Reinventing 2D: What a difference a matrix makes

Manhattan is a Cartesian plane brought to life. The original design plan for the streets of Manhattan, known as the Commissioners’ Plan of 1811, put in place the grid plan that defines Manhattan to this day. Using rectangular grids in urban planning is common practice but Manhattan went as far as naming its streets and avenues with numbers: 1st Street, 42nd Street, Fifth Avenue and so on.

The idea of a system of coordinates was first published in 1637 by René Descartes (hence Cartesian) and revolutionised mathematics by providing a link between geometry and algebra. According to legend, school boy René was lying in bed, sick, when he noticed a fly on the ceiling. He realised that he could describe the position of the fly using two numbers, each measuring the perpendicular distance of the fly to the walls of the room. Voilà! The Cartesian plane was born.

It is only natural then that we interpret the Cartesian plane in terms of spatial coordinates but what happens if we take the same old Cartesian plane and re-invent the 2D space as a space of functions?

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