What Is Geometry? From Measuring Fields to the Shape of Space
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
The word means earth-measuring, and that is what it was: the Egyptian surveyors who re-marked field boundaries after each Nile flood, and the rope-stretchers who laid out right angles for temples with a knotted cord of twelve equal spaces, were doing geometry without a theorem in sight. The Greeks turned the practice into the first system of deduction from stated assumptions, a book that was the model of rigorous thought for two thousand years, and in the nineteenth century mathematicians found that its most contested assumption could be dropped, and that the geometry of the universe was not the one in the book.
Euclid
Around 300 BC Euclid of Alexandria collected the geometry of his predecessors into the Elements, thirteen books that begin with definitions, five postulates and five common notions and derive from them, step by step, some 465 propositions, from the construction of an equilateral triangle to the proof that there are exactly five regular solids. The method was the point. Nothing was asserted that had not been proved from what came before, and the Elements became the textbook of logical argument for every educated person from Archimedes to Lincoln, who read it to learn what a demonstration was. It was printed in more editions than any book except the Bible, and it was taught in schools as geometry, more or less unchanged, until the 1960s.
What it contains
The Elements is the geometry of the flat plane and of space built from it, and its results are the ones everyone half remembers:
- •The angles of a triangle add up to two right angles, 180 degrees
- •The square on the hypotenuse of a right-angled triangle equals the sum of the squares on the other two sides, Pythagoras's theorem, proposition 47 of book one
- •Similar triangles have their sides in proportion, which is the basis of all measurement by angles
- •The area of a circle is proportional to the square of its radius, and the circumference to the radius, with the same constant, pi
- •There are five and only five regular solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron
- •Constructions with straightedge and compass alone, which cannot trisect an angle, double a cube or square a circle, as was finally proved in the nineteenth century
The fifth postulate
Four of Euclid's postulates are simple: a line can be drawn between any two points, extended indefinitely, a circle drawn with any centre and radius, all right angles are equal. The fifth is different. It says, in its usual form, that through a point not on a line exactly one parallel line can be drawn, and it looked to every reader from Euclid onward like something that ought to be provable from the other four rather than assumed. For two thousand years mathematicians tried to prove it and failed, and in the 1820s two of them, Nikolai Lobachevsky in Kazan and Janos Bolyai in Hungary, working independently and with Gauss having reached the same conclusion privately, saw why: it cannot be proved, because it can be denied without contradiction. Assume that through the point there are many parallels, and a consistent geometry results in which triangles have angles adding to less than 180 degrees and there are no similar figures of different sizes; assume there are none, as Riemann did in 1854, and the geometry of the surface of a sphere results, where every pair of lines meets and triangles add up to more than 180. Euclid's geometry was one of three, and which one described the world was a question for physics.
Descartes and the joining
In 1637 Descartes attached numbers to points, so that a point in the plane became a pair of coordinates and a curve became an equation, and geometry and algebra, which had been separate subjects, became two languages for one thing. A circle is x squared plus y squared equals r squared; a straight line is a linear equation; the intersection of two curves is the solution of two equations. Analytic geometry made calculus possible, since a curve that is an equation can be differentiated, and it is the geometry that every graph, every computer screen and every satellite fix uses, in which position is a list of numbers and shape is a formula.
Where it went
Riemann's geometry of curved spaces, in which the rules vary from point to point and the curvature is a quantity that can be measured from inside without stepping out to look, was a pure abstraction for sixty years until Einstein, in 1915, used it to describe gravity as the curvature of space and time by mass, and the bending of starlight around the Sun, measured in 1919, showed that the universe was Riemannian and not Euclidean. Since then geometry has become the study of any space with a notion of shape: the topology that asks what is preserved when a shape is stretched without tearing, the fractal geometry of coastlines and clouds, the geometry of the ten dimensions in which string theory lives, and the computational geometry that lets a phone draw a face and a car see a pedestrian. The surveyors' cord and Euclid's compass are still in it; the fifth postulate is now a measurement.
The takeaway
Geometry began as the measuring of land and was made a deductive science by Euclid, whose Elements derived the properties of triangles, circles and solids from five postulates and modelled rigorous argument for two millennia. Descartes joined it to algebra with coordinates, the nineteenth century showed that Euclid's parallel postulate could be denied to give the curved geometries of Lobachevsky and Riemann, and Einstein found that the universe obeys the curved kind, since which geometry has been the study of shape and space in every form.