Challenge to the reader: Hold two questions in your head while you read — the whole post is built to answer them. (1) Which spiral looks exactly the same after you zoom in on it, and why does no other curve in the gallery share that property? (2) The Cornu spiral is defined by integrals with no closed form in elementary functions, yet a driver who turns the wheel at a constant rate traces one perfectly. By the end you should be able to answer both from memory — then hit Compare All in the widget and check the first answer with your own eyes.
Every spiral on this page is the same idea in a different costume: a point that turns while it travels outward. What separates the ten of them is a single function — how the radius $r$ depends on the angle $\theta$. Get that dependence right and the whole menagerie sorts itself into families at a glance: constant spacing here, exponential spacing there, and one curve that refuses a tidy formula entirely.
That is the promise of this post. Nine of the ten spirals are live in the interactive canvas below — click through them, drag the growth slider, and watch the growth law change the character of the curve. The tenth, Nielsen’s spiral, is the one that got away.
1. One Equation, Ten Spirals
Every curve in the gallery is a polar curve: a function that answers one question for every angle.
\[r = f(\theta)\]Two numbers do all the work. The first is how fast $r$ grows with $\theta$. The second is the spacing between one turn and the next, measured along a fixed ray:
\[r(\theta + 2\pi) - r(\theta)\]If that spacing is constant, you have an Archimedean spiral. If it grows in proportion to $r$, you have a logarithmic spiral — and it becomes self-similar. If it shrinks, you are in the hyperbolic family: hyperbolic, Fermat, lituus, Poinsot, cochleoid.
Challenge: The Archimedean spiral in the widget is drawn with $a = 0.05 + 0.25k$, where $k$ is the slider value divided by 100. Compute the gap between two successive turns at slider position 0.50, then at 0.90 (the gap is $2\pi a$). Does the slider change the shape of the spiral, or only its scale?
2. The Archimedean Spiral: The Practical One
\[r = a\theta\]Archimedes described his spiral in On Spirals (c. 225 BCE), using it to square the circle and trisect angles. Its defining feature is a constant gap:
\[r(\theta + 2\pi) - r(\theta) = 2\pi a\]That single constant is why the Archimedean spiral is everywhere in engineering: coiled springs, vinyl record grooves, and scroll compressors all use constant-pitch spirals, because a constant pitch is exactly what a tracking stylus, a spring, or a compressor needs. The distance between successive windings never changes.
3. The Logarithmic Spiral: Nature’s Favourite
\[r = a e^{b\theta}\]Descartes first described it, but Jacob Bernoulli made it famous. He called it spira mirabilis — the marvellous spiral — because advancing the angle by any fixed amount multiplies every radius by the same factor:
\[r(\theta + \Delta) = e^{b\Delta}\, r(\theta)\]That is what self-similarity means: zoom in or out by any factor and the curve looks exactly the same. It is why the nautilus shell, rams’ horns, spiral galaxies, and even the approach path of a hawk hunting prey all approximate logarithmic spirals. Bernoulli wanted one carved on his gravestone — but the mason carved an Archimedean spiral by mistake.
Challenge: The widget’s Fibonacci spiral uses $b = \ln\phi / (\pi/2)$ with $\phi = 1.618\ldots$. Using the self-similarity relation above, by what factor does the radius grow over a quarter turn, and over a full turn? (Answer: $\phi$, and $\phi^4 \approx 6.854$.)
4. The Cornu Spiral: Saving Lives on the Highway
The Cornu spiral — also called the clothoid or Euler spiral — is the one curve in the gallery whose definition is not a tidy formula for $r$. Its curvature increases linearly with arc length, and its coordinates are the Fresnel integrals:
\[C(t) = \int_0^t \cos\left(\frac{\pi u^2}{2}\right)\, du, \qquad S(t) = \int_0^t \sin\left(\frac{\pi u^2}{2}\right)\, du\]These integrals have no closed form in elementary functions — a reminder that some of the most practical curves resist simple formulas. The payoff is physical: when you turn a steering wheel at a constant rate, your car traces a Cornu spiral, because the curvature of the path grows linearly with the distance travelled. That is precisely the shape engineers want for highway transition curves, the graceful entry and exit of freeway ramps. Before clothoids were used in railway design in the 19th century, trains had to slow dramatically for curves; the smooth transition allowed much higher speeds.
5. Fibonacci: The Celebrity Spiral
\[\phi = \frac{1+\sqrt{5}}{2} \approx 1.618\]The Fibonacci spiral is really a logarithmic spiral whose radius grows by a factor of $\phi$ every quarter turn. It is constructed by drawing quarter-circles inside squares whose side lengths follow the Fibonacci sequence, and it appears — sometimes genuinely, sometimes wishfully — in sunflowers, pinecones, and the Parthenon. One thing is certain: $\phi$ appears wherever optimal packing meets angular growth.
