# The presets Each opens in the app with `p=`, and each page answers in Markdown too. ## Wave · y = f(x, t) - [AM Carrier](https://www.wavelace.com/presets/am-carrier): A fast wave is multiplied by a slow bulge, so the loud stretches stand nine times taller than the quiet ones. Crests and bulge move at different speeds. - [Beat Interference](https://www.wavelace.com/presets/beat-interference): Two sine waves close in wavelength combine into a rhythm that swells and nearly flattens every 3.70 seconds, like two piano strings a hair apart in pitch. - [Bessel-ish](https://www.wavelace.com/presets/bessel-ish): Two parts make the curve: a cosine whose phase is quadratic in x, and an envelope that shrinks it towards the edges. - [Breathing Gauss](https://www.wavelace.com/presets/breathing-gauss): A bell curve of fixed shape slides between −3 and 3 like a pendulum bob, fastest through the middle and briefly still at each end, in simple harmonic motion. - [Chain Rule](https://www.wavelace.com/presets/chain-rule): The curve plots the slope of a chirp wave, not the wave itself, so its ripples grow taller away from the centre. The chain rule causes that growth. - [Chirp Wave](https://www.wavelace.com/presets/chirp-wave): A sine whose phase grows like x squared meets a plain travelling cosine, so its crests spread out near the middle and crowd together further from it. - [Crest Doubling](https://www.wavelace.com/presets/crest-doubling): A travelling wave made of a fundamental and its second harmonic, where turning the harmonic up raises a second crest on the falling side of the first. - [Cusp Heart](https://www.wavelace.com/presets/cusp-heart): A cusp curve lifted by a wide envelope traces a heart with a dimple at the top and a tip at the bottom, filled by a fast zigzag that shimmers as time passes. - [Damped Pulse](https://www.wavelace.com/presets/damped-pulse): A wave travels through a bell shaped window that stays fixed, growing as it enters and fading as it leaves. Five crests fit its half-height width. - [Depth Sheet](https://www.wavelace.com/presets/depth-sheet): A rippling curve trails copies whose heights are scaled by a second wave along depth, turning a plain ribbon into a sheet with leaning ridges. - [Dispersion](https://www.wavelace.com/presets/dispersion): Eight harmonics of a sawtooth, each travelling at a speed the slider sets, from rigid sliding to a shape that never repeats. - [Fractal Cosine](https://www.wavelace.com/presets/fractal-cosine): Five cosines are added, each three times as fine and half as tall as the one before it. - [Gamma Function](https://www.wavelace.com/presets/gamma-function): The curve passes through the factorials at the whole numbers, one step along from each, and fills in a value for every real number between them. - [Gamma Poles](https://www.wavelace.com/presets/gamma-poles): The gamma function extends the factorial smoothly, exact at every positive integer, before breaking into four alternating poles from zero downward. - [Gaussian Blur](https://www.wavelace.com/presets/gaussian-blur): A square wave convolved with a bell curve, worked out as an integral at every point: the corners round off as the bell widens. - [Heartbeat](https://www.wavelace.com/presets/heartbeat): A narrow spike and shallow trough repeat every 4 units, travelling right in a shape the eye reads as a heartbeat, though it is only a drawing. - [Heat Kernel](https://www.wavelace.com/presets/heat-kernel): A square wave is blurred by a widening average that rounds its corners first, smooths it into a fading sine, then starts over. - [Launch Angle](https://www.wavelace.com/presets/launch-angle): A projectile leaves the origin at a fixed speed while its launch angle swings between 15 and 75 degrees, drawing every arc in between as the clock runs. - [Living Wave](https://www.wavelace.com/presets/living-wave): Crests pile up near the origin, where the curve has no value, and drift outward over time. Frozen still, the wave matches the topologist's sine curve. - [Living Wave (cos)](https://www.wavelace.com/presets/living-wave-cos): A cosine wave leaves a gap where its phase runs to infinity, and its crests crowd together approaching that point, matching the topologist's sine curve. - [Living Wave (tan)](https://www.wavelace.com/presets/living-wave-tan): A tangent wave has poles cut flat at a common height, so each reads as a plateau. The plateaus crowd near the origin and slide to the right over time. - [Logarithmic Integral](https://www.wavelace.com/presets/logarithmic-integral): The curve estimates how many primes lie below a given number, an idea Gauss had as a teenager in 1792. - [Normal Distribution](https://www.wavelace.com/presets/normal-distribution): The curve traces the cumulative probability of a normal distribution, rising from