Anyone who's seen my [You Used Python For **What**?](http://www.youtube.com/watch?v=NLRWUQDaApw) talk knows that I often try to implement stuff in Python to better understand it. 

I've often found that writing implementation for computer often forces a greater understanding of concepts in part because you have to be so explicit.

Gerry Sussman wrote:

> It is surprisingly easy to get the right answer with unclear and informal symbol manipulation. To address this problem we use computer programs to communicate a precise understanding of the computations... Expressing the methods ... in a computer language forces them to be unambiguous and computationally effective. The task of formulating a method as a computer-executable program and debugging that program is a powerful exercise in the learning process.

(elided to make the general point)

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One example from physics is my little attempt at quantum computing in Python: [quantumpy](https://github.com/jtauber/quantumpy).

Back in 2005, I bought Sussman's [Structure and Interpretation of Classical Mechanics](http://www.amazon.com/Structure-Interpretation-Classical-Mechanics-Sussman/dp/0262194554/) (SICM) which is an extensive exploration of Lagrangian and Hamiltonian classical mechanics using Scheme.

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Recently I found out Sussman had been working on doing the same for General Relativity. I eagerly awaited [Functional Differential Geometry](http://www.amazon.com/Functional-Differential-Geometry-Gerald-Sussman/dp/0262019345/), co-authored, like SICM, by Jack Wisdom. My copy arrived early August (although versions had been available online for a while).

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I decided to try implementing the same ideas myself in Python starting with some of the foundational mathematical stuff in Appendix B.

The initial work is available at <https://github.com/jtauber/functional-differential-geometry>.

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The first thing I tackled what was Sussman and Wisdom call "tuples".

Their tuples come in two varieties, **up** and **down**. These basically correspond to contravariant and covariant vectors in differential geometry except that they are recursive (a component of a vector may be another vector). 

A literate doctest explaining how the `tuples` modules works is at <https://github.com/jtauber/functional-differential-geometry/blob/master/tuples.rst>

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As I mention in the README, the tuples module (or at least up tuples) are how I wish tuples worked in Python. 

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Next up was an implementation of functions such that you can do things like take `f(x)` and `g(x)` and do `f+g` such that `(f+g)(x) == f(x) + g(x)`.

See the `functions` module literate doctest at <https://github.com/jtauber/functional-differential-geometry/blob/master/functions.rst>.

There is also a `compose` function there that I guess *could* be associated with an operator if I could think of a binary operator that wouldn't otherwise make sense to apply to the results of two functions.