Recently, R. T. Worley succeeded in determining the Voronoi cells of the E_{6}^{*} and E_{7}^{*} lattices. These turned out to be a 6-dimensional polytope with 720 vertices, and a 7-dimensional polytope with 576 vertices, respectively. These two polytopes had been described in 1930 by H. S. M. Coxeter, who called them the 0_{221} and the 1_{23}. They belong to a large class of convex polytopes known as uniform polytopes. In this paper we will use Wythoff's construction and a new result concerning the lengths of edges of orthogonal trees to identify these polytopes. We will also discuss the symmetry groups of these two polytopes, which are of order 103,680 and 2,903,040 respectively. |
In the last five years, R. T. Worley [7,8] succeeded in determining the Voronoi cells of the E_{6}^{*} and E_{7}^{*} lattices. These turned out to be a 6-dimensional polytope with 720 vertices, and a 7-dimensional polytope with 576 vertices, respectively. Worley was primarily interested in determining exact values for the normalized second moments of these two polytopes. However, he failed to mention that these polytopes had been described as early as 1930 by H. S. M. Coxeter [2, pp. 414-417], who called them the 0_{221} and the 1_{23}. They belong to a large class of convex polytopes known as uniform polytopes, and they can both be derived by a procedure called Wythoff's construction.
In this paper we will independently identify these polytopes using Wythoff's construction, aided by a new formula for the lengths of the edges of a kind of simplex known as an orthogonal tree. The chief virtue of this new derivation is that it does not require the use of any coordinates whatsoever. We will also briefly discuss the symmetry groups of these two polytopes, which are of order 103,680 and 2,903,040 respectively.
A polytope is uniform if all its facets are uniform and it is ``vertex-regular,'' that is, if it can be rotated so that any vertex can be mapped onto any other vertex. To start the recursion off, a polygon is defined to be uniform if it is regular.
In 1918, W. A. Wythoff [9] found almost all the uniform polytopes in four dimensions by truncating the regular polytopes. His method was generalized by Coxeter for higher dimensions [3]. At the same time, Coxeter found a way to represent uniform polytopes as graphs with certain nodes circled. The uniform polytope represented by a diagram G is a cell of the uniform polytope or honeycomb represented by the diagram H if and only if G is a subdiagram of H. The rest of this section is essentially a summary of relevant parts of [3] and [4, chap. 11]; the reader is referred there for details.
Coxeter found every example of a simplex that tiles either Euclidean or spherical space by reflections. These simplices (called fundamental simplices) have the property that any two bounding hyperplanes meet each other at an angle which is a divisor of 180 degrees. An important example of a simplex that tiles 6-dimensional Euclidean space is shown in Figure 1.
Each node of this graph represents a vertex of the simplex. If two nodes share an edge then the two hyperplanes which are opposite the corresponding vertices meet at an angle of 60 degrees; otherwise the hyperplanes meet at an angle of 90 degrees. (Since the graph is a tree, this simplex is also called an orthogonal tree [5].) An easy geometric argument shows that each node can be labeled with an integer so that each node's label is half the sum of its neighbors' labels. A vertex represented by a node labeled 1 is called a special vertex. (The subscripts are purely for ease of reference.)
Each node can also be interpreted as a reflection in the corresponding bounding hyperplane. These seven reflections generate an infinite group, sort of a 6-dimensional kaleidoscope. The image of any special vertex, say 1_{a}, under the action of this reflection group is the lattice shown in Figure 2, which is defined to be the lattice E_{6}. (Hence Coxeter's name 2_{22}. In general, if a graph has three branches of length p, q, and r, then p_{qr} is obtained by circling the last node of the branch of length p, and 0_{pqr} is obtained by circling the center node.) The images of any one of the three special vertices 1_{a}, 1_{b}, or 1_{c} forms an E_{6} lattice. The lattice E_{6}^{*} consists of the images of all three special vertices, and is therefore the union of the three copies of E_{6}:
The E_{6}^{*} lattice is not uniform since it requires the union of more than one diagram to represent it, although we will soon see that its Voronoi cell is uniform.
