Shape and space
1637
Shapes become equations
René Descartes (and, independently, Pierre de Fermat)
Can every shape be turned into an equation?
Descartes showed that any point can be named by two numbers, and any curve by an equation linking them. Geometry and algebra became one subject. A circle is now just x² + y² = r².
- Eigenvalues λ
- 1.93 and 0.37
- Meaning
- Stretches along the gold and pink lines
A matrix moves every arrow in the plane. Most arrows get turned. The special ones, along the dashed lines, only stretch or shrink: those are eigenvectors, and the stretch factor is the eigenvalue. Rotation has no such arrows at all, so its eigenvalues are complex numbers, the language of spinning.
How they figured it out
The story goes that Descartes, lying in bed one morning, watched a fly crawl across the ceiling and realised he could describe its position by its distance from two walls. It may be a legend, but the idea is real: any point is a pair of numbers, and any curve is a rule linking them.
He published it in 1637 as an appendix, La Géométrie, to his Discourse on Method. Pierre de Fermat had the same idea independently but didn’t publish. Overnight, centuries of geometry could be done with algebra, and algebra could be drawn.
The equation
Every point on a circle is the same distance r from the centre, written as numbers.
See it in real life
Every pin on a map
Any place on a map app is two numbers, latitude and longitude. A video game character is an (x, y) that changes every frame. The board game Battleship is Cartesian coordinates with explosions.
Where engineers use it
CAD and graphics
Every point in a 3D model, game or CNC tool path is a set of coordinates.
Plotting data
Every graph of a measurement against time uses his axes.
Where it came from
- Euclid’s Elements c. 300 BC
Built on
Descartes turned Euclid’s shapes into equations.
- Algebra gets its name c. AD 820
Built on
Descartes drew algebra’s equations as curves.
What it led to
- The invention of calculus 1665–1684
Built on
Calculus works on curves written as equations, which Descartes supplied.
- Eigenvalues 1829
Built on
Eigenvectors are directions in Descartes’ coordinate plane that a matrix leaves pointing the same way.