Discover self-contained mathematical algorithms rendered in real time on HTML Canvas. Inspect the formulas, remix parameters, and copy the code for your own projects.
Homage to Hamid Naderi Yeganeh
A(k) = 3k/20000 + sin(0.5πu^7)cos^6(41πu) + 0.25cos^16(41πu)cos^12(0.5πu)sin(6πu), B(k) = -cos(0.5πu^7)(1 + 1.5cos^6(0.5πu)cos^6(1.5πu))cos^6(41πu) + 0.5cos^10(0.3πu)cos^10(0.9πu)cos^10(1.8πu), R(k) = 0.02 + 0.1sin^2(41πu)sin^2(0.9πu) + 0.05cos^2(41πu)cos^10(0.5πu) Homage to Dominikus Baur (@dominikus)
r(u,v) = ((R + r cos v)cos u, (R + r cos v)sin u, r sin v) Terminal velocity descent on twin light rails
x = xc ± (w_void + dist * u^2.2), y -= v * dt Monochrome optical thought loop and impossible Escher staircase
r(s, t) = R * ((s/N + t/T) % 1)^1.6, θ = 4π * norm Force-directed semantic network with data packet propagation
F_i = sum(k_r / d^2) - sum(k_s * (d - d0)) - μ * v Infinite psychedelic sacred geometry hyperspace
r = R * z^2.2 * (1 + α*cos(N*θ)) Volumetric oceanic caustic web with chromatic dispersion
I = sum(sin(k_i · x + ω_i * t + δ(λ))) Isometric sinusoidal hexahedral lattice
Z = A * sin(ω*t - k * hypot(x, y)) Infinite spiraling character matrix
f(z) = [R cos(θ + ρ), R sin(θ + ρ)] * fov / z Leonardo da Vinci Branching & Moyogi Curvature
B(u) = (1-u)^3 P_0 + 3(1-u)^2 u P_1 + 3(1-u) u^2 P_2 + u^3 P_3, \quad w(u) = w_0 (1 - 0.78u)^{1.4} Volumetric Moonbeams, Silver Water Wake & Bioluminescent Waves
I_lunar(r, θ) = I_0 exp(-σr) P(θ) · [Σ A_k cos(k θ + ωt)], wake = exp(-Δx²/2w²) · (N · L)^m Magnetohydrodynamic Plasma Loops & Coronal Mass Ejection
B_dipole = μ_0 m / (4π r³) · (2 cos θ r_hat + sin θ θ_hat), I_corona(r) ∝ r^(-2.5) · pulse(t) Henyey-Greenstein Scattering & Cumulus Cloudburst
P_HG(cos θ) = (1 - g²) / [4π (1 + g² - 2g cos θ)^(3/2)], I(beam) = I_0 · P_HG(θ) · [1 - O_cloud] Snell's Law Refraction, Surface Caustics & Marine Cathedral Light
n_air sin(θ_i) = n_water sin(θ_t), I(d) = I_0 exp(-αd) · wave_caustics(x, t) Mie Volumetric Scattering & Ocean Wave Glitter Path
I(r,θ) = I_0 exp(-σr) P(θ) · [Σ A_k cos(f_k θ + ω_k t)]^p, glitter = exp(-Δx²/2w²) · (N · L)^m Pterois Radiata Spines & Fan Pectoral Kinematics
spine = line(root, angle, len) + wave(t), pectoral_fan = 10_rays(angle, t) Disciform Geometry & Neon Striations
disc_body = ellipse(r_x=0.52, r_y=0.54), heckel_bars = vertical(b) Carangiform Serpentine Kinematics & Kohaku Patterns
spine = sin(3t - s*2.4)*amp(s), skin = ivory + kohaku_patches(s) Volumetric Veil Fins & Hydrodynamic Undulation
spine = (s/N)^1.5 * amp * sin(ks - ωt), fin_veil = cubicBezier(p0, cp1, cp2, p3) Counter-Phase 4-Wing Flight & Slender Abdomen
forewing = sin(8t), hindwing = sin(8t - π/2), abdomen = 10_segments Apis Mellifera Kinematics & Hexagonal Honeycomb Matrix
wing_stroke = rotate(-π/2 + sin(18t)*0.35), honeycomb = regularHexagon(r) Superellipse Chitinous Elytra & Articulated Legs
elytra = superellipse(w, h, p=2.4), leg = coxa -> femur(t) -> tibia(t) Transcendental Polar Curve & Iridescent Wing Venation
r = exp(sin θ) - 2*cos(4θ) + sin^5((2θ - π)/24), x = sin(θ)*r*flap_x Vogel Model Seed Packing & Golden Ray Petals
θ = n * 137.508°, r = c * sqrt(n), floret = [cos(θ)*r, sin(θ)*r] Zygomorphic Bilateral Floral Symmetry & Neon Spurs
dorsal = bezier(base, ctrl, tip), lip = bezier(base, cyan_ctrl, nectar_spur) Paul Bourke High-Harmonic Foliation Curve
r = 5*(1 + sin(11θ/5)) - 4*sin^4(17θ/3)*sin^8(9θ - π/2) Translucent Multi-Tier Petal Foliation & Golden Pod
petal = cubicBezier(base, ctrl1, tip, ctrl2), whorl_offset = w*π/N Harmonic Polar Rose Curves & Crystalline Chords
r = R*sin(k*θ + φ), chord_point = [r*cos(i*d°), r*sin(i*d°)] Acanthus Leaf Scrolls & Coiling Golden Buds
