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

TL;DR

This paper introduces a visualization technique for four-dimensional polytopes using parallel hyperplane slices arranged in an oval display, enabling intuitive understanding of 4D structures through 3D graphics and rotations.

Contribution

It presents a novel method to visualize 4D polytopes by slicing with parallel hyperplanes and displaying the slices in an oval arrangement, enhancing comprehension of 4D geometry.

Findings

Slices form an oval shape in the display window

Rotations in 4D space synchronously transform all slices

The method facilitates intuitive grasp of 4D polytopes

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.