A toolkit to describe and interactively display three-manifolds embedded in four-space

TL;DR

This paper introduces a data structure and toolkit for describing, manipulating, and interactively visualizing three-manifolds embedded in four-dimensional space using OpenGL, enabling exploration of complex topological structures.

Contribution

It presents a novel data structure and algorithm for intersecting 4-space manifolds with 3-flats, supporting interactive visualization and analysis of 3-manifolds in 4D.

Findings

Enables interactive exploration of 3-manifolds in 4-space.

Supports extension to higher Euclidean dimensions.

Provides a practical tool for visualizing complex topological structures.

Abstract

A data structure and toolkit are presented here that allow for the description and manipulation of mathematical models of three-manifolds and their interactive display from multiple viewpoints via the OpenGL 3D graphics package. The data structure and vector math package can be extended to support an arbitrary number of Euclidean spatial dimensions. A model in 4-space is described by its bounding pure simplicial 3-complex. By intersecting a 3-flat with this 3-manifold, the algorithm will extract the requested closed pure simplicial 2-complex surface enclosing the desired 3D slice. The user can interactively rotate, pan, zoom, and shade arbitrary 3D solid or wire-frame views of the revealed 3D object created by intersection, thus exploring both expected and unexpected symmetries or asymmetries in the world of 3-manifolds in 4-space.