To Get Overall Shapes and New Data of the 120-Cell and the 600-Cell

TL;DR

This paper develops 2D projective models of complex 4D polytopes, specifically the 120-Cell and 600-Cell, by deriving transformation matrices and calculating vertex coordinates to facilitate visualization and analysis.

Contribution

It introduces a method to display 2D projections of 4D regular polytopes using transformation matrices and detailed vertex data, advancing visualization techniques in higher-dimensional geometry.

Findings

Successfully displayed 2D models of 120-Cell and 600-Cell

Calculated all vertex coordinates and adjacency relationships

Provided a pattern for visualizing 4-variable problems

Abstract

This research will be helpful for people to display the 2-dimensiona projective models of 4-variable actual problems in many fields, in order to investigate deeply those actual problems. By using the theory of N-dimensional finite rotation group of the regular polytopes, the author established the 2-dimensional projective model of 4-dimensional rectangular coordinate system, and deduced a transformation matrix, and adopt it to display successfully the 2-dimensional overall shapes of two most complicated regular polytopes 120-Cell and 600-Cell. In the meantime, the author calculated all the vertex coordinates and determine the joint relationships between adjacent vertices of the regular polytopes 120-Cell and 600-Cell. Also, this provided a pattern for displaying the 2-dimensional projective model of 4-variable actual problem. (The tables of vertex coordinates of the 120-Cell and the 600-Cell and the table of joint relationship between adjacent vertices of the 120-Cell and the 600-Cell are listed below)