The sine of a figure eight

A sine wave is a circle seen from the side. A point walks round a circle of radius one at a steady pace, and its height, read against the distance it has walked, rises and falls in the familiar wave. Everything about the sine — its period, its symmetries, the formula for the sine of a sum — is a fact about the circle, read through that one act of walking and measuring.

The act can be repeated on any closed curve, and on almost every curve it produces nothing worth naming: a wave with no formula, no symmetry beyond the curve’s own, and no algebra connecting one value to another. There is one curve on which it produces something as rich as the sine, and in some ways richer. It is the figure eight that Jacob Bernoulli wrote down in 1694 while studying the shape of a bent elastic rod, the curve of points whose distances from two fixed points multiply to a constant:

$$(x^2 + y^2)^2 = x^2 - y^2.$$

Bernoulli called it the lemniscus, a ribbon. It is the same curve whose crossing point has no single slope, and that crossing is where every walk on it begins. Walk along it from its centre at a steady pace and read the distance from the centre against the distance walked. The function that reading defines is the lemniscatic sine, written $\mathrm{sl}$, and the story of what it does is the story of how the subject of elliptic functions began.

The walk, and the wave it makes #

The right-hand panel of the figure is the lemniscatic sine over one circuit. It starts at nought at the centre, climbs to one where the walker reaches the far end of the right-hand loop, falls back to nought at the centre, and then goes negative — the convention is that the left-hand loop counts as distance in the opposite direction — before returning. It looks almost exactly like a sine, and the dashed sine of the same period behind it confirms how close: the two agree at every zero and every peak, and between them the lemniscatic one runs slightly ahead on the way up and slightly behind on the way down.

The stations on the left are what make the reading honest. They are spaced at equal arc length, not at equal angle, and they bunch visibly near the centre crossing and spread out round the ends of the loops. That is because the figure eight is not traced at a uniform angular rate by anything natural; the only uniform thing about the walk is the walking. The circle hides this distinction, because on a circle equal arcs and equal angles are the same thing — which is exactly why the sine can be introduced through angles at all, and why the angle is really an arc once anybody looks closely.

The curve’s whole length, measured by adding eight thousand short chords, is $5.244115$ — a measurement and not a formula, since length on a curve is defined by exactly such sums. Half of that, the distance from the centre round one loop and back, is the number that plays the role $\pi$ plays for the circle, and it has its own symbol: $\varpi$, a variant of the Greek letter pi, the lemniscate constant, equal to $2.622058$ to six places.

Where the fourth power comes from #

Why should this curve, of all the curves in the plane, behave like the circle? The answer is in the arc integral, and it takes three lines.

In polar coordinates the figure eight is $r^2 = \cos 2\theta$. A small step along it changes the distance from the centre by $dr$ and the angle by $d\theta$, and the length of the step is $\sqrt{dr^2 + r^2\,d\theta^2}$. Differentiating the curve’s equation gives $2r\,dr = -2\sin 2\theta\,d\theta$, and since $\cos 2\theta = r^2$, $\sin 2\theta = \sqrt{1 - r^4}$. So $d\theta = -r\,dr/\sqrt{1 - r^4}$, and the step’s length is

$$ds = \sqrt{dr^2 + \frac{r^4\,dr^2}{1 - r^4}} = \frac{dr}{\sqrt{1 - r^4}}.$$

The distance walked from the centre to a point at distance $r$ is therefore

$$s = \int_0^r \frac{dt}{\sqrt{1 - t^4}},$$

and that should be read beside the circle’s. On a circle of radius one, the arc from the bottom of the circle up to height $h$ — or equivalently the angle whose sine is $h$ — is

$$\arcsin h = \int_0^h \frac{dt}{\sqrt{1 - t^2}}.$$

The two integrals differ by one exponent. That single change, $t^2$ to $t^4$, is the whole difference between the circle and the figure eight, as far as walking is concerned.