zero to one as its mean drifts back and forth, crossing one half at the mean. - [Partial Sums](https://www.wavelace.com/presets/partial-sums): A travelling sine gains one odd harmonic a second, up to twelve, flattening into plateaus joined by near-vertical walls that keep a fixed overshoot beside them. - [Pendulum Period](https://www.wavelace.com/presets/pendulum-period): The curve gives the period of a swinging pendulum against its starting angle, barely rising for small swings before shooting up as the angle nears vertical. - [Phase Fold](https://www.wavelace.com/presets/phase-fold): A sine wave has a faster sine added to its phase, folding its crests over, while a slow envelope pinches the whole line flat and lets it swell again. - [Piecewise](https://www.wavelace.com/presets/piecewise): A parabola on the right meets a straight line on the left at the origin, where both are zero. Their slopes disagree there, giving the curve a kink. - [Residue Staircase](https://www.wavelace.com/presets/residue-staircase): The integral of a function with two poles around a growing circle, plotted against its radius. It steps up by each pole's residue as the circle takes it in. - [Resonance Curve](https://www.wavelace.com/presets/resonance-curve): The steady amplitude of a damped oscillator against the frequency driving it, with a peak that sharpens as the damping falls and drains away as it grows. - [Saw Cascade](https://www.wavelace.com/presets/saw-cascade): Straight ramps climb and drop vertically in a pattern sliding steadily right, the standard example of a wave needing every harmonic. - [Sinc Splash](https://www.wavelace.com/presets/sinc-splash): A ripple sliding sideways sits inside an envelope tall in the middle and fading like a sinc function, so the peak stays put as crests pass. - [Square Fourier](https://www.wavelace.com/presets/square-fourier): Four sine harmonics sum into a square wave that slides sideways, and the horn beside each jump, the Gibbs phenomenon, grows narrower but never shorter. - [Standing Wave](https://www.wavelace.com/presets/standing-wave): A sine curve swells and collapses in place, formed from two identical waves travelling in opposite directions that cancel at fixed points called nodes. - [Taylor Sine](https://www.wavelace.com/presets/taylor-sine): A polynomial approximation to sine gains a term every second, matching the curve near the centre but drifting further out until more terms accumulate. - [Travelling Ripple](https://www.wavelace.com/presets/travelling-ripple): A wave slides sideways while a bell shaped curve limits how tall each crest may grow. A crest is born small, peaks in the middle and fades as it travels on. - [Twin Solitons](https://www.wavelace.com/presets/twin-solitons): Two secant hyperbolic humps, 2.63 units wide at half height, slide right side by side, eight units apart, unlike real solitons that can pass through each other. - [Uncertainty](https://www.wavelace.com/presets/uncertainty): The curve shows how narrowing a wave packet in position widens its spread in frequency, illustrating the trade-off behind the uncertainty principle. - [Viete Product](https://www.wavelace.com/presets/viete-product): Cosines of an angle halved again and again, multiplied together, settle on one value. Viète published it in 1593, the first infinite product written down. - [Wave Packet](https://www.wavelace.com/presets/wave-packet): Fine ripples, crests 0.42 apart, sit inside a fixed hump while only the ripples slide through, unlike a real packet, which would travel and spread. ## Polar · r = f(θ, t) - [Alain's Curve](https://www.wavelace.com/presets/alains-curve): Two loops meeting at the centre swell and shrink as a parameter breathes between 0.4 and 1.0, in a curve named after Alain Juhel. - [Antenna Beam](https://www.wavelace.com/presets/antenna-beam): The curve plots the direction pattern of a line of antennas driven in step, whose length breathes over time so the beam narrows, widens and grows side lobes. - [Breathing Rose](https://www.wavelace.com/presets/breathing-rose): The rose turns slowly while a finer eleven-fold ripple rides over its petals, swelling one while thinning the next as the two turn in opposite senses. - [Butterfly Curve](https://www.wavelace.com/presets/butterfly-curve): The curve stands still and draws four lobed wings every turn, closing into a butterfly only after twelve. Temple H. Fay published it in 1989. - [Fermat Spiral](https://www.wavelace.com/presets/fermat-spiral): The spiral grows so that equal turns add equal area rather than equal radius, the law behind the seed pattern in a sunflower head, and its size gently pulses. - [Focal Conic](https://www.wavelace.com/presets/focal-conic): A single eccentricity slider carries the curve through circle, ellipse, parabola and