To find the Voronoi cell of E_{6}^{*} we must find the point on
the simplex which is furthest from the three special vertices 1_{a},
1_{b}, 1_{c}. Formally, we want to find the
point P belonging to the simplex that maximizes
Proof: To find the lengths of the edges of the simplex, label each edge of its graph with the inverse of the product of the labels of that edge's two endpoints:
This new diagram is to be read as follows: to find the distance between two vertices of the simplex, take the square root of the sum of the numbers on the edges of the unique path between the two nodes representing the two vertices. (This is the new formula mentioned in the introduction. It was discovered by the author in 1990, and then generalized to arbitrary orthogonal trees with angles other than 60 and 90 degrees by Coxeter in [5].) For instance, the distance from 1_{a} to 1_{c} is . This means that the triangles {1_{a}, 2_{a}, 3}, {1_{b}, 2_{b}, 3}, and {1_{c}, 2_{c}, 3} are all congruent right triangles lying on three orthogonal planes.
So we see immediately that P = 3 is at least a local maximum for equation (1) since any infinitesimal adjustment would move P closer to some 1_{N}. But since the images of vertex 3 form the honeycomb 0_{222},
The proof of the next theorem is very similar.
Proof: The graph for the fundamental simplex for the E_{7} and E_{7}^{*} lattices is:
Again, we have labeled each node with a number in such a way that each node's label is half the sum of its neighbor's labels, and again we have labeled each edge with the inverse of the product of the labels of its endpoints. The rule for determining edge length is the same as in theorem 1. The E_{7}^{*} lattice consists of the images of the two special vertices 1_{x} and 1_{y}. We want to find the point P which maximizes min_{N=x,y}dist(P,1_{N}). This time, the desired point P is easily seen to be the vertex 2_{z}, the images of which form the honeycomb 1_{33}, which has only one kind of cell, the 1_{23} = 1_{32}. This completes the proof of theorem 2.
The rotation and symmetry groups of the 0_{221} and 1_{23} and the relationships between them are summarized in the following chart:
The groups in the diagram surrounded by thick boxes have a central subgroup of order 2 containing the central inversion (which is a rotation in even dimensions); those surrounded by thin boxes do not. Indeed, each group surrounded by a thick box is the direct product of the group to its lower left and the group of order two containing the central inversion. Groups are connected by single lines to normal subgroups, and by double lines to other subgroups.
The symmetry group of the 2_{21} was originally studied in the 19th century as the group of automorphisms of the 27 lines on the cubic surface, and the rotation group of the 1_{23} was originally known as the automorphism group of the 28 bitangents of a non-singular quartic curve. Two good modern references for the study of these polytopes, their groups, and their histories are [6, pp. 21-33] and [1, pp. 26, 46].
[2] H. S. M. Coxeter. The Polytopes with Regular-Prismatic Vertex Figures, I. Phil. Trans. Royal Soc. London (A), 229 (1930), 329-425.
[3] H. S. M. Coxeter. Wythoff's Construction for Uniform Polytopes. Proc. London Math. Soc. (2), 38 (1935), 327-339. Reprinted as Chapter 3 of Twelve Geometric Essays, Southern Illinois University Press, 1968.
[4] H. S. M. Coxeter. Regular Polytopes, 3rd ed., Dover, 1973.
[5] H. S. M. Coxeter. Regular and Semi-Regular Polytopes, III. Mathematische Zeitschrift, 200:1-45, 1988.
[6] H. S. M. Coxeter. Orthogonal Trees. Proc. of the 7th Annual Symposium on Computational Geometry, ACM Press, 1991, pp.89-97.
[7] R. T. Worley. The Voronoi Region of E_{6}^{*}. J. Austral. Math. Soc. (A), 43 (1987), 268-278.
[8] R. T. Worley. The Voronoi Region of E_{7}^{*}. SIAM J. Disc. Math., 1.1 (1988), 134-141.
[9] W. A. Wythoff. A Relation between the Polytopes of the C_{600} Family. Proc. Royal Acad. of Sci., Amsterdam, 20 (1918) 966-970.