The circle’s integral has an antiderivative that can be written down, because $\arcsin$ is defined to be one; the reason this is satisfying is that $\arcsin$ is the inverse of something simpler, the sine. The figure eight’s integral has no antiderivative in terms of the usual functions — that was proved in the nineteenth century — so the arc length of the lemniscate cannot be written in closed form. What can be done is what was done for the circle: name the integral’s inverse. The lemniscatic sine $\mathrm{sl}$ is defined by $\mathrm{sl}(s) = r$ exactly when $s = \int_0^r dt/\sqrt{1 - t^4}$, just as $\sin s = h$ exactly when $s = \arcsin h$.

The figure in this essay builds $\mathrm{sl}$ a third way and checks it against the other two. Differentiating the definition gives $\mathrm{sl}'(s) = \sqrt{1 - \mathrm{sl}(s)^4}$, and differentiating again, $\mathrm{sl}'' = -2\,\mathrm{sl}^3$ — an equation of motion, the lemniscatic counterpart of $\sin'' = -\sin$. Integrating that equation step by step from $\mathrm{sl}(0) = 0$, $\mathrm{sl}'(0) = 1$ gives the curve in the hero figure, and the stations it places on the figure eight lie on the curve to nine decimal places and are spaced by equal measured arcs.

Fagnano doubles an arc #

The first sign that this function has an algebra came from an Italian nobleman, Giulio Carlo de’ Toschi di Fagnano, who in 1718 found something that should not have been possible. The arc length of the lemniscate cannot be computed in closed form; yet Fagnano showed how to double an arc, exactly, with arithmetic and a single square root. If an arc from the centre ends at distance $r$, the arc twice as long ends at

$$r_2 = \frac{2r\sqrt{1 - r^4}}{1 + r^4}.$$

In terms of the function, that is the doubling formula $\mathrm{sl}(2s) = 2\,\mathrm{sl}(s)\,\mathrm{sl}'(s)/(1 + \mathrm{sl}(s)^4)$, the lemniscatic counterpart of $\sin 2s = 2\sin s\cos s$. The consequence Fagnano drew was geometric. Since the formula uses only arithmetic and square roots, a doubled arc’s endpoint can be constructed with compass and straightedge from the original’s; and since the formula can be solved for $r$ given $r_2$ — a quartic equation in $r^2$ that is really a quadratic — so can a halved arc’s. Fagnano could cut a quarter of the figure eight into two, four or eight equal arcs by construction, and into three and five as well, without ever knowing how long any of the arcs were.

Leonhard Euler was sent Fagnano’s collected papers on 23 December 1751, a date Carl Jacobi later called the birthday of elliptic functions. Euler saw at once that the doubling was the special case of an addition formula, and found it:

$$\mathrm{sl}(a + b) = \frac{\mathrm{sl}(a)\,\mathrm{sl}'(b) + \mathrm{sl}(b)\,\mathrm{sl}'(a)}{1 + \mathrm{sl}(a)^2\,\mathrm{sl}(b)^2}.$$

Set the denominator to one and this is exactly the sine’s addition formula, with $\mathrm{sl}'$ in the role of the cosine. The denominator is what the fourth power costs.

Gauss finds the length inside an average #

The most remarkable fact about the figure eight’s length was found by Carl Friedrich Gauss at the age of twenty-two, and it is a fact about an operation that seems to have nothing to do with curves.

Take two positive numbers, here $1$ and $\sqrt2$. Replace them by their ordinary average and their geometric mean — the square root of their product. Repeat. The two numbers close on each other with extraordinary speed, and their common limit is called the arithmetic–geometric mean.

On 30 May 1799 Gauss wrote in his diary that he had established, to the eleventh decimal place, that the arithmetic–geometric mean of $1$ and $\sqrt2$ equals $\pi/\varpi$ — and that the proof of it would surely open an entirely new field of analysis. It did. The proof came within the year, and it rests on a change of variables that turns the arc integral of the figure eight into itself with the two numbers replaced by their two means. Each step of the averaging is a transformation of the integral that leaves its value alone, and in the limit the integral becomes trivial.