hyperbola, with the attracting focus held at the origin. - [Fourfold Mandala](https://www.wavelace.com/presets/fourfold-mandala): Twelve near-copies of a four-lobed butterfly curve, each stretched by a different amount, stack into the mandala seen from above. - [Gielis Supershape](https://www.wavelace.com/presets/gielis-supershape): One polar formula that covers circles, rounded squares and stars. Its letters set the symmetry count and how sharp the corners come out. - [Lemniscate](https://www.wavelace.com/presets/lemniscate): The curve is a pair of lemniscates of Bernoulli, first described in 1694, crossed at right angles and rotating steadily as time passes. - [Polar Mandala](https://www.wavelace.com/presets/polar-mandala): The curve sums a strong sixfold ripple in the angle. A finer seventeenfold ripple rides on it, so the shape only repeats itself after a full turn. - [Pulsing Cardioid](https://www.wavelace.com/presets/pulsing-cardioid): The heart shape is a cardioid, drawn by a point on a circle rolling around an equal circle, with a single cusp. A pulse grows and shrinks it, keeping its shape. - [Rational Rose](https://www.wavelace.com/presets/rational-rose): Giving the rose a fraction of the angle instead of a whole number draws petal counts no whole number reaches, over several turns. - [Rose 20](https://www.wavelace.com/presets/rose-20): A cosine positive at ten bearings draws ten petals, and its negative values fill the gaps. Each is eighteen degrees wide. - [Square Petal](https://www.wavelace.com/presets/square-petal): Odd harmonic terms added to a five-petalled rose curve approximate a square wave, giving the petals flat ends with a persistent small overshoot. - [Trefoil Bloom](https://www.wavelace.com/presets/trefoil-bloom): A bulging ring is reshaped by two ripples drifting around it in opposite directions, one with three lobes, the other with six. ## Surface · z = f(x, y, t) - [Charged Rod](https://www.wavelace.com/presets/charged-rod): The surface is the electrostatic potential of a uniformly charged rod, ridged along its length and ringed by elliptical contours with foci at its ends. - [Checker Waves](https://www.wavelace.com/presets/checker-waves): Two perpendicular waves multiply into square cells that bulge up or down in alternating pairs, sliding diagonally and repeating roughly every six seconds. - [Chirp Transform](https://www.wavelace.com/presets/chirp-transform): The height is the Fourier transform of a chirped Gaussian pulse, each row a spectrum whose width grows as its peak falls, so every row carries the same energy. - [Chladni Plate](https://www.wavelace.com/presets/chladni-plate): Two standing modes of a square are subtracted, and the lines where they cancel are the pattern sand settles along on a bowed plate. - [Cross Ripple](https://www.wavelace.com/presets/cross-ripple): Two plane waves along crossing diagonals add into ridges sliding sideways between straight nodal lines that never move, an exact solution of the wave equation. - [Cusp Catastrophe](https://www.wavelace.com/presets/cusp-catastrophe): A quartic potential whose single resting place splits into a pair with a barrier between them as one slider passes zero, the standard pitchfork. - [Diffraction](https://www.wavelace.com/presets/diffraction): A square hole casts a bright central square on a distant screen, crossed by fainter lobes that dip below the plane where the wave is out of step. - [Diffusion Front](https://www.wavelace.com/presets/diffusion-front): The sheet stacks every instant of a spreading edge: a sharp step at the near edge softens into a long ramp as diffusion widens with the square root of time. - [Drumhead](https://www.wavelace.com/presets/drumhead): Each point's height is the Bessel function of its distance from the centre, so a drum's rippling rings rise and fall together in one standing mode. - [Egg Carton](https://www.wavelace.com/presets/egg-carton): A still sine profile in x multiplies a cosine that slides along y, so the surface forms a chessboard of hills and hollows that repeats as the clock runs. - [Gaussian Bump](https://www.wavelace.com/presets/gaussian-bump): One bell over the middle of the sheet, breathing. The height is e raised to −(x² + y²)/4, times cos t. - [Guitar String](https://www.wavelace.com/presets/guitar-string): Each row across the sheet is the string's shape at one instant, a single arch whose ends stay fixed while its crest swings between 3.5 and −3.5. - [Huygens Slit](https://www.wavelace.com/presets/huygens-slit): Every point across the slit acts as a wave source under Huygens' principle, summing into a beam that dims to zero near 52 degrees off axis. - [Huygens Slits](https://www.wavelace.com/presets/huygens-slits): Waves spreading from two narrow slits overlap to create bright and dark rays fanning