Circle

• 1. The Shape Before Writing

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  Before any human wrote a single word, the circle already existed in the world. The full moon hung in the sky. A slice of round fruit showed its perfect curve. Prehistoric people built stone circles and timber circles, and they pressed circular forms into petroglyphs and cave paintings. The shape was known long before the beginning of recorded history. A circle is every point in a plane that sits the same distance from one fixed point, the centre. That single rule of equal distance is all it takes. From it came the wheel, and from the wheel came gears, and from gears came much of modern machinery. The same shape that inspired the study of geometry also pushed forward astronomy and calculus. How does a rule this simple generate a constant that can never be written down completely? Why did a Greek word for a hoop end up naming a circus? And why were geometers willing to chase one impossible square for thousands of years? The answers begin with the names people gave to the parts of a circle.

• 2. A Vocabulary Of Parts

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  Radius is the line segment that joins the centre to any single point on the circle, and its length is half the diameter. The diameter runs from one point on the circle through the centre to another, and it is the longest distance between any two points on the shape. The diameter is a special case of a chord, the longest one a given circle allows. A chord is any line segment whose endpoints lie on the circle, splitting it into two segments. The circumference is the length of one full circuit around the circle, the distance all the way round. A disc is the region of the plane that a circle bounds. In strict mathematical usage the circle is only the boundary, while everyday speech lets the word circle mean the disc too. An arc is any connected part of the circle. A sector is bounded by two equal radii and one of the arcs between their endpoints, while a segment is bounded by a chord and one of the arcs joining the chord's ends. A secant is an extended chord, a straight line that cuts the circle at two points, and a tangent is a straight line that touches the circle at exactly one point. Two concentric circles bound an annulus, a ring. Two overlapping discs share a lens. These names carry into the etymology of the shape itself.

• 3. Hoops, Rings, And Names

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  The word circle descends from the Greek kirkos and kuklos, which were themselves a metathesis of the Homeric Greek krikos, meaning hoop or ring. The same root sits close to the origins of the words circus and circuit. The naming did not stop at language. Disc-shaped artifacts survive from prehistory, including the Nebra sky disc and the jade discs called Bi. The circle also gathered meaning far beyond its geometry. From the time of the earliest known civilisations, the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Greeks and Romans of classical Antiquity, the circle appeared in visual art to carry an artist's message. Worldview changed what people saw in it. Some emphasised the perimeter to express a democratic manifestation, while others focused on the centre to symbolise cosmic unity. In mystical doctrines the circle stood for the infinite and cyclical nature of existence, and in religious traditions it represented heavenly bodies and divine spirits. It signified unity, infinity, wholeness, divinity, balance, stability and perfection. Cultures conveyed these ideas through the compass, the halo, the vesica piscis, the ouroboros, the Dharma wheel, the rainbow, mandalas and rose windows. Magic circles belong to some traditions of Western esotericism.

• 4. The Number That Will Not End

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  The ratio of a circle's circumference to its diameter is pi, an irrational constant approximately equal to 3.141592654. The Egyptian Rhind papyrus, dated to 1700 BCE, already gave a method to find the area of a circle, with a result that corresponds to roughly 3.16049 as an approximate value of pi. Archimedes proved, in his Measurement of a Circle, that the area enclosed by a circle equals the area of a triangle whose base is the circle's circumference and whose height is its radius. That comes to pi times the radius squared. The enclosed area is about 79 percent of the circumscribing square whose side equals the diameter. Archimedes also used the circle as the limit of a series of regular polygons, where the number of sides grows toward infinity, to approximate pi. The circle holds a further distinction. For a given arc length, it encloses the maximum possible area, a fact tied to the isoperimetric inequality in the calculus of variations. Among all shapes with a given perimeter, the circle wins on area, and that quiet superiority would later collide with a problem no compass could solve.

