Drawing math art: 12 equations that look like flowers
One of the joys of mathematics is how the same compact equation can produce something cold-and-dry (a textbook chart) or stunning (a flower, a butterfly, a cathedral rose window). Polar coordinates are particularly forgiving — small changes to integers in r = f(θ) can flip a curve from boring to glorious.
Here are twelve equations we've collected. Open each in DesmosGraph (links provided), play with the constants, and find one you fall in love with.
1. Five-petal rose
r = 3·sin(5θ)
The classic. Odd k means k petals.
2. Eight-petal rose
r = 2·cos(4θ)
Even k means 2k petals.
3. Cardioid
r = 1 + sin(θ)
The heart curve of polar coordinates.
4. Limaçon with inner loop
r = 0.5 + cos(θ)
A snail with an extra coil inside.
5. Logarithmic spiral
r = e^(0.15θ)
Found in nautilus shells and galaxies.
6. Heart curve (Cartesian)
(x² + y² − 1)³ = x²·y³
The implicit version of a heart.
7. Lemniscate of Bernoulli
(x² + y²)² = 2(x² − y²)
An infinity sign carved by algebra.
8. Folium of Descartes
x³ + y³ = 3xy
A loop with two infinite arms.
9. Astroid
x^(2/3) + y^(2/3) = 1
A four-pointed star.
10. Star polygon (rose with high frequency)
r = cos(11θ/2)
Half-integer multipliers create star-like overlapping petals — striking when you let θ run for several turns.
11. Petal flower with double frequency
r = 1 + 2·sin(2θ) + 0.3·sin(11θ)
Mix two frequencies for ornate ripples on top of larger petals.
12. Butterfly curve
r = e^(sin(θ)) − 2cos(4θ) + sin((2θ − π)/24)^5
Temple Fay's celebrated 1989 curve. Worth a slow zoom.
How to make your own
The recipe is simple: take a known curve like r = sin(kθ), then add a small perturbation with a different frequency. Use sliders to tune. Soon you'll be making originals nobody has seen before.
Math art is one of the friendliest entry points to deeper geometry. You don't need to follow a curriculum — you just need to stay curious.