outward from the pair, both above and below them. - [Laplace Transform](https://www.wavelace.com/presets/laplace-transform): The surface computes an integral at every point and rocks between the transform of a sine wave and of a cosine wave as the clock runs. - [Membrane Modes](https://www.wavelace.com/presets/membrane-modes): Six standing waves of shrinking weight combine into a checkerboard of humps and hollows that rocks in place and repeats every 6.28 seconds. - [Monkey Saddle](https://www.wavelace.com/presets/monkey-saddle): The cubic surface is scaled by a clock that flips it over and back every few seconds, while the point where every ridge meets every valley stays level. - [Paraboloid Family](https://www.wavelace.com/presets/paraboloid-family): One slider turns a bowl into a trough and then a saddle by changing the sign of a single term, the second derivative test drawn out. - [Radial Pond](https://www.wavelace.com/presets/radial-pond): The height depends only on distance from the centre, so the surface forms concentric rings. An envelope shrinks their swell as the rings spread outward. - [Ripple Tank](https://www.wavelace.com/presets/ripple-tank): Two point sources send out circular waves that overlap, reinforcing on some paths and cancelling on others, paths that are hyperbolas with the sources as foci. - [Saddle (elliptic)](https://www.wavelace.com/presets/saddle-elliptic): The height is x²/9 − y²/4: a rise along the x axis, a fall along the y axis, the plainest saddle there is. The two divisors set how fast each happens. - [Saddle Swirl](https://www.wavelace.com/presets/saddle-swirl): A saddle swelling and flattening is crossed by ridges on one diagonal and a wave sliding along the other, and the crossings seem to spin though nothing turns. - [Spinning Drum Mode](https://www.wavelace.com/presets/spinning-drum-mode): A circular drum vibrates with a hump on one side and a hollow on the other, and the whole pattern turns round the rim once every 6.28 seconds. - [Spiral Wave](https://www.wavelace.com/presets/spiral-wave): Three spiral arms rotate around a point where the wave's phase has no value at all. ## Curve · (x, y, z) = f(u, t) - [Breathing Ring](https://www.wavelace.com/presets/breathing-ring): A ring at distance 3 from the axis with a ripple laid over it, six waves to the turn, breathing as the clock runs. - [Euler Spiral](https://www.wavelace.com/presets/euler-spiral): A curve whose curvature grows with the distance travelled coils into a fixed point called an eye, and spins slowly about the origin as the clock runs. - [Fresnel Phasor](https://www.wavelace.com/presets/fresnel-phasor): The Fresnel integrals lifted along a third axis form a static coil, loose in the middle and tight at each end, read as a chain of diffraction phasor arrows. - [Helix](https://www.wavelace.com/presets/helix): The curve climbs at a constant angle, and its curvature and its twist through space both stay the same all the way along. - [Hypotrochoid 3/4](https://www.wavelace.com/presets/hypotrochoid-3-4): A circle rolls inside one a third larger and a pen on it draws four crossing lobes. Moving the pen out makes a four-pointed star, then a rounded square. - [Hypotrochoid 7/13](https://www.wavelace.com/presets/hypotrochoid-7-13): A small circle rolls inside a larger one and a pen on it draws a flower of thirteen petals. Moving the pen out turns them into a star, then loops. - [Hypotrochoid 7/5](https://www.wavelace.com/presets/hypotrochoid-7-5): A large circle rolls round a smaller one inside it, and a pen on it draws five overlapping lobes. Moving the pen out sharpens them into five cusps. - [Knot Family](https://www.wavelace.com/presets/knot-family): Two sliders set how often a strand winds around a ring and around its tube, which decides which knot it ties and how often it crosses itself. - [Lissajous Knot](https://www.wavelace.com/presets/lissajous-knot): All three coordinates are sines of the same u, at frequencies 3, 4 and 5 and amplitude 2.5. - [Living Lissajous](https://www.wavelace.com/presets/living-lissajous): The curve is a Lissajous figure, drawn by two perpendicular oscillations of different frequency. It repeats its cycle about every 0.571 seconds. - [Lyapunov Descent](https://www.wavelace.com/presets/lyapunov-descent): A damped oscillator's path lifted to the height of its energy spirals down a bowl to rest. Without damping it circles on one level. - [Magnetic Field Line](https://www.wavelace.com/presets/magnetic-field-line): One line of a fusion machine's magnetic field winds round a torus, turning a little round the tube each time it goes round the long way. - [Rabi Oscillation](https://www.wavelace.com/presets/rabi-oscillation): A spin in a magnetic field, driven by a weak turning field, spirals from up to down and