We start out on our expedition.  We began by doing some “guided mathematical visualization” with our eyes closed, led by Dr. Gunn.  At the other extreme Dr. Gunn asked whether a crystal he brought with him represents “mathematical visualization”. We chose to leave unanswered the question of whether some being could have thought the crystal into existence as a kind of “mathematical visualization”.  In any case we decided that neither example qualified as a mathematical visualization (MV) in the sense of this course.  To be a MV in the sense of this course means to combine something of the thought quality of pure mathematics with the sense-perceptible, material nature of the crystal.  It must be an “embodiment” of mathematics.

Next, Dr. Gunn introduced 3 types of “product” that qualify as mathematical visualization:

1. 3D prints (the ones he passed around he had made in the last 6 years at the 3D Lab at TU Berlin.) Please also check out this blog post from Franziska Lippoldt — a student in last year’s course — describing her experience making a 3D print from her semester project.
2. Mathematical movies, where the content was generated directly from mathematical models in the computer.
   1. Outside In: A movie from the Geometry Center on turning the sphere inside out.
   2. The Borromean Rings: A movie from TU Berlin introducing the new logo (new in 2006!) of the International Mathematical Union.
3. Interactive applications which allow the user to explore and experience a mathematical domain.  Example of the Archimedean solids app.

We noted that these three categories fit in sequence between the crystal and pure math, as they progress from more “thing-like” to more “mind-like”, the latter being characterized by pure freedom, the former by pure necessity — in philosophical terms.

We established that most if not all of the course participants plan to go into some “applied” area when they graduate from the TU.  Hence they will not be working in “pure math” mode but in a mode in which mathematics is brought into some “embodied” form in the real world.  Hopefully the visualization course can be a preparation for this direction of work and will develop relevant non-mathematical skills  such as design, communication, and teamwork.