroup of H_{4}.

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

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 H

**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 E_{8}.

+---+------+-----+-----+-----+------+-----+-----+-----+ | 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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