back. Off resonance it only partly flips. - [Spherical Spiral](https://www.wavelace.com/presets/spherical-spiral): A curve loops from pole to pole eight times around a sphere, a pattern called a Clelia curve. - [Spirograph](https://www.wavelace.com/presets/spirograph): A pen inside a smaller circle rolling in a larger one traces five lobes that swell and shrink over time. - [Square Orbit](https://www.wavelace.com/presets/square-orbit): The path starts as a circle, gains a harmonic each second and sharpens toward a square standing on one corner, before snapping back to begin again. - [Tornado](https://www.wavelace.com/presets/tornado): A curve spirals ten times up a cone that widens from a point at the bottom to a wide rim at the top, turning slowly about its own axis. - [Torus Knot](https://www.wavelace.com/presets/torus-knot): The strand traces a trefoil knot on the surface of a torus, and the whole curve turns rigidly in place as the clock advances. - [Trefoil Knot](https://www.wavelace.com/presets/trefoil-knot): The strand combines two circular motions turning opposite ways. A third motion lifts it out of that plane, turning crossings into over and unders. - [Viviani's Curve](https://www.wavelace.com/presets/vivianis-curve): The curve lies where a sphere of radius 2 meets a cylinder touching it from inside. Its lobes were posed as a dome-window puzzle by Vincenzo Viviani in 1692. ## Shape · (x, y, z) = f(u, v, t) - [Breathing Sphere](https://www.wavelace.com/presets/breathing-sphere): The poles of a sphere stay fixed while its waist swells and narrows between 1.1 and 0.9 times its resting radius, briefly a true sphere twice a cycle. - [Einstein–Rosen Bridge](https://www.wavelace.com/presets/einstein-rosen-bridge): The curved space around a black hole at one instant, drawn as a surface: two sheets joined by a throat that closes before anything can cross it. - [Enneper Surface](https://www.wavelace.com/presets/enneper-surface): A minimal surface whose edge grows and shrinks. Once its radius passes the square root of three, its folds pass through each other. - [Heart](https://www.wavelace.com/presets/heart): The parameter u goes round the outline of a heart, and v sweeps that outline from the back of the figure to the front. - [Helicoid to Catenoid](https://www.wavelace.com/presets/helicoid-to-catenoid): A spiral ramp bends into a waisted tube and back without stretching, every stage a minimal surface like a soap film. It measures the same all the way. - [Horn Torus](https://www.wavelace.com/presets/horn-torus): The tube radius has grown until it equals the ring radius, so the hole has shut to a single point. One slider carries the surface from ring to spindle. - [Klein Bottle](https://www.wavelace.com/presets/klein-bottle): This is the figure-eight form of the Klein bottle. A tube of centre radius 2 runs round the vertical axis with u, and v runs round its cross-section. - [Kuen Surface](https://www.wavelace.com/presets/kuen-surface): A surface bent like a saddle by exactly the same amount at every point, as the pseudosphere is. It cannot stay smooth, so it folds along sharp creases. - [Möbius Strip](https://www.wavelace.com/presets/mobius-strip): A band sweeps once around a circle while twisting through 180 degrees, so its two edges swap places and one curve forms the whole boundary instead of two rims. - [Quadric Family](https://www.wavelace.com/presets/quadric-family): One slider reshapes a surface of revolution: a waisted hyperboloid, a double cone when the waist closes, then two separate bowls. - [Seashell](https://www.wavelace.com/presets/seashell): A tapering tube coils around a central axis and closes to a point, following the shell geometry Henry Moseley described in 1838. - [Superellipsoid](https://www.wavelace.com/presets/superellipsoid): A single exponent carries one surface through a rounded cube, an exact sphere, a regular octahedron and a star with six spikes. - [Supershape Flower](https://www.wavelace.com/presets/supershape-flower): A five-armed star from one formula, laid out in three thin layers by a steep second formula taken a whole turn: a flower of ten arms. - [Supershape Pagoda](https://www.wavelace.com/presets/supershape-pagoda): A round section turned about a profile that swings in and out six times: a superformula surface of revolution, stacked rims and a cup. - [Supershape Urchin](https://www.wavelace.com/presets/supershape-urchin): One formula taken around and from bottom to top, the two radii multiplied: sixteen spikes around, eight tiers high, a spiny urchin. - [Tangent Tube](https://www.wavelace.com/presets/tangent-tube): A tube of varying thickness wraps around a flat ellipse, swelling where the curve is crossed quickly and narrowing where it slows down. - [Torus](https://www.wavelace.com/presets/torus): The