# Mural – Mathematical Connections

## Mural

### Mathematical Connections

Wave equation

The equation graffitied by OctoPi in green and yellow is the wave equation, a second-order linear partial differential equation (or PDE for the cognoscenti) describing wave propagation. The ripples created by the green paint dripping from OctoPi’s bucket are described by this equation!

### Examples of flows developing swirls

Two Views of Von Kármán Vortices

Water strider by David Hu, Brian Chan, and John Bush

Octonion

The triangular figure on the leftmost part of the wall is a graphic representation for the multiplication rules for octonions. It is one of the many ways in which this Fano plane mnemonic can be drawn.  There are only 7 “bubbles”  in this multiplication table despite the “octo” root (signifying eight) in “octonion” because the eighth octonion is simply the number 1, and no mnemonic is needed to know that if you multiply anything with 1, you still have the same anything.  The Fano plane is also an example of a projective plane – the same drawing, but with arrows replaced by lines, is the projective plane of order 2. The location of this octonion multiplication table on the back of a wall of Conway’s Curios is not fortuitous: it alludes to the book On Quaternions and Octonions by John Horton Conway and Derek A. Smith.

Wavelet

The “ferocious wavelet” nearby, pointed to by OctoPi’s spraypaint bottle, is indeed the graph of a wavelet, that is, a function of which a discrete family of scaled translates constitutes an orthonormal basis for the space of all square integrable functions on the line. It looks similar to but is not identical to the wavelet generating the Daubechies-4 or D4 wavelet basis; it has only one vanishing moment and has slightly less smoothness.