Projection is the union of 8 concentric projections of the 30-vertex orthoplex (also called Using Geometer's Sketchpad to draw it is fairly simple. This is a reflection of the fact that the Coxeter element hĭespite the apparent complexity of this object, the process of The sketch has the same symmetry as the regular 30-sided polygon, or One feature of this projection is clear, specifically that the figure in This sheds some light on how to obtain the projection: Simply line upĪll the orbits in one plane along their common center of symmetry. Thus, every orbit under the subgroup H acting on Gosset's figure isĮither a point or a figure with the symmetry of a regular polygon. Since theĪction of the Coxeter group on R^8 is real, so is the action of Generated by h is isomorphic to the 30-element cyclic group. The sketch in the applet represents an orthogonal projection withĪ Coxeter element h in the Coxeter group for E(8). While there are many different ways to "view" Gosset's figure, Gosset's 8-dimensional figure is merely the convex hull of these For any three integers i < j < k lying inĪll these points taken together with their negatives comprise a set of 240 Let e(i) denote the ith element of the standard basis for R^8,īe half of the sum of all of these. To understand this object is by becoming familiar with the root system of the exceptional One canĬhange the scale and the location of the center to get (slightly) different views Of Gosset's semi-regular polytope in 8 dimensions. This JavaSketchpad applet is a depiction of a projection Sorry, this page requires a Java-compatible web browser.
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