ring's centre circle has radius three. Its tube has radius one, so every point on the surface sits exactly that far from the centre circle. - [Twisted Torus](https://www.wavelace.com/presets/twisted-torus): The surface is an ordinary torus of centre radius 3. Its grid lines wind twice round the tube each lap, and pushing the mesh to its top stop removes the twist. ## Complex · w = f(z, t) - [Breathing Julia](https://www.wavelace.com/presets/breathing-julia): The filled Julia set is redrawn as its constant moves round a circle, so the shape swings between a single connected body and scattered dust. - [Burning Ship](https://www.wavelace.com/presets/burning-ship): The fractal reworks the Mandelbrot formula by folding each point positive before squaring, trading smooth spirals for straight edges and plumes of smoke. - [Complex Exp](https://www.wavelace.com/presets/complex-exp): Colour bands from the exponential's angle cross bands from its size, rising to a cliff on one side and flat on the other, and the plaid turns about the origin. - [Complex Sine](https://www.wavelace.com/presets/complex-sine): Three zeros sit spaced along the real axis, each a dip where the colour wheel turns once around, and the surface climbs into two rising walls either side. - [Julia Dials](https://www.wavelace.com/presets/julia-dials): Two sliders set the constant of a filled Julia set, so the picture can be walked from a single plump body to a dendrite to scattered dust. - [Mandelbrot](https://www.wavelace.com/presets/mandelbrot): Squaring a number and adding a constant, repeated over and over, sorts points into those that stay bounded or escape, with a crumpled boundary between them. - [Möbius Turn](https://www.wavelace.com/presets/mobius-turn): A dark dimple orbits a fixed bright spire on the sheet, which flattens to one level tone for an instant whenever the dimple meets the spire, then grows back. - [Multibrot](https://www.wavelace.com/presets/multibrot): One slider sets the power that is applied over and over, and each whole value turns the best known fractal into one with a different rotational symmetry. - [Newton Basins](https://www.wavelace.com/presets/newton-basins): Repeating one step of Newton's method for the cube roots of one sorts points into three colours, with a pit at the centre where nearby points shoot away. - [Newton Petals](https://www.wavelace.com/presets/newton-petals): Newton's method for the roots of one paints a basin around every answer, and a single slider sets how many roots there are and how many petals appear. - [Orbiting Poles](https://www.wavelace.com/presets/orbiting-poles): A map with two poles circling its centre is applied again and again, and the fractal edge between points falling inward and points drifting away turns. - [Rational Map](https://www.wavelace.com/presets/rational-map): The colouring turns anticlockwise at each zero of the fraction and backwards at each pole. An extra pole at infinity balances the extra degree of the numerator. - [Simple Pole](https://www.wavelace.com/presets/simple-pole): 1/z is the smallest example of a pole: one point where the function has no value and near which it grows without bound. - [z squared](https://www.wavelace.com/presets/z-squared): Squaring doubles the argument, so walking around the origin turns the colour wheel through two full cycles. ## Quantum · ψ under V(x) - [Barrier or Well](https://www.wavelace.com/presets/barrier-or-well): One slider lifts a rectangle of potential into a wall the packet must tunnel through, or sinks it into a hole that reflects the packet anyway. - [Double Well](https://www.wavelace.com/presets/double-well): A wave packet rests in one of two wells separated by a hill, and although its energy stays below the hill's top, it tunnels through until both wells share it. - [Free Packet](https://www.wavelace.com/presets/free-packet): A Gaussian hump drifts along a line with no potential in its way, spreading as it travels and leaving a faint tail behind before it meets a wall and bounces. - [Harmonic Well](https://www.wavelace.com/presets/harmonic-well): A wave packet released off centre in the potential swings back and forth and returns to its original shape instead of spreading out. - [Kronig–Penney](https://www.wavelace.com/presets/kronig-penney): A wave packet meets seven identical barriers spaced evenly along the axis, and interference between the reflections opens a gap where it mostly bounces back. - [Potential Step](https://www.wavelace.com/presets/potential-step): A wave packet hits a step in the potential and splits. Part reflects at full speed while part slows and crosses, so the transmitted share settles at 0.44. - [Quartic Well](https://www.wavelace.com/presets/quartic-well): A quantum packet sloshes in a well with stiffened walls, then