Seeing in four dimensions
By Julie Rehmeyer
Mathematicians create videos that help in visualizing four-dimensional objectsThree dimensions can be so limiting. Mathematicians, freed in their imaginations from physical constraints, can conjure up descriptions of objects in many more dimensions than that. Points in a plane can be described with pairs of numbers, and points in space can be described with triples. Why not quadruples, or quintuples, or more?
There is the minor difficulty that our nervous systems are only equipped to conjure images in three dimensions. But that doesn’t stop Étienne Ghys of the École Normale Supérieure in Lyon, France, from visualizing the four-dimensional dynamical systems he studies: “I live in dimension four,” he says. And you can too. Ghys has now created a series of videos teaching others to visualize four dimensions the way he does. His work is in collaboration with Jos Leys, a Belgian graphic artist and engineer, and Aurélien Alvarez, a mathematics graduate student at ENS Lyon.
The mathematician Ludwig Schläfli was one of the first to take higher-dimensional objects seriously, though he probably had little idea how to visualize them. In three dimensions, there are five regular solids, shapes for which every face is identical. In the mid-1800s, Schläfli figured out that in four dimensions, there are six regular solids, including one with 600 faces! Each of those 600 faces is itself a three-dimensional tetrahedron, and 20 of them meet at each vertex.
