Challenge to the reader: Two different curves in the widget below change character at the same slider setting. Find it twice: turn the Rose Curve to the smallest setting that draws exactly five petals, and turn the Limaçon to the setting where its inner loop pinches shut into a cardioid. (Answers at the end of the post.)
When mathematicians moved beyond polynomials — into sines, cosines, and exponentials — they unlocked a new world of curves. Unlike their algebraic cousins, curves built from these transcendental ingredients can cross themselves infinitely many times, spiral without end, and trace shapes of heart-stopping beauty. From the hypnotic dance of Lissajous figures to the improbable butterfly curve, here are six curves that transcend algebra.
Why this matters: Every MP3 file, every JPEG image, and every oscilloscope trace is a conversation between these curves and the world. Fourier’s insight — that any repeating signal is a sum of sine waves — is why your phone can compress music. Lissajous figures are how physicists compare frequencies. Rose curves describe the radiation patterns of antennas. And the butterfly curve is a reminder that even in an age of instant computation, a simple formula can still surprise the person who plots it.
Controls: Select any curve to see it animated. On the Harmonic Sum, drag the A₁, A₂, and A₃ sliders to weight each sine component — the three components are drawn in their own colors, and their sum is superimposed as the thick white line. On the other curves, move the Variation slider to adjust frequency, petal count, or inner radius. Click Gallery to compare all curves at once.
1. The Harmonic Sum: Three Sines, One Shape
The simplest Fourier series that is not just a single sine:
\[f(a) = A_1 \sin a + A_2 \sin 2a + A_3 \sin 3a\]The first term is the fundamental; the second and third are its harmonics, at twice and three times the frequency. Fourier’s insight — developed in 1822 while studying heat diffusion — was that every repeating shape, from a violin tone to a jagged voltage trace, is a recipe of terms like these, each with its own weight.
The widget plots all three components in their own colors and superimposes their sum as a thick white line. Drag the A₁, A₂, and A₃ sliders and watch the sum rebuild itself from its parts: zero two weights and the thick line collapses onto the lone survivor; restore a harmonic and the line reshapes, growing steeper peaks and new dips — exactly as an instrument’s tone changes when you add overtones.
Challenge: Set A₁ = 1 and A₂ = A₃ = 0, and count the complete oscillations across the window. Repeat with only A₃ = 1. (Answer at the end of the post.)
2. Lissajous: The Dancing Figures
In 1857, Jules Lissajous made sound visible. He attached mirrors to tuning forks, bounced a light beam off them, and projected the resulting patterns onto a screen. The curves traced by two perpendicular harmonic motions —
\[x = A \sin(at + \delta), \quad y = B \sin(bt)\]— produce an infinite variety of looping, tangled figures. The ratio $a:b$ fixes the fundamental shape; the phase $\delta$ controls the twist. At integer ratios you get stable closed loops; at irrational ratios the curve never repeats, eventually filling a rectangle entirely.
You have seen Lissajous figures: they are the classic oscilloscope music visualiser, and the Australian Broadcasting Corporation’s logo is a 1:1 figure. In the 1:1 case the phase does all the work — at one extreme you get a circle, at the other a straight line.
The widget fixes a 3:4 ratio and sweeps the phase shift $\delta = k\pi$.
3. The Rose Curve: Petals from the Polar World
Luigi Guido Grandi named the rhodonea (rose) in 1723. In polar coordinates:
\[r = \cos(k\theta)\]When $k$ is an integer, the curve has $k$ petals if $k$ is odd, and $2k$ petals if $k$ is even. When $k$ is rational, you get overlapping petals; when irrational, the curve fills an annulus densely. Grandi was so pleased with his roses that he sent them to Leibniz, who was duly impressed.
The widget takes a shortcut through all of that. It reads a whole-number petal count $p = \lfloor 1 + 8k \rfloor$ off the slider, then plots $r = \cos(p\theta)$.
Challenge: Find the smallest slider value that draws exactly five petals — then explain why the setting $k = 0.40$, where the formula gives $p = 4$, draws eight petals instead.
4. The Butterfly Curve: A Computer-Age Discovery
Temple H. Fay discovered this curve in 1989 — not in the 17th or 18th century, but in the age of personal computers. Its equation looks implausibly messy:
\[r = e^{\cos \theta} - 2\cos(4\theta) + \sin^5(\theta/12)\]But plot it over $[0, 12\pi]$ and a butterfly emerges, complete with antenna-like spirals at the tips. The $\sin^5(\theta/12)$ term is the key to the wing shape: it is the slowest-moving term in the equation, with a period twelve times longer than the swing that drives $e^{\cos\theta}$. Without it, you just get a lumpy blob. Even in an age where computers can plot anything instantly, a beautiful curve can still surprise us.
Watch the widget while the butterfly is on screen: the slider does nothing, because this curve’s equation has no $k$ in it at all.
5. The Limaçon: Pascal’s Snail
Étienne Pascal (Blaise’s father) studied the limaçon around 1650. Its polar form is $r = b + a\cos\theta$. Depending on the ratio $b/a$, it morphs through four forms:
- $b \gt a$: a dimpled oval
- $b = a$: the cardioid (heart-shape, with a cusp)
- $a/2 \lt b \lt a$: an inner loop appears
- $b \le a/2$: the inner loop dominates
The name “limaçon” comes from the Latin limax (snail), which is fitting — the curve does resemble a snail shell when the inner loop is present.