smears out and does not reform, because its faster and slower parts drift out of step. - [Reflectionless Well](https://www.wavelace.com/presets/reflectionless-well): A hole in the potential deep enough to turn a fifth of the packet back, which at certain depths lets the whole packet pass as though it were not there. - [Smooth Step](https://www.wavelace.com/presets/smooth-step): A wave packet carries enough energy to clear this gentle rise in the potential, yet part of it still reflects. - [Squeezed Packet](https://www.wavelace.com/presets/squeezed-packet): A wave packet released off centre in a parabolic trap swings at a fixed rate while its width narrows and fattens twice as often. - [Tunnelling Barrier](https://www.wavelace.com/presets/tunnelling-barrier): A wave packet meets a barrier taller than its energy, yet 0.375 of it appears on the far side, the same leak by which alpha particles escape a nucleus. ## Quantum 2D · ψ under V(x, y) - [Double Slit](https://www.wavelace.com/presets/double-slit): A wave packet crosses a thin wall pierced by two openings and spreads into a broad lobe with a dark line each side, though much of it tunnels through the wall. - [Free Packet 2D](https://www.wavelace.com/presets/free-packet-2d): A Gaussian bump drifts across a flat plane with no potential in its way, spreading into a circle as it travels until it meets a wall and bounces back. - [Orbiting Packet](https://www.wavelace.com/presets/orbiting-packet): A quantum wave packet circles the centre of a harmonic bowl once every 6.28 seconds without spreading, its width holding steady like a coherent state. - [Scattering Disc](https://www.wavelace.com/presets/scattering-disc): A wave packet aimed above a disc wraps around it, leaving a shadow that diffraction fills in. About a third of the packet is thrown back the way it came. - [Single Slit](https://www.wavelace.com/presets/single-slit): A wave packet passes through a narrow gap in a wall and spreads into a broad fan instead of a narrow beam. - [Soft Corral](https://www.wavelace.com/presets/soft-corral): A circular wall traps a wave packet on a flat floor, where it spreads out, reflects and briefly refocuses. ## Flow · (ẋ, ẏ, ż) = f(x, y, z, t) - [Aizawa Attractor](https://www.wavelace.com/presets/aizawa-attractor): Specks rise through a tube on the axis, spill out over a round shell and fall back, never twice the same way. Made symmetric, the chaos goes. - [Burgers Vortex](https://www.wavelace.com/presets/burgers-vortex): An exact steady flow of a viscous fluid: specks drawn in along a plane spiral out along a narrow core, whose width the viscosity sets. - [Cubic Spiral Sink](https://www.wavelace.com/presets/cubic-spiral-sink): Seeds spiral inward around a point that looks like a centre once linearised, but the cubic terms decide the answer, and no orbit quite arrives at the middle. - [Four-Wing Attractor](https://www.wavelace.com/presets/four-wing-attractor): A chaotic flow from 2009 whose seeds wander between four wings, two above the ground and two below, never settling and never repeating. - [Gradient Flow](https://www.wavelace.com/presets/gradient-flow): A landscape of hills and hollows flips every 6.28 seconds, so seeds sliding downhill toward a basin suddenly find themselves on a summit and slide again. - [Hopf Bifurcation](https://www.wavelace.com/presets/hopf-bifurcation): One number decides the picture. Below zero every thread winds into the middle, and above it they all settle onto a ring that widens as the number rises. - [Lorenz Attractor](https://www.wavelace.com/presets/lorenz-attractor): These are Lorenz's equations of 1963, written with his y and z exchanged, because the app is y-up and the textbook writes them z-up. - [Rössler Attractor](https://www.wavelace.com/presets/rossler-attractor): The plot traces Otto Rössler's 1976 chaotic system, a looping spiral that stretches outward before folding back near the centre and starting again. - [Thomas Attractor](https://www.wavelace.com/presets/thomas-attractor): A seed is caught by one of twenty seven rest points, wound around it and flung towards the next, in what its discoverer called labyrinth chaos. - [Van der Pol](https://www.wavelace.com/presets/van-der-pol): A vacuum tube oscillation that always settles at the same width, while one slider decides whether it glides around smoothly or creeps and jumps. - [Whirlpool](https://www.wavelace.com/presets/whirlpool): Three linear flow lines spiral seeds inward while their height decays faster, so they flatten onto the floor before winding into the centre. ## Swarm · f(r) between bodies - [Driven Springs](https://www.wavelace.com/presets/driven-springs): Every pair of bodies is joined by a spring whose natural length rises and falls in time, so the whole cloud swells and shrinks