Circle

Euclid defines a circle in Book I as a plane figure contained by one line, such that all straight lines drawn from a certain point within it to the boundary are equal to one another. That point is the centre. The definition is a condition on distance, and everything else about the circle follows from it.

His third postulate then grants that a circle may be described with any centre and any distance. This is the compass, stated as a permission rather than as a tool, and it is one of only five things Euclid asks to be granted. Together with the straightedge of the first two postulates it is the entire apparatus of classical construction.

There is a subtlety about which compass. Euclid's postulate has often been read as describing a collapsing compass, one that closes the moment it is lifted and so cannot carry a distance from one part of a diagram to another. Euclid does not need the stronger tool, because Propositions 2 and 3 of Book I show how to transfer a given length to a given point using only the collapsing kind. The result is that the two compasses are equivalent in power: anything the rigid compass can construct, the collapsing one can construct too, at the cost of extra steps. It is an early and elegant demonstration that a restriction can cost nothing at all.

The circle is also the source of nearly every new point in construction work. Bisecting a segment, erecting a perpendicular, copying an angle, bisecting an angle: each of these standard moves is done by striking arcs and using where they cross. The straightedge joins; the compass decides. When a construction turns out to be impossible, it is because no finite sequence of such crossings can reach the required point.

The problem is old and precisely stated: given a circle, construct with compass and straightedge a square of exactly equal area. Plutarch reports that Anaxagoras worked on it in prison in the fifth century BC, which makes it one of the oldest recorded open problems in mathematics. It stayed open for more than two thousand years.

It is now closed, and closed by proof rather than by exhaustion. The chain runs as follows. Squaring the circle requires constructing a length of sqrt pi from a unit length, which in turn requires constructing pi. In 1761 Johann Heinrich Lambert proved that pi is irrational: it is not the ratio of two whole numbers. That alone does not settle the question, because many irrational lengths are perfectly constructible; sqrt 2 is simply the diagonal of the unit square. The decisive result came in 1882, when Ferdinand von Lindemann proved pi transcendental, meaning it is not a root of any polynomial equation with rational coefficients.

That finishes it. Every length reachable by compass and straightedge is algebraic, and its degree over the rationals is a power of 2, because each new point arises from intersecting lines and circles and so satisfies at worst a quadratic over what came before. A transcendental number satisfies no polynomial at all. Therefore pi is not constructible, and the circle cannot be squared. The proof is not that nobody has managed it. It is that the target lies outside the set of things the tools can reach.

This deserves stating plainly, because sacred-geometry writing often treats squaring the circle as a standing mystery or an unfinished quest. As a symbol of the union of heaven and earth, the circle set against the square, the image is a legitimate and ancient one. As a mathematical problem it was answered in 1882, and the answer is no. Good approximations remain useful in practice, and purported exact solutions have continued to arrive in mathematics departments for well over a century.

A circle has no corners, no beginning, and no distinguished point anywhere on it. Rotate it about its centre by any amount whatever and it is unchanged, which is a stronger symmetry than any polygon possesses. That property, sameness under any turn, is what most of its symbolic use rests on.

Circumambulation turns the figure into an action. Muslim pilgrims perform tawaf, seven circuits of the Kaaba. Hindu and Buddhist practice includes pradakshina and the circling of stupas, and the same movement is made at Tibetan holy mountains. Walking a circle holds a centre fixed and makes the walker's relation to it the only thing that changes, which is close to what the figure is taken to mean in each case.

In Zen the enso is a circle brushed in one or two strokes, usually left slightly open, and it is valued precisely for revealing the state of the person who drew it rather than for geometric accuracy. Mandalas in Indian and Tibetan traditions set circles and squares in fixed relation to one another, and the Tibetan sand mandalas are swept away on completion, which is part of the point being made.

Christian, Buddhist and Iranian art all place a disc behind the head of a holy figure. The nimbus appears in Hellenistic and Iranian art before it is Christian, and the route of its transmission is a matter of continuing art-historical argument rather than settled fact. The wheel carries the Buddha's teaching, and turns the year in agricultural calendars.

Prehistoric stone circles are the largest and least explained instances. Stonehenge, Avebury, Callanish and hundreds of lesser sites are laid out on circular or near-circular plans across Britain, Ireland and Brittany. Some alignments to solstice sunrise and sunset are securely established by measurement. What the builders intended by the circular plan itself is not recoverable, and confident accounts of it are interpretation rather than evidence.