Graph Paper Showcase — Featured plots and visualizations

Classic math

Six phase-shifted sine waves overlaid in rainbow colors

Rainbow Trigonometry

Six offset sinusoids.

A polar rose whose petal count steps as its frequency changes

Polar Roses

A polar rose r = sin(kθ) whose petal count steps as the integer k animates through a discrete set.

A 3:2 Lissajous curve sweeping through its phase offsets

Lissajous Family

A 3:2 Lissajous curve whose phase offset animates, braiding through the whole one-parameter family.

Superellipse Family

An implicit superellipse |x|^n + |y|^n = 1 whose exponent animates, morphing from astroid through circle to squircle.

A polar limaçon morphing from inner loop through cardioid to dimple

Limaçon Family

An animated polar limaçon r = a + cos(θ) whose offset a sweeps through the inner-loop, cardioid, and dimpled forms.

Five colored polyhedra with dark edges standing on a grid: a tetrahedron, a cube, an octahedron, a dodecahedron and an icosahedron

Platonic Solids

The five regular convex polyhedra, side by side, each built from a vertex list and a face table.

Analysis and calculus

Mandelbrot Set

The classic fractal defined by iterating complex quadratic polynomials.

Julia Set

Iterating z² + c with the constant c animating around a circle, sweeping the whole family of Julia fractals.

Domain Coloring

A complex power z^a whose exponent animates, winding the phase more times around the origin.

Contour map of two Gaussian peaks with a saddle

Contour Landscape

Elevation lines of a surface with two peaks and a saddle point.

Composite Fourier wave with two component harmonics

Fourier Harmonics

A composite wave with its individual harmonic components shown separately.

Curves and geometry

Implicit Heart

A classic heart shape defined by the zero set of an algebraic equation.

Spiral Galaxy

Logarithmic spirals winding outward — the blueprint of spiral galaxies.

Butterfly Curve

A delicate parametric curve discovered by Temple Fay.

Conchoid of de Sluze approaching a vertical asymptote

Conchoid of Sluze

An implicit algebraic curve with a cusp and asymptote at x = 1.

Bouquet of folium, lemniscate, and astroid

Implicit Bouquet

Folium of Descartes, Bernoulli lemniscate, and astroid composed in one implicit plot.

A mosaic of sixteen polygonal cells in blue, green, yellow, orange and red, separated by thin dark edges, each with a dark dot at its site

Voronoi Diagram

Sixteen drifting sites split the plane into colored cells: every point takes the color of its nearest site, with time animating.

A large circle filled with smaller touching circles of every size, coloured orange, yellow, green, blue, purple and teal by generation

Apollonian Gasket

Circles packed inside a circle without end, each new one touching three others; Descartes' theorem gives its size, and every curvature is a whole number.

A red five-lobed curve traced inside a grey ring by a pen on a smaller rolling wheel

Spirograph

A wheel rolling inside a ring, with a pen that traces a hypotrochoid; the wheel animates, and the sizes of the ring and wheel and the pen's distance from the wheel's centre can be edited.

A curve made of an oval and an open branch, a blue line through a blue point on each part, an orange point where the line meets the oval again, and a red point under it at the end of a dashed vertical segment

Adding Points on an Elliptic Curve

The chord construction on the curve y² = x³ − 2x + 1: the line through two points meets the curve in a third point, and its mirror image is their sum.

3D surfaces and curves

Möbius Strip

A surface with only one side — follow the edge and you return flipped.

Saddle Surface

A hyperbolic paraboloid — curves up in one direction and down in the other.

Klein Bottle

A non-orientable surface that passes through itself in 3D — a bottle with no inside.

Boy's Surface

An immersion of the projective plane — a one-sided surface with three-fold symmetry and a single triple point.

Roman Surface

Jakob Steiner's quartic model of the projective plane — four lobes with tetrahedral symmetry meeting at a triple point.

Cross-Cap

The simplest self-intersecting model of the projective plane — a pinched cap with one double-point segment and two pinch points.

Trefoil Knot

The simplest non-trivial knot — a closed curve that cannot be untied without cutting.

Seashell

A logarithmic spiral shell — nature’s own parametric surface.

Three torus knots with different winding ratios

Torus Knot Family

Three torus knots at different winding ratios stacked in one scene.

