A homotopy is a (smooth) map which continuously deforms a space nicely (without introducing any singularities).
I was part of illiMath2006: an NSF funded Research Experience for Undergraduates (REU) summer program at the University of Illinois at Urbana-Champaign. One of the objectives of the program was to deliver an academia-industry collaborative project centered around a student principal investigator (me). My mentors for the project were George Francis at UIUC, Stuart Levy at the National Center for Supercomputing Applications, and Ulises Cervantes-Pimentel at Wolfram Research.
Our project Sakubo was such a success that Wolfram Research decided to continue the project the following year when I worked there as a summer 2007 intern. We built szgMathematica (= Syzygy + Mathematica) which enables users to use Mathematica’s graphical functions, such as Plot3D or ParametricPlot3D, and view the resulting surfaces in the CUBE.
We even built a function that simulated a rollercoaster where a user could input any closed parametrized curve for the rollercoaster track. The parametrization controlled the speed of the motion. We tried many closed curves from Mathematica’s new (at the time) Knot library and found that crazier knots were not as fun to ride as the standard trefoil. Perhaps there was just too much going on.
The second iteration of the project involved many more people. I worked with George Francis at UIUC, Stuart Levy, Camille Goudeseune at the NCSA, Jim Crowell at ISL, and Ulises Cervantes-Pimentel, Joshua Martell at WRI.