Symmetric four-dimensional polytope and visualization method in four, eight, and sixteen dimensions using Hopf maps

Area of Science:

• Geometry
• Computational Mathematics
• Theoretical Physics

Background:

• Classical three-dimensional electrostatics provides a foundation for exploring higher-dimensional geometric structures.
• Optimization methods in physics can inspire novel mathematical discoveries.

Purpose of the Study:

• To discover and characterize a new four-dimensional polytope.
• To analyze the symmetry properties of this novel polytope.
• To develop a geometric method for visualizing high-dimensional configurations.

Main Methods:

• Utilizing optimization methods derived from classical three-dimensional electrostatics.
• Applying geometric principles to analyze vertex neighbor distances and symmetry.
• Developing a visualization technique for four, eight, and sixteen-dimensional spaces.

Main Results:

• Identification of a previously unknown four-dimensional polytope with 80 vertices.
• Detailed analysis of vertex symmetry: 64 vertices have 12 nearest neighbors with 4 distinct distances, while 16 vertices have 10 nearest neighbors with 2 distinct distances.
• Development of a simple geometric method for visualizing this polytope and related high-dimensional configurations.

Conclusions:

• The discovered four-dimensional polytope exhibits significant, quantifiable symmetry.
• The geometric visualization method provides an accessible way to understand complex high-dimensional structures.
• This work bridges concepts from electrostatics optimization and pure geometry, opening avenues for further research in high-dimensional spaces.