The figure’s table shows the practical meaning. Four rounds of averaging and one square root per round compute the length of the figure eight to the limit of the arithmetic, where adding up strips under the integrand would take thousands of terms for six places. Variants of the same iteration are how $\pi$ itself has been computed to trillions of digits, since the Gauss–Legendre algorithm of the 1970s is the arithmetic–geometric mean applied to the circle’s own integrals.

The same constant in a swinging pendulum #

The figure eight’s half-length turns up in a place with no figure eight in it. A pendulum swinging through small angles has a period that does not depend on how far it swings; that is the approximation every clock relies on, and it rests on replacing $\sin\theta$ by $\theta$ in the equation of motion. For large swings the approximation fails, the period grows, and the exact period is an arc integral of the same family as the lemniscate’s — a length that has to be walked rather than computed from a formula.

Release the pendulum from the horizontal, a quarter-turn from the bottom, and the integral is exactly the figure eight’s. Its period is longer than the small-swing period by the factor $\varpi\sqrt2/\pi = 1.180341$: eighteen per cent longer, and the eighteen per cent is the lemniscate constant in disguise. The arithmetic–geometric mean computes it in four steps from $1$ and $1/\sqrt2$, which is how the factor above was checked. Galileo believed the period of a pendulum was the same for every swing, and for the swings he could time it nearly is; the correction he could not see is a ratio of the two curves’ lengths, the circle’s $\pi$ against the figure eight’s $\varpi$.

The constant has one more property shared with $\pi$. Theodor Schneider proved in 1937 that $\varpi$ is transcendental — the root of no polynomial with whole-number coefficients — so, like the circle’s, the figure eight’s length cannot be constructed from a unit with compass and straightedge, even though so many of its divisions can. The figure eight can be cut into seventeen equal pieces by construction, and the length of any one of them cannot be drawn.

A sine with two periods #

The circle’s sine, extended to complex numbers, has one period: $\sin(z + 2\pi) = \sin z$ for every complex $z$, and no other independent shift leaves it unchanged. In the imaginary direction it does not repeat at all; it grows exponentially, since $\sin(iy) = i\sinh y$.

The lemniscatic sine is different, and the difference is a single identity. Because the integrand $1/\sqrt{1 - t^4}$ is unchanged when $t$ is replaced by $it$ — the fourth power of $i$ is one — the function satisfies $\mathrm{sl}(iz) = i\,\mathrm{sl}(z)$. Rotating its input by a right angle rotates its output by a right angle. So whatever the function does along the real line, it does along the imaginary line as well, and its real period becomes an imaginary one.

The figure computes the function at every complex point from its real values alone, using the addition formula with $b = iy$, and the picture that emerges is a crystal. Zeros sit at every point $\varpi(m + in)$ for whole numbers $m$ and $n$; poles, where the function is infinite, sit at the centres of the squares between them. Shifting by $\varpi$ in either direction changes only the sign, and shifting by $(1 + i)\varpi$ or $(1 - i)\varpi$ changes nothing: the function is doubly periodic. Shading the whole plane with it is shading one tilted square, of side $\sqrt2\,\varpi$, over and over.

A function that repeats in two independent directions and has only poles as singularities is an elliptic function, and the name is inherited from the arc length of the ellipse, whose integral is of the same kind. Niels Henrik Abel and Carl Jacobi made the theory in the 1820s by doing for the general integral $\int dt/\sqrt{\text{quartic}}$ what had been done for the figure eight: invert it, extend it to complex values, and find the second period. The lemniscate is the case where the second period is the first turned through a right angle, which is why its picture is a square crystal rather than a slanted one.

The surprising connection: the figure eight divides like the polygons #

Gauss proved in 1796 that a regular polygon with $n$ sides can be drawn with compass and straightedge exactly when $n$ is a power of two times a product of distinct Fermat primes — $3, 5, 17, 257, 65537$ — or, equivalently, when the number of whole numbers below $n$ that share no factor with $n$ is a power of two. Dividing a regular polygon’s circle into $n$ equal arcs is the same problem, so the theorem is about dividing a circle.