6. The Inward Winds: Hyperbolic, Fermat, Lituus — and Their Cousins
Not every spiral grows. The remaining members of the gallery either tighten outward or wind inward forever:
\[r = \frac{a}{\theta}, \qquad r = a\sqrt{\theta}, \qquad r = \frac{a}{\sqrt{\theta}}\]- Hyperbolic (Pierre Varignon, 1704): winds inward from infinity to a central point — the inverse of the Archimedean.
- Fermat (Pierre de Fermat, 1636): a parabolic spiral, with the radius growing like a square root — tighter than the Archimedean, with two symmetric arms.
- Lituus (Colin Maclaurin, 1722): named after a Roman staff, it winds infinitely around the origin but never reaches it.
- Poinsot (Louis Poinsot, 19th century): a generalisation of the hyperbolic spiral, shifted off the origin.
- Cochleoid (Roger Cotes, 1722): snail-shaped — cochlea is Latin for snail shell — with the radius following a sinc pattern.
7. Reference Table: The Ten Spirals at a Glance
| Spiral | Who / when | Spacing behaviour | Radial equation |
|---|---|---|---|
| Archimedean | Archimedes, c. 225 BCE | Constant — every turn sits the same distance from the last | $r = a\theta$ |
| Logarithmic | Descartes; made famous by Bernoulli | Grows exponentially — self-similar at every scale | $r = a e^{b\theta}$ |
| Hyperbolic | Pierre Varignon, 1704 | Decays as the angle grows — winds in from infinity | $r = a/\theta$ |
| Fermat | Pierre de Fermat, 1636 | Grows like a square root — tighter than Archimedean | $r = a\sqrt{\theta}$ |
| Lituus | Colin Maclaurin, 1722 | Compresses as the angle grows; never reaches the origin | $r = a/\sqrt{\theta}$ |
| Poinsot | Louis Poinsot, 19th century | A hyperbolic spiral shifted off the origin | $r = a/(\theta + c)$ |
| Cochleoid | Roger Cotes, 1722 | Snail-shaped — peaks near the start, then decays | $r = a\sin\theta/\theta$ |
| Cornu (clothoid) | Marie Alfred Cornu, 1874 | Curvature grows linearly with arc length — no elementary closed form | Fresnel integrals only |
| Fibonacci | Leonardo Fibonacci, c. 1200 | Each quarter-turn multiplies the radius by the golden ratio | $r = a\phi^{2\theta/\pi}$ |
| Nielsen | Niels Nielsen, 1865–1931 | Defined through special functions — no elementary parameterisation | special functions only |
8. Nielsen’s Spiral: The One That Got Away
Niels Nielsen (1865–1931) discovered a spiral based on special functions that resists simple parameterisation. Unlike the others in our gallery, it cannot be expressed with elementary functions alone — it involves integral representations related to Bessel functions. Nielsen made fundamental contributions to the theory of the gamma function and generalised hypergeometric series; his spiral is a footnote, but a reminder that not every beautiful curve yields to a tidy formula.
9. Why the Growth Law Is Everything
Step back, and the whole menagerie collapses into one sentence: every spiral here is an answer to “how does the radius respond to the angle?” Change that response and the entire geometry changes — that is the root cause of every difference in the table above.
- Nature chooses the logarithmic family because living things grow by adding material in proportion to what is already there. A growth law that multiplies rather than adds is exactly what keeps the shape unchanged at every size, which is why shells, horns, and galaxies all land in the same family. A shell that has to keep working while it grows has no other option.
- Engineering chooses the Archimedean family because machines want constant tolerances: a constant-pitch groove, a constant-rate spring, a smoothly varying ramp. Constant spacing is a manufacturing requirement, not an aesthetic one.
- The Cornu spiral is what a human hand produces. Curvature that grows linearly with distance travelled is the natural output of a steadily turned wheel, which is why the clothoid feels right on a highway and why the same Fresnel integrals also turn up in the diffraction of light at the edge of a shadow — one curve, two completely different scales.
The unification is the real point: a single exponent in $r(\theta)$ decides whether a curve spirals outward like a machine, outward like a living thing, or inward forever — and the two curves with no exponent at all are the ones we still cannot draw in closed form.
Challenge: Run Compare All. Which spiral’s turns crowd most tightly around the origin regardless of the slider, and which one races to the edge of the canvas first? Then explain, using the table, why moving a single slider can never turn the Archimedean spiral into the logarithmic one.
10. The Final Challenge
Final challenge. The widget draws every curve from $\theta = 0.2$ to $\theta = 7\pi$ — three and a half turns.
(a) At the end of the draw, which radius is larger: the Archimedean spiral with $a = 0.175$, or the logarithmic spiral with $b = 0.175$ — and by roughly what factor? (Use the radial equations in the table; a calculator is allowed.)
(b) Zooming into the logarithmic spiral gives the same curve back, while zooming into the Archimedean gives a different spiral with a different pitch. Explain why in one sentence using the two growth laws, then confirm it with the slider.
(c) Name the one curve in the gallery with no closed form in elementary functions — and the one discussed in the prose that resists parameterisation altogether. What do those two have in common?
Try it: Click each spiral button to see it alone. Hit Compare All for a grid view. The slider adjusts the growth rate — see how each spiral’s character changes.