Mural – Mathematical Connections

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!

The other equation (in red and orange) is also a famous PDE, the Navier-Stokes equation. It describes turbulent flow of a “fluid” – this can be what we commonly think of as a fluid, like water, but it also works for gas flow, like air flow around the wing of an airplane. When a flowing fluid encounters an obstacle, the flow develops little swirls, like those in the smoke from the snuffed-out candle when it has to move around the mahlstick; these have been observed in nature, at many scales as well as in simulations of solutions of the Navier-Stokes PDE.

Examples of flows developing swirls

Two Views of Von Kármán Vortices

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

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.

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.

The graffiti “tag” (on the lower right) of self-referential artist OctoPi has mathematical meaning too!  The tag includes the lower-case Greek letter, , which is the mathematical constant defined in Euclidean geometry as the ratio of a circle’s circumference to its diameter (approximately equal to 3.14159). 

The French phrase on her tag, “Ceci n’est pas un poulpe”, reflects both her artistic background and sensitivity to the popularity of octopus in the diets of many humans.  Surrealist painter Rene Magritte’s famous painting, “Ceci n’est pas une pipe.” (This is not a pipe.) has been interpreted many ways, including that the pipe in the painting is not a pipe but a drawing of a pipe.  Also, there are two words in French for octopus: le poulpe and la pieuvre. Le poulpe generally refers to food, and la pieuvre refers to the bright, resourceful cephalopod in the ocean.