Triality with Zometool


A "duality" in mathematics is a correspondence between two objects, exhibiting some kind of equivalence beween them. For instance, one says the cube and regular octahedron are dual because one can switch from one to the other by interchanging the words "face" and "vertex". The cube has 6 faces, with 3 meeting at every vertex, and the regular octahedron has 6 vertices, and there are 3 vertices on each face. The cube and regular octahedron, indeed, are similar because the have the same symmetry group. There is a duality in the projective plane because to each statement there is associated a dual statement where the words "point" and "line" are interchanged. In topology, there is a duality between open and closed subsets of a space; a set is open exactly when its complement is closed. In differential/algebraic geometry/topology, there is a duality between cycles and cocycles. The list goes on and on and on.

Likewise, a "triality" is a correspondence between three objects, showing that they are essentially equivalent. However, whereas dualities in mathematics are extremely common, trialities are harder to find. One of the more famous instances of triality occurs in the representation theory of the spin group Spin(8), the universal covering group of the special orthogonal group SO(8). The phenomenon of triality for Spin(8) is an essential consequence of the symmetry of its Coxeter graph D(4):

The Coxeter Graph D(4)

It is clear that the automorphism group of this graph is isomorphic to the 6-element group S(3) of permutations on three letters, in other words, the symmetry group of an equilateral triangle. The triality for Spin(8) works as follows: To each irreducible representation of Spin(8) is associated a particular labeling of the vertices of the Coxeter graph D(4). By permuting the branches of this graph one obtains as many as 5 other corresponding representations of Spin(8). These 6 representations, in general, are not isomorphic as Spin(8)-modules, but nevertheless share various critical properties.

Many Zome models are intimately related to the geometry of Hamilton's quaternions, and quaternions provide a useful paradigm for studying the phenomenon of "triality". It is therefore not a surprise to see the phenomenon of triality occur with some Zome models. Consider the following Zome models of the 16-cell:

Zome Projections of the 16-Cell

Upon completion of any of these three models, one should be able to find a total of 16 tetrahedra arranged so that each edge is shared by 4 tetrahedra and the vertex figure is a regular octahedron. Hence each is a model of a 16-cell.

As usual, each of these models represents the image of the projection

p : (w,x,y,z) |---> (x,y,z).
of the vertices and edges of a 16-cell. Notice that each of these models is constructed from 8 balls, 6 R1 struts, 6 R2 struts, 6 Y2 struts, and 6 B2 struts, and that the construction of each differs only slightly from the others. Each model has two B2 equilateral triangles, each of which uses three balls. For each model, the two remaining balls lie on an axis of trihedral symmetry. For each model, there is a unique strut type from {R1,R2,Y2} which does not emanate from this trihedral axis. Truncating the 16-cell to its edge midpoints yields the 24-cell, an arrangement of 24 regular octahedra in 4-dimensional space. The remarkable fact here is that truncating any one of these three models yields the same Zome model of the 24-cell.

Zome Projection of the 24-Cell

Notice again that this model has the same trihedral symmetry as the three projections of th 16-cell given earlier. (The black balls represent intersections between some of the projected edges, and do not represent vertices of the 24-cell.)

Based on this evidence, one may suspect there is a triality between these three different Zome models, and indeed this is the case. In fact, this triality is merely another manifestation of the triality provided by the D(4) Coxeter graph.

Quaternionic Definitions of Regular Polychora

The vertex sets for the three models of the 16-cell pictured above are imbedded in a particular way in the vertex set for the 600-cell, and, in discussing the 600-cell, one quickly encounters quaternions and the binary icosahedral group. First, let Q denote the usual 8-element group of quaternions generated by i and j. Next, define the "binary tetrahedral group" T as the smallest group of quaternions containing i and ½(1+i+j+k). Notice that T is the union of Q with the 16 quaternions which can be produced from

by any choice of signs in the four coordinates. Next, define the "binary icosahedral group" I to be the smallest group of quaternions containing i and ½(ai+j+bk), where b is the Golden Ratio and a is its field-theoretic conjugate. Then I contains T as a subgroup, and it also has the 96 quaternions which can be produced from
by applications of an arbitrary choice of signs and/or an even permutation of the four coordinates.