Tube around a five-lobed torus knot, in five colored segments, seen from above

Knotted Tube

A rainbow tube around the (3, 5) torus knot, built on its Frenet frame.

A smooth surface shaded from orange at the top to blue at the bottom, full of round openings like a sponge, cut to a cube

Gyroid

Schoen's gyroid, the minimal surface that divides space into two intertwined labyrinths, approximated by the level set of sin x cos y + sin y cos z + sin z cos x; the level animates.

Physics and engineering

A damped sine wave whose envelope tightens as its damping coefficient changes

Damped Oscillation

A decaying sinusoid whose damping coefficient animates — the exponential envelope tightens and loosens as c sweeps.

Wave Interference

Two point sources create ridges of constructive and destructive interference.

Spherical Harmonics

The angular shape of atomic orbitals — four lobes of a d-orbital.

Buck-Boost Converter

Normalized input resistance vs. duty ratio for buck, boost, and buck-boost DC-DC converters.

Kernel Functions

Common kernels for density estimation: Gaussian, Epanechnikov, quartic, tricube, and uniform.

Double Helix

Two intertwined helices joined by rungs — the iconic shape of DNA.

Space curve with Frenet-Serret tangent, normal, and binormal arrows

Frenet Frame

Tangent, normal, and binormal vectors riding along a space curve.

Electric Dipole

The vector field of a 2D electric dipole, with field lines flowing from positive to negative charge.

Lagrange Points

The five points where a satellite can hold still relative to the Sun and Earth — found where two force-component curves cross — with the Earth mass animating.

The horseshoe-shaped CIE 1931 chromaticity diagram, each point painted with the sRGB color of its chromaticity, with the sRGB triangle drawn inside it

CIE Chromaticity Diagram

The horseshoe of all visible colors, each point painted with the sRGB color of its chromaticity by a color formula, with the sRGB gamut triangle inside it.

A dark planet crossing a limb-darkened orange star, with two light curves above it showing the dip in brightness during the transit

Exoplanet Transit

A planet crossing its limb-darkened star, with the light curve an observer would record: a flat-bottomed dip for a uniform disc, a rounded, deeper one for the real star.

A horizontal grey ray entering a pale blue circle, reflecting inside it, and leaving down and to the left as a narrow fan of rainbow-colored rays

Rainbow Refraction

One ray of white light through a water droplet: refracted in, reflected once, refracted out, and split into a fan of colors, with the impact height animating.

A V-shaped pattern of light and dark waves behind a small dark ship, with dashed lines along the arms of the V

Kelvin Wake

The V-shaped wave pattern behind a ship in deep water: its wavelength changes with the ship's speed, but its half-angle stays 19.47°.

Pale green bright lines that cross in a blue water column below a wavy surface

Underwater Caustics

Sunlight refracted by a wavy water surface gathers into moving bright lines: the patterns on the floor of a swimming pool.

A circular drumhead with a mesh, bulging up in magenta on one side and down in blue on the other, inside a brown rim

Vibrating Drum

A circular drumhead with a fixed rim, vibrating in three normal modes at once: Bessel functions in motion.

A dark wing shape with streamlines flowing around it over a yellow to blue shaded background

Joukowsky Airfoil

Air flowing past a wing, found by mapping the flow around a circle with Z = z + 1/z; the angle of attack animates.

Simulations

Three colored dots on a figure-eight curve, each followed by a trail of its own color

Three-Body Problem

Three equal masses under Newton's gravity, stepped live by an action: the figure-eight orbit, and an equilateral triangle that turns and then breaks up into chaos.

Lorenz Attractor

Edward Lorenz's model of convection, integrated live in 3D: the orbit never repeats, yet it stays on a butterfly-shaped attractor.

A red ball in flight with its red trail, a faint red path that lands early and steeply, and a blue parabola that goes higher and lands much farther along the ground

Projectile with Air Drag

A ball thrown through air, computed step by step with a drag force that grows as the square of the speed, beside the parabola it would follow without air.

A field of grey arrows that turn counterclockwise around a grey dot, a blue closed curve shaped like a rounded triangle around the dot, and a red point with a red trail on the curve

Predators and Prey

The Lotka–Volterra model in the plane of the two populations: a direction field, the closed orbit of a solution, and a state that an action moves around it.