together on the beat. - [Gravity Disc](https://www.wavelace.com/presets/gravity-disc): Forty masses pulling by an inverse square law spin as one disc, until close slingshot encounters scatter pairs outward, the way real star clusters evaporate. - [Molecules](https://www.wavelace.com/presets/molecules): Thirty bodies pulled together by a Morse force fall into a knot, rebound, and settle into a ragged droplet held together indefinitely. - [Springs](https://www.wavelace.com/presets/springs): Twelve bodies, each pulled or pushed by a spring to every other, swell and shrink together as one breathing cloud that turns slowly on the spot. ## Bodies · mechanical systems - [Bead String](https://www.wavelace.com/presets/bead-string): Five beads on a stretched chain of springs between two walls, one plucked sideways. The pulse runs down the chain and back; a slider sets the tension. - [Coupled Oscillators](https://www.wavelace.com/presets/coupled-oscillators): Two masses on springs between walls, joined by a weaker spring. Set one swinging and the motion walks across to the other and back; a slider sets the coupling. - [Double Pendulum](https://www.wavelace.com/presets/double-pendulum): Two hanging rods swing chaotically from a wide starting angle, and a tiny difference in that angle quickly grows into total disagreement. - [Elastic Chaos](https://www.wavelace.com/presets/elastic-chaos): A weight on a spring released nearly upright swings and stretches unpredictably. A slider sets the starting angle, and runs a hair apart soon disagree. - [Figure-8](https://www.wavelace.com/presets/figure-8): Three equal masses chase each other around one closed curve shaped like a flattened eight, each a third of a period behind the last. - [Hierarchical Triple](https://www.wavelace.com/presets/hierarchical-triple): Two close masses orbit each other quickly while a third circles the pair far more slowly, a difference that keeps the group stable. - [Horseshoe Orbit](https://www.wavelace.com/presets/horseshoe-orbit): A particle near a planet's orbit drifts round and turns back each time it nears the planet. A slider sets how far out it starts; past a threshold it slips by. - [Lagrange Points](https://www.wavelace.com/presets/lagrange-points): Five particles parked where two orbiting bodies' pulls balance. Three drift off at once, two hold; a slider sets the mass ratio past which those two let go. - [Lagrange Triangle](https://www.wavelace.com/presets/lagrange-triangle): Three equal masses at the corners of a triangle spin together as one rigid shape, an exact solution Lagrange found in 1772 that breaks apart after a few turns. - [Ring Trap](https://www.wavelace.com/presets/ring-trap): Eight fixed charges on a ring push a light charge of the same sign back whenever it nears them. A slider sets its speed; past a threshold it slips out. - [Roche Lobe](https://www.wavelace.com/presets/roche-lobe): A particle orbits the heavier of two turning bodies. A slider sets its speed; past a threshold the neck between the two opens and it slips across to the other. - [Rutherford Scattering](https://www.wavelace.com/presets/rutherford-scattering): A light charge fired past a heavy one of the same sign bends away on a hyperbola. A slider sets how close it aims, and the angle follows Rutherford's formula. - [Spring Pendulum](https://www.wavelace.com/presets/spring-pendulum): A weight on a spring bounces and swings. Tuned so one swing takes two bounces, the bounce feeds the swing and takes it back every ten seconds. - [Sun and Planets](https://www.wavelace.com/presets/sun-and-planets): Two lightweight planets trace clean circles around a central star, their periods related by Kepler's third law of planetary motion. - [Triple Pendulum](https://www.wavelace.com/presets/triple-pendulum): Three equal rods hang from a pivot in a chain, released together from rest at 114.6°. Two runs a hair apart soon swing completely out of step. ## Bellman · v(s) = maxₐ [R(s, a) + γ v(s′)] - [Bellman 3×3](https://www.wavelace.com/presets/bellman-3-3): A wall and two terminal rewards sit on a 3 by 3 grid whose cells rise to their value under Bellman's equation, settling after four sweeps. - [Cliff Walk](https://www.wavelace.com/presets/cliff-walk): A seven by four grid holds a start and a goal on a cliff row of five −100 terminals, and nine sweeps of value iteration trace the walk that hugs the edge. - [Maze](https://www.wavelace.com/presets/maze): A seven-by-seven maze. The start S is in the top-left corner, the goal, a terminal worth +1, in the bottom-right, and the walls are the # cells. - [Russell–Norvig 4×3](https://www.wavelace.com/presets/russell-norvig-4-3): A terminal worth +1 sits in one corner. A terminal worth −1 sits below it. Slipping sends the optimal route the long way round the pit.