In the Disquisitiones, after proving it, Gauss remarked that the principles of his method applied not only to the circle but to many other transcendental functions, for example to those depending on the integral $\int dt/\sqrt{1 - t^4}$. He never published what he meant. In 1827 Abel did, and the statement is as clean as anyone could wish.

The figure eight can be divided into $n$ equal arcs with compass and straightedge exactly when the regular $n$-gon can be drawn. Five equal arcs: yes, as the figure shows. Seven: no. Seventeen: yes, by a construction as intricate as Gauss’s for the seventeen-sided polygon and in the same way. The list along the bottom of the figure is the polygon list, number for number.

Why the same list? The circle’s division points are values of the sine at rational multiples of its period, and those values generate fields whose symmetry group is commutative — which is what makes a tower of square roots reach them when the group’s size is a power of two. Abel’s proof shows the same thing for the lemniscatic sine, with the Gaussian integers $m + in$ in place of the ordinary whole numbers: the identity $\mathrm{sl}(iz) = i\,\mathrm{sl}(z)$ makes multiplication by $i$ an operation on division points, the symmetry group of the $n$-division points is again commutative, and its size is again governed by a count that comes out a power of two in exactly Gauss’s cases. The second period, the one the complex picture showed, is what makes the arithmetic work. A curve whose sine had only one period could not have carried the theorem.

Division points computed, constructions not drawn #

Constructibility is a statement about every step being a square root, and no figure here draws a construction. The division figure shows where the five and seven division points are, computed by integration; it does not show compass and straightedge reaching the first set and failing to reach the second. That is Abel’s theorem, quoted. The figure’s list is the theorem’s condition evaluated at each $n$, which is a check that the condition picks out Gauss’s numbers and not a check of the theorem itself.

The complex picture computes the function; it does not prove there are no further zeros or poles. The shading was evaluated on a grid of nine thousand cells, the marked zeros and poles were checked to vanish and blow up, and the periodicity was confirmed at sampled points. A zero hiding between grid cells would not be seen. That there are exactly two zeros and two poles in each period square follows from the theory of elliptic functions, and the picture is consistent with it rather than a demonstration of it.

Gauss’s identity is shown to fourteen places, not proved. The table’s agreement between $\pi$ over the mean and the arc length measured by integration is the same kind of evidence Gauss had on 30 May 1799, at three more decimal places. The proof is the transformation of the integral under one averaging step, which is algebra and not picturable.

Still open: the functions that build the rest #

The circle’s sine and the figure eight’s sine each do one more thing, and it is the thing that turned them into a programme. The values of the circle’s sine and cosine at rational multiples of $\pi$ — the coordinates of the vertices of regular polygons — generate every field extension of the rational numbers whose symmetry group is commutative. That is the Kronecker–Weber theorem, and it says that one function, the exponential, evaluated at simple points, builds all such fields. The values of the lemniscatic sine at the figure eight’s division points do the same for the field of Gaussian rationals $m/k + i\,n/k$: they generate every commutative extension of it. Leopold Kronecker called the hope that this pattern would continue — that every number field has its own special functions whose values build its commutative extensions — the dream of his youth.

David Hilbert put it on his list of problems for the new century in 1900, as the twelfth. For fields like the Gaussian rationals, the answer is the theory of complex multiplication, of which the lemniscate is the first example. For general number fields Hilbert’s twelfth problem is open. Real progress has come in this century: Samit Dasgupta and Mahesh Kakde announced in 2021 a construction for the totally real fields, using special values of a different kind of function. What the right functions are for every field, and whether any single construction covers them all, is not known.

The next reading of the circle this subject owes is not a new curve but an old function seen from its zeros: the sine as an infinite product of the factors its roots force, where the same zeros at multiples of $\pi$ that the figure eight’s crystal generalised turn out to determine the circle’s function completely — and to hand over the sum of the reciprocal squares as a by-product.