# Math 22a Fall 2018

## 22a Linear Algebra and Vector Analysis

# Three dimensional Torus

This video visualizes the three dimensional torus T^{3}in R

^{4}. The manifold is parametrized by three angles φ, θ, ψ. Each frame shows a stereographic projection into three dimensional space R

^{3}after rotating in four dimensional space (applying an orthogonal 4x4 matrix). Similarly as with the visualization of the three dimensional sphere, we draw a two parameter family of circles on the torus and encode one of the parameters with color. In the second movie below we attempt to draw the projected parametrized surfaces which have been made transparent. The parametrisation given appears in this document. The trick to see the 2D tori is to focus on one color and connect the dots.

**Direct Media Links:**Webm, Ogg Ipod ;

Background Music from Alan Walker and K-391, Julie Bergen and SeungRi: Ignite.