The 120 elements of I coincide with the vertices of a regular 600-cell in R^4. Two vertices are joined by an edge iff their dot product is b/2. Also, the 8 elements of Q coincide with the vertices of a regular 16-cell, and the 24 elements of T coincide with the vertices of a regular 24-cell. Call these three polytopes, as defined here, the "canonical" 600-cell, 16-cell, and 24-cell, respectively.

Although the preceding construction of the 24-cell may seem a bit contrived, the object fits quite naturally into the quaternion geometry of 4-dimensional space. Suppose q is any non-zero quaternion, and let

the set obtained by multiplying q by the usual basis for the quaternions. Then E is automatically an orthogonal basis for R^4, and one can immediately say that the 16-cell is the convex polytope whose vertices lie in E or the negatives -E. Call this 16 cell "induced by q on the right". One can similarly obtain the left-induced 16-cell by first multiplying the quaternion basis by q on the left, etc. The 24 midpoints ½(±e(i)±e(j)) of the left-induced 16-cell coincide with the vertices of a 24-cell. Correspondingly, there is now a 24-cell "left-induced" by q and a 24-cell "right-induced" by q.

The geometry of quaternions serve, again quite naturally, to illustrate the self-duality of the 24-cell. Considering the vertices as quaternions, one may obtain the centers of the 24 octahedra, coinciding with the vertices of the dual polytope, by multiplying by the quaternion ½(1+i). Multiplying by this quaternion again, one obtains the original 24 vectors, although they are now scaled down by a factor of ½. As the careful reader may have noticed by now, the particular scale of the quaternions representing vertices does not play a great role in determining whether one has a 24-cell; the only thing that matters is their relative placement in 4-dimensional space. Thus, multiplication by 1+i represents a sort of "projective dualization" transformation, sending a 24-cell to its dual.

One may have realized by now that the groups Q, T, and I serve to provide some convenient notation to designate vertex sets for many polychora. For example, if q is a non-zero quaternion, then 2Qq is the vertex set for the 16-cell left-induced by q. Similarly, qT is the vertex set for a 24-cell, and q(1+i)T is the vertex set for the dual 24-cell. Indeed, since these (convex) polychora are determined uniquely by their vertex sets, one may use this notation to designate the polychora themselves. Thus, for example, one may say qT is "the" 24-cell left-induced by q

Exploration of the Zome Models

In order to see the triality, one studies the vertex set 2(1+i)I, obtained by multiplying each element of I by 2(1+i). This transformation dilates the vectors in I by a factor of sqrt(8). (The factor 2 is used to clear a lot of the fractions.) Moreover, quaternion multiplication on the left affects an orthogonal transformation which is a composition of two commuting rotations. If one studies the elements of 2(1+i)I, one notices that it has 4 orbits of the vectors

(-bb,a,a,a), (32 elements),
(a-b,1,1,1), (32 elements),
(aa,b,b,b), (32 elements),
(2,2,0,0), (24 elements),
under the group generated by even numbers of sign changes and arbitrary permutations of the four coordinates. (Here bb and aa respectively denote the squares of b and a.) The first of these orbits is another image of the 24-cell; in fact, these 24 vectors coincide with the centers of the 24 octahedra of the 24-cell represented by T, thus exbihiting its self-duality.

One can see the triality by examining the first three orbits described above. Denote

[a]=(-bb,a,a,a)Q, [1]=(a-b,1,1,1)Q, and [b]=(aa,b,b,b)Q,
the 16-cells left-induced by (-bb,a,a,a), (a-b,1,1,1), and (aa,b,b,b). Projecting down to 3-space using the mapping p, one obtains the three Zome models pictured above. This is where one should scrutinize the models very carefully. Only one of these three models has a pair of balls where only R1's and R2's meet. This model is the projection of [1]. The model with R2's and Y2's emanating from the trihedral axis is the projection of [b], and the remaining model is the projection of [a]. Evidently, p[b] has the vertex (b,b,b), p[1] has the vertex (1,1,1), and p[a] has the vertex (a,a,a).

Next, one can check that the union of the three vertex sets of [a], [1], and [b] coincide with the vertices of a 24-cell. Indeed, one has

Since T is the union of Q with ½(1,1,1,1)Q and ½(1,-1,-1,-1)Q, the union of [a], [1], and [b] yields a 24-cell. In fact, one can check this with Zome, for truncating any one of these polychora to their edge midpoints yields the same projection of the 24-cell.

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