Gosset's Figure in 8 Dimensions

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This JavaSketchpad applet is a depiction of a projection of Gosset's semi-regular polytope in 8 dimensions. One can change the scale and the location of the center to get (slightly) different views by moving the highlighted points.

Structure.

The simplest way to begin to understand this object is by becoming familiar with the root system of the exceptional Lie algebra E(8). Here is one presentation of these roots: Let e(i) denote the ith element of the standard basis for R^8, and let
      r=(1/2)[e(1)+e(2)+...+e(8)]
be half of the sum of all of these. For any three integers i < j < k lying in {1,2,3,4,5,6,7,8}, denote
          a(i) = r-e(i),
        b(i,j) = e(i)+e(j),
      c(i,j,k) = r-[e(i)+e(j)+e(k)],
        d(i,j) = e(i)-e(j).
All these points taken together with their negatives comprise a set of 240 points in R^8. Gosset's 8-dimensional figure is merely the convex hull of these 240 points.

While there are many different ways to "view" Gosset's figure, the sketch in the applet represents an orthogonal projection with some interesting properties. Choose a Coxeter element h in the Coxeter group for E(8). Then the subgroup H generated by h is isomorphic to the 30-element cyclic group. Since the action of the Coxeter group on R^8 is real, so is the action of H. Thus, every orbit under the subgroup H acting on Gosset's figure is either a point or a figure with the symmetry of a regular polygon. This sheds some light on how to obtain the projection: Simply line up all the orbits in one plane along their common center of symmetry. One feature of this projection is clear, specifically that the figure in the sketch has the same symmetry as the regular 30-sided polygon, or the "triacontagon". This is a reflection of the fact that the Coxeter element h has order 30.

Rendering the Gosset Figure.

Despite the apparent complexity of this object, the process of using Geometer's Sketchpad to draw it is fairly simple. This particular projection is the union of 8 concentric projections of the 30-vertex orthoplex (also called the "cross-polytope"). Thus, in building this with Geometer's Sketchpad, one must first construct a tool which draws a projection of an orthoplex with a given center and radius.

Although the 8 projected orthoplexes are centered at the same point, difficulties arise when determining the 8 different radii and the angles by which each is offset. The following applet is a guide to locating these radii:

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In the figure, the outer ring of vertices are those of a regular triacontagon. The inner three rings are obtained by placing an equilateral triangle and two regular pentagons, as they appear in the figure. Orbiting these under the action of the 30-element cyclic group on the plane, this yields four of the rings appearing in the projection of Gosset's figure. (As it turns out, the 120 vertices in this set of four rings coincide with those of a particular projection of the regular 600-cell.) The remaining four sets of vertices are obtained by dilating the first four by the golden ratio b=(1+sqrt(5))/2. With 8 concentric rings and 30 vertices in each ring, this yields all 240 vertices of Gosset's figure.


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