✦ Mathematical Beauty ✦

Where Equations
Become Art

Beautiful expressions derived using mathematics & plotted on graphs

Every curve here is born from a mathematical formula — pure logic that unfolds into breathtaking visual poetry. From ancient spirals to fractal infinities, these are equations that paint with numbers.

r = a(1 − sinθ) z(n+1) = z² + c r = a·√θ x = a·cos³t r = e^(cosθ) − 2cos(4θ) r = sin(kθ)
19
Visualizations
∞
Mathematical Depth
5
Categories
∀
Open & Free
01 / 05

Polar & Parametric Curves

Curves defined in polar coordinates or parametric form — where r and θ weave intricate symmetries impossible to describe in Cartesian terms.

Polar
Cardioid
r = a(1 − sinθ)
A heart-shaped curve traced by a point on a circle rolling around an equal circle. Named from the Greek word for heart — καρδιά.
Open Visualization →
Parametric
Astroid
x = a·cos³t   y = a·sin³t
A hypocycloid with four cusps — the path traced when a circle of radius a/4 rolls inside a circle of radius a.
Open Visualization →
Polar · k=5
Rose Curve
r = a · sin(kθ)
Petals unfold from the origin in perfect symmetry. When k is odd, there are k petals; when even, 2k petals bloom.
Open Visualization →
Polar
Lemniscate of Bernoulli
r² = a² · cos(2θ)
The ∞ symbol made rigorous — this figure-eight curve is the locus of points where the product of distances to two foci equals a constant.
Open Visualization →
Polar · Special
Heart Curve
r = 2−2sinθ + sinθ·√|cosθ| / (sinθ+1.4)
A romantic polar equation that traces a perfect heart. Pure mathematics expressing what words sometimes cannot.
Open Visualization →
Animated
Butterfly Curve
r = e^cosθ − 2cos(4θ) − sin⁵(θ/12)
Discovered by Temple H. Fay (1989), this extraordinary polar curve strikingly resembles a butterfly — emergent beauty from transcendental functions.
Open Visualization →
Radial Variant
Butterfly Curve — Radial
r = e^cosθ − 2cos(4θ) − sin⁵(θ/12)
The same butterfly equation drawn as radiating lines from the origin — revealing the hidden structure of the curve's angular sweep.
Open Visualization →
Parametric
Epicycloid
x = (R+r)cosθ − r·cos((R+r)θ/r)
The beautiful gear-tooth curves that inspired Ptolemy's planetary models. A small circle rolling outside a larger one traces these star-like forms.
Open Visualization →
02 / 05

Spirals

Spirals appear everywhere in nature — galaxies, shells, flowers, DNA. Each mathematical family captures a different law of growth.

Spiral
Archimedean Spiral
r = b · θ
The classic spiral of equal spacing — described by Archimedes in 225 BC. Wind evenly spaced coils of wire and you draw this curve naturally.
Open Visualization →
Spiral · Nature
Logarithmic Spiral
r = a · e^(bθ)
Also called the equiangular spiral or spira mirabilis. Found in nautilus shells, spiral galaxies, and the seeds of sunflowers. Self-similar at every scale.
Open Visualization →
Spiral
Fermat's Spiral
r = a · √θ
Unlike other spirals, Fermat's spiral has two branches — one positive and one negative — giving it a unique symmetry about the pole.
Open Visualization →
03 / 05

Advanced Forms

Generalized mathematical shapes that unify dozens of known curves under single elegant equations.

Generalized
Superformula
r = (|cos(mθ/4)/a|ⁿ² + |sin(mθ/4)/b|ⁿ³)^(−1/n₁)
Johan Gielis's 2003 generalization that describes an astonishing range of natural shapes — from starfish to flowers to snowflakes — with six parameters.
Open Visualization →
Superformula Variant
Superformula Variant
n₁=0.3, n₂=0.3, n₃=0.3, m=6
A specific parameter set of the superformula that produces organic, amoeba-like shapes with six-fold symmetry and smooth undulating lobes.
Open Visualization →

"Mathematics is the language in which God has written the universe."

— Galileo Galilei
04 / 05

Fractals & Chaos

Self-similar structures born from simple recursive rules. Zoom into infinity and find the same patterns repeating endlessly.

z²+c
Fractal · Classic
Mandelbrot Set
z(n+1) = z²n + c
The most famous object in mathematics. Infinitely complex, yet born from the simplest quadratic iteration. Points are colored by how fast they escape to infinity — creating the universe's most iconic boundary.
Open Visualization →
ZOOM
Animated Zoom
Mandelbrot Zoom
z(n+1) = z²n + c
The Mandelbrot set rendered with a continuously zooming camera — revealing nested spirals and miniature copies of the whole set at every scale.
Open Visualization →
COLUMN SCAN
Reveal Animation
Mandelbrot Reveal
z(n+1) = z²n + c
Watch the Mandelbrot Set materialize column by column — experiencing how each pixel's color is individually computed from the iteration formula.
Open Visualization →
05a / 05

Complex Analysis

Visualizing functions of complex numbers — where hue encodes argument and brightness encodes magnitude.

eᶻ iterated 10×
Domain Coloring
Complex Exponential Domain Coloring
f(z) = eᶻ applied 10 times
Domain coloring maps each complex number to a color based on its argument and magnitude. Iterating the complex exponential creates spectacular fractal-like patterns of swirling hues across the complex plane.
Open Visualization →
A
Created & Curated By
Ankit Chaubey
Inspired by many mathematical works and open-source explorers. This gallery is a personal journey through the visual poetry hidden within equations — where calculus meets canvas and functions become art. Every curve here is a story told in the language of mathematics.
github.com/ankit-chaubey