The widget fixes $a = 1$ and lets the slider set $b = 2k$. One diplomatic footnote: though it keeps company with transcendental curves, the limaçon is secretly algebraic — square the equation and it becomes the polynomial $(x^2 + y^2 - ax)^2 = b^2(x^2 + y^2)$. It earns its place in this gallery as the polar family’s gateway, and because the cardioid at $b = a$ is the shape everyone means when they say “heart”.
Challenge: Find the slider setting where the inner loop pinches shut into the cardioid, then predict the smallest $b$ for which the curve has no inner loop at all — and say whether the widget’s slider can smooth the dimple away completely. (Hint: the dimple survives all the way up to $b = 2$.)
6. The Heart Curve: A Valentine from Popular Culture
Pop culture’s heart has no patience for the cardioid’s single cusp. The widget’s version is parametric —
\[x = \sin^3 t, \qquad y = \frac{13\cos t - 5\cos 2t - 2\cos 3t - \cos 4t}{16}\]— up to a sliding scale, and it is a trigonometric polynomial: a truncated Fourier series wearing a valentine costume. Every term is a cosine, so the machinery of Fourier analysis applies even to the shape at the bottom of a love letter. The slider scales the heart without changing its shape, and no physical principle chose it: the heart curve is a product of the graphics age, found by asking what a handful of cosines can be made to look like.
7. The Family at a Glance
| Curve | Equation | What fixes the shape | Signature feature |
|---|---|---|---|
| Harmonic sum | f(a) = A₁ sin a + A₂ sin 2a + A₃ sin 3a | The weights A₁, A₂, A₃ | The first three terms of a Fourier series |
| Lissajous | x = sin(3t + δ), y = sin(4t) | The ratio 3:4 and the phase δ | Closes for rational ratios; fills a rectangle otherwise |
| Rose | r = cos(pθ) | Petal count p | p petals for odd p, 2p petals for even p |
| Butterfly | r = exp(cos t) − 2 cos(4t) + sin^5(t/12) | Nothing — one fixed shape | The slow wing term needs a 12π window |
| Limaçon | r = b + cos θ | The ratio b/a | Cardioid at b = a; inner loop when b drops below a |
| Heart | x = sin³t, y = (13cos t − 5cos 2t − 2cos 3t − cos 4t)/16 | Fixed shape; the slider only scales it | A trigonometric polynomial valentine |
8. Deeper Significance: Why These Curves Never Run Out of Surprises
Why do sines and exponentials produce curves that polynomials cannot match? The answer is periodicity with incommensurable periods. A polynomial’s shape is rigid: it can cross a given line only finitely many times, so it can never revisit a pattern endlessly. A sum of sines with rationally related frequencies is periodic and closes; make one frequency irrational and the curve never repeats — yet it never escapes either. It wanders densely through its rectangle forever. That is ergodic theory in miniature: the Lissajous figure with an irrational ratio is the simplest picture of a trajectory that becomes evenly spread over its whole domain.
The same duality — closing versus filling — runs through all of these curves. The rose closes for integer petal counts and fills an annulus otherwise. The butterfly, built from three frequencies locked in a ratio of small integers, traces its wings exactly once every $12\pi$ of parameter. And the harmonic sum makes that alphabet visible — three sines in three colors, their sum one thick white line: Fourier’s theorem says every repeating function, however jagged, is a sum of sine waves. These curves are not a random gallery at all — they are different sentences in one language.
9. Final Challenge
Synthesis challenge: Three parts, all resting on one parity argument.
(a) Prove the rose petal rule. For the curve $r = \cos(p\theta)$ with $p$ a whole number, compare the point at angle $\theta$ with the point at angle $\theta + \pi$. When $p$ is odd the cosine flips sign, so the curve retraces the same petals; when $p$ is even it does not flip, so a second set of petals is drawn. Explain why this yields $p$ petals for odd $p$ and $2p$ petals for even $p$.
(b) Use the widget’s formula $p = \lfloor 1 + 8k \rfloor$ to give the smallest slider setting that produces exactly 1, 3, 5, and 7 petals.
(c) The same closing argument governs the Lissajous figure. Explain why $x = \sin(3t + \delta)$ and $y = \sin(4t)$ close into a loop, while a frequency ratio like $3:\pi$ never does, no matter how long you watch.
Answers to the challenges:
- Five petals: $p = \lfloor 1 + 8k \rfloor = 5$ first happens at $k = 0.50$. At $k = 0.40$ the formula gives $p = 4$, and because even petal counts double, the curve shows eight petals.
- Cardioid: the widget’s limaçon is $r = 2k + \cos\theta$, which becomes the cardioid when $2k = 1$, i.e. $k = 0.50$. The inner loop needs $b \lt 1$; the dimple survives up to $b = 2$, and the widget’s largest is $b = 1.9$ — so the dimple never fully disappears.
- Harmonic sum: the window is $4\pi$ wide, so $A_1 \sin a$ alone draws $2$ complete oscillations while $A_3 \sin 3a$ alone draws $6$ — three times as many, because its frequency is three times higher.
- Final challenge (b): the settings are $0.05$, $0.25$, $0.50$, and $0.75$, for 1, 3, 5, and 7 petals.