Square

Euclid constructs the square at Proposition 46 of Book I: on a given straight line, to describe a square. The method is a perpendicular and two transfers of length. Raise a perpendicular at one end of the segment, cut it to the segment's own length, then use the compass to fix the fourth vertex from the two ends. The perpendicular itself comes from the vesica construction: two arcs of equal radius about the two endpoints cross at two points, and the line through those crossings is perpendicular to the original.

The square is constructible for the simplest possible reason. Gauss-Wantzel requires a side count of the form: a power of 2 multiplied by a product of distinct Fermat primes. Four is 2 squared, a pure power of two, needing no Fermat prime at all. The same holds for 8, 16, 32 and every further doubling, so bisecting the arcs of a square gives the octagon, then the 16-gon, indefinitely.

Euclid needs the square immediately, because Proposition 47 is the theorem of Pythagoras, stated as an equality between the areas of squares raised on the three sides of a right triangle rather than as an algebraic formula. Book II then proves what are now read as algebraic identities in the form of statements about rectangles and squares.

The square is also the standard instrument for constructing surds. The diagonal of a unit square is sqrt 2. Erect a unit perpendicular at the end of that diagonal and the new hypotenuse is sqrt 3, and repeating the move produces the spiral of Theodorus, which yields the square root of every whole number in turn using nothing but a straightedge and a right angle.

The most consequential fact about the square is that its side and its diagonal cannot both be measured by any common unit, however small that unit is made. The diagonal is sqrt 2 times the side, and sqrt 2 is irrational.

The classic proof is short. Suppose sqrt 2 equals a fraction in lowest terms, p over q. Then p squared is twice q squared, so p squared is even, so p is even, so p squared is divisible by four, so q squared is even, so q is even. Both being even contradicts the fraction being in lowest terms. A version of this argument appears in Aristotle, and it circulated as an appended proposition in the Elements.

The discovery is traditionally credited to the Pythagoreans in the fifth century BC and is usually described as a crisis for them, since their programme took the world to be intelligible through ratios of whole numbers. The story that Hippasus was drowned at sea for revealing it is repeated constantly and is not reliable: the sources are centuries late, they disagree over whether the offence concerned irrationality or the dodecahedron, and some present the drowning as divine punishment rather than as murder. Historians of mathematics treat it as legend. That incommensurability was known and that it mattered is not in doubt; the melodrama around it is.

The real effect was on method. Greek mathematics after this point handles magnitude geometrically rather than numerically, and Book V of the Elements sets out the theory of proportion credited to Eudoxus, which treats ratios of incommensurable magnitudes rigorously without ever needing a number for them. It is a workaround of extraordinary quality, and it held for close to two thousand years, until the nineteenth-century construction of the real numbers.

Plato's Meno turns the same fact into a scene. Socrates asks an untutored slave boy to double a square, watches him guess that doubling the side will do it, and leads him to see that the answer is the square raised on the diagonal.

Four-sidedness is used, with striking consistency across unrelated traditions, for the earth, the settled and the ordered.

Chinese cosmology states it outright: tian yuan di fang, heaven is round and earth is square. The Temple of Heaven and the Altar of Earth in Beijing are built on that distinction, round and square respectively, and the same pairing governs much older ritual jade, the disc and the tube. The principle is at least two thousand years old and shaped Chinese cartography and city planning for most of that time.

Indian architecture works from the Vastu Purusha mandala, a square grid, commonly of 64 or 81 cells, with a cosmic figure laid within it and deities assigned to the compartments. Temple plans, and in the prescriptive texts town plans, are set out upon it. The square here is not a symbol applied to the building; it is the instrument by which the building is measured out.

Islamic garden design uses the chahar bagh, the fourfold garden divided by water channels into quarters, a form running from Persian precedent through Mughal India and connected within its own tradition to the four rivers of paradise described in the Quran. The Kaaba is a roughly cubic structure, and Muslims orient prayer toward it from anywhere on earth.

The four cardinal directions, the four classical elements, the four humours, the four rivers of Eden, the four winds and the four evangelists all sit within the same family of associations, and a good deal of medieval European diagrammatic thought consists of mapping one set of four onto another.

Geometrically the square tiles the plane, meets itself at right angles, and is the shape that a plumb line and a level naturally produce. That practical fact, that squareness is simply what careful building yields, probably underwrites more of the symbolism than any doctrine does.