A Visualization Method of Four Dimensional Polytopes by Oval Display of Parallel Hyperplane Slices

Abstract: A method to visualize polytopes in a four dimensional euclidian space $(x,y,z,w)$ is proposed. A polytope is sliced by multiple hyperplanes that are parallel each other and separated by uniform intervals. Since the hyperplanes are perpendicular to the $w$ axis, the resulting multiple slices appear in the three-dimensional $(x,y,z)$ space and they are shown by the standard computer graphics. The polytope is rotated extrinsically in the four dimensional space by means of a simple input method based on keyboard typings. The multiple slices are placed on a parabola curve in the three-dimensional world coordinates. The slices in a view window form an oval appearance. Both the simple and the double rotations in the four dimensional space are applied to the polytope. All slices synchronously change their shapes when a rotation is applied to the polytope. The compact display in the oval of many slices with the help of quick rotations facilitate a grasp of the four dimensional configuration of the polytope.