- Mathematica was used for most experiments and illustrations.
Initially, many graphs were built with Geomag.
A combination of math, concrete models and virtual implementation works well.
Here are some pictures at work in June, 30, 2009:
- This project website is written in Perl. It allows to edit things
locally with my standard editor and with a simple "make" update the entire website.
Contrary to content management systems, classical websites
are fast, easy to edit and update, especially if many files have to be added. They also are
always a "document", which can be read locally and disconnected from the web.
A document must also remain readable in the future without being locked to a particular operating system,
programming language, database or vendor. About Perl: 24 years of track record make it
a stable programming language. I use Perl since 1994. Mathematica started at about the same time
and I program it in regularly since 1988. Not being consistent, one is locked to a vendor with Mathematica.
But Mathematica is also a good "Pseudo code". It will be readable also if the language will have evolved
more. A good sign for stability is that most Mathematica and Perl code written 20 years ago still works
now without or minor modifications. Having only a few hours per month for this project, I can not afford
to waste time with rewriting things if a developer on an ego trip thinks it is time to streamline the language.
- The top background photo was shot in the summer of 2009. It shows part of the amazing
MyZeil mall in Frankfurt which illustrates
how much architecture is influenced by topology.
- I learned linear algebra and logic from Ernst Specker, a student of Heinz Hopf, one of the giants in topology.
In Specker's classes, also the role, interpretation and use of mathematics would come up
like the role of the connections between discrete and the continuum.
Specker introduced us for example to the work of Kustaanheimo, who investigated the use
of finite fields in physics. Kustaanheimo had estimated for example the size of the prime to be used
in order that one could do all the physics, we know. Specker wondered aloud in class how curious it is
that we often model discrete phenomena (in biology or business for example)
with continuous models (using calculus) and then in turn numerically compute
them by discretizing the model again (using numerical methods).
The theme Discrete/Continuum is intriguing: we model discrete processes like financial transactions
with stochastic differential equations, but then discretize these differential equations to simulate
them. We model finite particle systems like the motion of atoms in the air with partial differential
equations like the Navier-Stokes equations, but then in turn numerically integrate them
on a computer with finite element methods. This illustrates that the steps discrete -> continuum, continuum -> discrete
can be of utterly different nature.
[Added in January, 2012: Ernst Specker passed away on December 10, 2011.
Obituary ETHZ [PDF]]