Metaballs.
Liquid blobs that drift, merge and pull apart. There are no meshes or models: every pixel is computed from a formula describing the shapes.
How it works
Each blob is a sphere described by a signed distance function, which gives the distance from any point to its surface. A smooth minimum blends nearby spheres so they melt into one another instead of overlapping. For every pixel, a ray steps forward by that distance until it touches the surface (raymarching), and the surface normal drives the lighting. Rays that pass just beside a blob are blended into the background, which keeps the edges smooth.
A* versus Dijkstra.
Two pathfinding algorithms race through the same maze. Both find the shortest route; watch how much of the maze each one has to explore to get there.
How it works
Both algorithms always expand the cheapest cell next. Dijkstra only knows how far it has come, so its wavefront spreads evenly in every direction. A* adds an estimate of the distance left (the Manhattan distance, the same heuristic from my 8-puzzle work), so its wavefront leans toward the goal. Colour shows distance from the start, the dots are the frontier still waiting to be explored, and the counters show how many cells each has expanded so far.
Black hole.
Hot gas streaming round a black hole, inspired by Gargantua in Interstellar. Gravity bends the light so strongly that you see the far side of the disc arching over the top, and the stars behind smear into rings.
How it works
Every pixel is a ray of light traced backwards from the camera and bent toward the hole at each step, following a Schwarzschild black hole. Rays that fall in stay black: that is the shadow. The disc is a thick volume of gas, its swirls made by domain warping (noise fed through noise), glowing hotter near the centre and brighter on the side spinning toward you (Doppler beaming, toned down as in Interstellar). Inner gas orbits faster, so the streaks shear into rings. Rays that escape sample the stars, so stars behind the hole appear around its edge. More on how it works.
Aurora.
Northern lights folding slowly over a mountain lake. The light is built up from dozens of thin layers stacked in the sky, so the curtains have real height and ripple as they drift.
How it works
The sky is sliced into about forty horizontal layers, from low to high. For each pixel, the ray from the camera is followed up through every layer, and at each one a slowly drifting noise pattern decides whether a curtain passes through that spot. Because the same pattern repeats in every layer, the curtains line up into tall sheets, with a sharp, bright lower edge that fades and turns violet with height, as oxygen and nitrogen glow at different altitudes. Fine vertical streaks come from a second, faster noise along each curtain. The lake below simply looks up the same sky in the mirrored direction, nudged by moving ripples.
Ink in water.
Drops of coloured ink sinking through still water, curling into rings and trailing threads as they fall. Where inks overlap they mix like real dye, by each soaking up its own colours of light.
How it works
A drop of ink falling into water rolls up into a vortex ring: a doughnut of ink that spreads and slows as it sinks, trailing a thread from where it entered. Each drop here is that shape, swept through a slowly churning 3D noise field so it breaks into wisps. A ray from every pixel crosses the tank in small steps, and the ink it passes through absorbs light following the Beer–Lambert law: each ink removes the colours it isn't, so blue over pink darkens toward violet, just as dye does. The noise is computed once into a 3D texture, so each step costs only a few lookups.
Soap bubble.
A film of soapy water a few hundred nanometres thick. Light bouncing off its inside and outside surfaces interferes, so the colours swirl as the film flows and thins, and dark spots open where it is thinnest.
How it works
Light reflects off both surfaces of the film. The two reflections travel different distances, so for some wavelengths they add up and for others they cancel: thin-film interference. For each of red, green and blue, the brightness is 2R(1 − cos(4πnd·cosθ / λ)), where d is the film's thickness, n its refractive index and λ the wavelength. The thickness is a swirling noise pattern that drains downward over time, so the top thins first; below about 30 nm both reflections cancel completely and the film turns black. The bubble's wobbling shape is raymarched, and both its front and back walls reflect the room.
Water ripples.
A pool you can touch. Ripples spread, bounce off the edges and interfere, bending the tiles and light beneath them.
How it works
Each pixel stores the water's height now and a moment ago. The next height is the average of its four neighbours, times two, minus the previous height: a discrete form of the wave equation, with slight damping so waves fade. The surface slope bends the image underneath (refraction), and where the surface curves, light concentrates into bright caustics.
Reaction–diffusion.
Two virtual chemicals spread out and react. From that one rule grow coral, dividing cells and labyrinths, the same maths behind leopard spots and fingerprints.
How it works
This is the Gray–Scott model, running on your GPU. Every pixel holds two concentrations, A and B. Each step, both diffuse to their neighbours, and wherever they meet, B converts A into more of itself: A + 2B → 3B. A is fed in at rate F and B is removed at rate K. Those two numbers alone decide whether you get coral, cells or a maze, and switching pattern reshapes what is already growing. The canvas runs 14 simulation steps every frame.