The various musical items on this page are generated from a variety of mappings that produce strange attractors. The programmes are written in Logo, and the decisions about where the music goes comes from the iterated mapping. Decisions about timing, about instruments and harmony are made from musical considerations, but alwys have a link to the underlying mapping. This is very much work in progress. I am pleased with the Piano Sonata, and less so with others, but with time there are things I can improve.
Logistic Piano Sonata
The Verhulst Process or Logistic Map has been used as a model for population growth in a setting where there is a fixed upper possible limit. What is interesting is that, is we change the parameter we get convergence to a single value, a pair, a set of four, then 8 and more as period doubling sets in. The length of each section of one, two, four etc. limits reduces by geometric ratio. At the limit of the ratio chaos occurs, but with interesting windows of order. The common ratio os the Feigenbaum number.
This piece iterated x-> rx(1-x) and gradually increases r from 2 to 4 and back to 2. The piano plays the values.
Lorenz Butterfly map
This is the original mapping that Lorenz used to start questioning some of our deterministic assumptions. he was modelling weather changes, and set up the iterations based on a set of starting values to see how the weather would develop after hours or days. He then reran his model using numerical values that were rounded off to a certain number of decimal places as the starting point. They were very close to the original values, but the outcome diverged rapidly and massively. It led to the question Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? The idea of the effect of a butterfly flap has been used frequently in literature i=since then as a metaphor for unexpected large change. This piece starts two butterflies near each other playing different instruments as they go.
The play described on the Metamathematics page cannot really be described as Theatre as it is basically talking heads with a few pictures. Even when performed it is much more about the content of the text than the performance. However I suppose it represents a link between thinking about Maths and Theatre.
At the other extreme is a Samuel Becket play Quad, which has no words, only movement. From a Maths point of view it is all about geometry and combinations. You can show the version below and ask them to find the Maths in it. There is another version here, which I think is the original performance. The music is better, but the lighting is worse.
Alternatively you can look at Beckett’s instructions for the play, found half way through the article here. What is fun, is to ask students to write and perform a version that uses 5 players. i will append a video of that soon.
There are so many references here, that I only list a couple of my favourites.
Robert Heinlein – “And he built a crooked house” on the four dimensional hypercube as a house.
Jorge Luis Borges – Most of it really, but “The Library of Babel” in Labyrinths is wonderful. There is, astonishingly a pdf of this book here! There is also a great book “The Unimaginable Mathematics of Borges’ Library of Babel” full of different viewpoints from combinatorics to geometry to topology.
When is Maths Art? A theory of knowledge question of itself. Is the Mandelbrot set a work of Art? If so download Xaos, and zoom in to your own masterwork. Or is the Art of Maths better represented by Suman Vaze? The painting is based on the four colour theorem and proudly adorns the walls of my flat. The one at the top shows the fourth routes of -1, and their four different multiplications and at the bottom, their sum.