• 5. Angles, Arcs, And The Radian

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  One radian is the central angle subtended by a circular arc whose length equals the radius. Place a circle of radius r at the vertex of an angle, let the angle cut off an arc of length s, and the radian measure is simply the ratio of the arc length to the radius. The angle a complete circle subtends at its centre measures a full turn, which is 360 degrees. The relationships between angles and chords run deep. If a central angle and an inscribed angle stand on the same chord and the same side, the central angle is exactly twice the inscribed angle. Every inscribed angle that subtends a diameter is a right angle, a result known as Thales' theorem. Chords carry their own rules. Chords are equidistant from the centre if and only if they are equal in length, and the perpendicular bisector of any chord passes through the centre. When two chords cross and divide one chord into lengths a and b and the other into c and d, the products ab and cd are equal. A tangent line through a point on the circle stands perpendicular to the diameter through that point. From any point outside the circle, two equal tangents can be drawn. These properties feed the constructions that geometers performed with only a compass and a straightedge.

• 6. Compass And Straightedge

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  Place the fixed leg of a compass on a centre point, set the movable leg on a point of the circle, and rotate. That is the simplest and most basic construction of a circle. Through any three points not all on the same line, there lies exactly one circle. To build it, construct the perpendicular bisectors of two segments joining the points and mark where they intersect; that intersection is the centre, and it passes through all three points. This is the circumcircle. Every triangle has one circle circumscribed about it through its three vertices, and one incircle inscribed inside it tangent to all three sides. A tangential polygon is any convex polygon that can hold an inscribed circle tangent to every side, while a cyclic polygon is one a circle can be drawn around through every vertex. A polygon that is both is called bicentric. Apollonius of Perga showed another way to define a circle entirely. He proved that a circle can be the set of points with a constant ratio of distances, other than one, to two fixed foci A and B. The proof leans on the angle bisector theorem and on the fact that interior and exterior angles sum to 180 degrees, producing a right angle that traces a circle. This Apollonian view turns the circle into one member of a larger family of figures.

• 7. A Circle Among Its Relatives

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  An ellipse with an eccentricity of zero is a circle, its two foci collapsed into a single centre. The circle sits inside a wider catalogue of curves as a limiting or special case. A Cassini oval, the set of points whose distances to two fixed points have a constant product, becomes a circle when those two points coincide. A supercircle reduces to a circle when its exponent n equals 2. A curve of constant width keeps the same perpendicular distance between parallel tangent lines no matter the direction, and the circle is the simplest such figure. Distance itself can be redefined, and the shape called a circle changes with it. In taxicab geometry, where p equals 1, circles are squares with sides tilted at 45 degrees to the axes, and the geometric analog of pi becomes 4. For the Chebyshev distance, a circle is again a square, this time with sides parallel to the axes. Topology loosens the definition further. There the circle is the one-dimensional hypersphere, the 1-sphere, and it includes all of its homeomorphisms, so any shape that can be deformed into a loop counts. The geometry can also be oriented, with a circle carrying a clockwise or anti-clockwise value, except for a circle of radius zero, which is simply a point. With so many definitions stretching the idea, one ancient question kept its grip on geometers.

• 8. The Square That Cannot Be Built

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  Squaring the circle asked ancient geometers to construct a square with the same area as a given circle, using only a compass, a straightedge, and a finite number of steps. The problem outlasted civilisations. In 1880 CE, Ferdinand von Lindemann proved that pi is transcendental, meaning it is not the root of any polynomial with rational coefficients. In 1882, the task of squaring the circle was proven impossible as a consequence of the Lindemann-Weierstrass theorem. A challenge thousands of years old finally had its verdict, and it was a closed door. The circle had already entered new territory by then. With the advent of abstract art in the early 20th century, geometric objects became artistic subjects in their own right, and Wassily Kandinsky often used circles as elements of his compositions. The same shape that medieval scholars tied to the divine, the shape Plato described in his Seventh Letter as a perfect form distinct from any drawing or definition, still resists being squared. Despite the proof, the topic continues to draw pseudomath enthusiasts who keep trying.