Friday, June 12, 2009

Multiplication Table Rainbows

The multiplication table, much maligned as a symbol of rote learning, presents what should be considered one of the most accessible structures for mathematical exploration and recreation. In it you can find stars (described here), surprising relationships (here), and rainbows.

If you divide the range of the table into groups (for example, you could divide the range 0-100 into 7 groups for the ROYGBIV colouring of the rainbow), and colour in the table based on these, you'll find that the groups form rainbow-like curves.

It turns out that bands of colours form hyperbolic arcs. Think of the entries in the table in terms of y*x = k, where k is a constant representing a particular colour and x and y are the terms (column and row values) that are multiplied together to yield a particular k; solving for y gives y = k/x, the formula for a hyperbola.

These pictures were generated in Fathom (the bottom two) and in TinkerPlots (the top one). There is something subtly wrong with the rainbow shown below - it has the right range of colours, but Fathom puts them in the reverse order of what is observed in natural rainbows.

The Fathom and TinkerPlots files used to generate these graphs are here.

28-08-2012 Update - see this related post.