What Is Geometry? From Measuring Fields to the Shape of Space

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.

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

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.

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.

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.

One of the most elegant uses of Greek geometry was measuring the size of the Earth without leaving Egypt. Eratosthenes, librarian at Alexandria in the third century BC, learned that at Syene, to the south, the Sun stood directly overhead at noon on midsummer's day, so that it lit the bottom of a well. At the same moment in Alexandria, a vertical rod cast a shadow showing the Sun about one fiftieth of a full circle away from overhead. If the Sun's rays arrive parallel, that angle equals the angle between the two cities measured at the Earth's centre.

The rest is proportion. If the distance from Alexandria to Syene is one fiftieth of the way round, the whole circumference is fifty times that distance, which Eratosthenes took to be about five thousand stadia. His answer was 250,000 stadia, and although the exact length of the stadion he used is uncertain, most estimates put his result within a modest margin of the true figure of about 40,000 kilometres. The same reasoning with similar triangles is how surveyors once measured mountains, and how you can find the height of a tree today by comparing the length of its shadow with the length of your own.

The theorem named after Pythagoras was known long before him. A small clay tablet from Babylonia, catalogued at Yale as YBC 7289 and written roughly between 1800 and 1600 BC, shows a square with its diagonals drawn in and a number written along one of them in the Babylonian base-sixty system. Converted to decimals it reads 1.41421296, the length of the diagonal of a square with sides of 1, which is the square root of 2. The true value is 1.41421356, so the scribe's figure is correct to five decimal places, an accuracy that needed a reliable method rather than a lucky guess.

Another tablet of about the same age, known as Plimpton 322, lists fifteen rows of numbers that turn out to be built from Pythagorean triples, sets of whole numbers in which two squares add up to a third. The first row gives 119 and 169, and 119 squared plus 120 squared is exactly 169 squared. What the table was for is still argued over: some scholars see a teacher's aid for setting problems, others a kind of early trigonometric table. Either way, Babylonian scribes were using the relationship more than a thousand years before Pythagoras, though as a tool for calculation rather than as a theorem with a proof.

Honeycomb cells are hexagons, and the reason is geometry. If you want to divide a flat surface into equal cells using the least total wall length, hexagons are the most efficient shape. Squares and triangles also fit together without gaps, but they need more wall for the same area, which for a bee means more wax, and wax is costly to make.

Mathematicians suspected this for a very long time; it was discussed in ancient times. It was finally proved in 1999 by Thomas Hales, who showed that a hexagonal grid divides the plane into equal areas with the least possible perimeter. Bees, of course, did not need the proof: their wax cells appear to form roughly circular shapes that settle into hexagons as the warm wax is pulled by surface forces.

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.