stem = [cos(0.6ku)*r + warp, sin(0.6ku)*r], bud = tip + [cos(-θ)*r_bud, sin(-θ)*r_bud] Moorish Girih Inlaid Gold & Emerald Strapwork
ctrl = (p1 + p2)*0.5 + norm(a2 ± π/2)*offset, curve = quadraticBezier(p1, ctrl, p2) Rose Engine Cycloid Banknote Tracery
x = (R-r)*cos(θ) + d*cos(k*θ + φ), y = (R-r)*sin(θ) - d*sin(k*θ + φ) Logarithmic Spiral Volutes & Granulated Lace
r = r_0 + (R-r_0)*u^0.88, volute = [cos(0.45θ)*r + sin(θ)*offset, sin(0.45θ)*r] Einstein Field Equations & Gravitational Lensing Deflection
z(r) = 2 * Math.sqrt(rs * (r - rs)); defAngle = (4 * GM) / (c^2 * b); Orthogonal E & B Vector Fields & Poynting Energy Flow
E(z,t) = E_0 sin(kz - wt) j_hat; B(z,t) = B_0 sin(kz - wt) k_hat; Complex Phasor Harmonic Decomposition & Spectrum Analyzer
z(t) = sum(c_k * exp(i * k * omega * t)) Lagrangian Mechanics & Chaotic Phase Space Orbit
alpha1 = num1(th1,th2,w1,w2)/den1; alpha2 = num2(th1,th2,w1,w2)/den2; Non-Linear Dynamical Systems & Butterfly Chaos Theory
dx=s*(y-x)*dt; dy=(x*(r-z)-y)*dt; dz=(x*y-b*z)*dt; Schrödinger Wavefunction Probability Density Cloud & Spherical Harmonics
P = Math.pow(r*n, l) * Math.exp(-r*2) * Math.abs(Y_lm(theta, phi)) Cephalofoil Hydrodynamics & Serpentine Swimming Wave
spine_wave = (x/L)^1.6 * 36 * sin(3.2t - 4*(x/L)), head_t = quadraticCurve(-45, 0, -15, headSpan) Prehensile Tail Spiral & Foliaceous Camouflage
tail = tail_root + [35 - cos(θ)*r, 25 + sin(θ)*r], r = 55 * exp(-0.45*θ), leaf_sway = sin(2.5t + phase)*14 10-Joint Chebyshev Linkage Deep Sea Stride
knee = coxa + [75*reach, -80*reach + lift], tip = elbow + [35*reach + groundReach, 110*reach] Benthic Fin Stilts & Current Sensing
stilt_fin = quadCurve(pelvic_root, groundContact), prey_sensor = bezierCurve(pectoral_root, wave_tips) Gymnosome Parapodia Wings & Visceral Core
wing = bezierCurve(root, wingCurl*18, wingTip + wingWave*28), visc_core = radialGradient(5, 2, 22) Transparent Cranial Shield & Tubular Green Eyes
dome = ellipse(35, -25, 48, 34), tubular_eye = rotate(-π/2 + sin(1.5t)*0.35, green_lens) Distensible Pelican Jaw & Luminous Whip-Tail
pouch = bezierCurve(top_jaw, 95 + 85*pulse, -10 + 110*pulse, tail_root), tail_organ = radialGradient(tip, 24*pulse) Ear-Fin Flapping Kinematics & Cirrate Skirt
ear_flap = bezierCurve(root, earCurl*15, earTip + earFlap*26), skirt = quadCurve(arm_prev, midPt - 10, arm_cur) Velvety Webbed Cloak & Arm Tip Photophores
web_arch = quadCurve(armTip_a, midPt, armTip_next), photophore = circle(tip, 3.5*(1 + 0.3*sin(4t + a))) Metachronal Ctenes & Rainbow Light Diffraction
ctene_wave = sin(4t - 8s + 0.5r), hue_diffract = (360s + wave*40) % 360 Colonial Zooid Helical Chain & Bioluminescent Net
stem = [s*L + sin(1.2t + 4s)*25, sin(4πs + 0.8t)*R], tentilla = [stem.x + sin(2.5t - 6u)*15u, stem.y + u*H] Bathypelagic Esca Photophore & Needle Jaw Array
esca = root + [85 + sin(2.5t)*22, -95 + cos(2t)*16], glow = radialGradient(esca, 12 * (1 + 0.3*sin(4t))) Hexagonal Dendritic Stefan Phase Freezing
main_spine = L*(0.3 + 0.7*growth), prong = spine_pos + [cos(±π/3)*len, sin(±π/3)*len] Brownian Drift Flutter & Parallax Crystals
drift = sin(phase)*1.2 + wind*r*0.8, fy += speed*dt, crystallite = 6_arm_dendrite(r) Terminal Velocity Vectors & Impact Ripples
y += speed * dt, x += speed * sin(wind) * dt, ripple = ellipse(x, y, r, 0.35r) * exp(-γt) Capillary Crown Elevation & Rayleigh-Plateau Jet
crown_h = sin(age * π) * 110, jet_h = sin(jetAge * π) * 160, drop_y += vy*dt + 0.5*g*dt^2 Rankine Combined Vortex Dynamics & Updraft Physics
v_theta = speed / (0.3 + r*0.7), R_funnel = 20 + z^2.2 * 170, sway = sin(1.5t + 3z)*40z Pupillary Sphincter & Fractal Collagen Meshwork
pupil_r = R * (0.28 + 0.12*sin(1.5t)), trabecula = quadCurve(pupil, furrow, limbus) 3D Revolving Molecular Helical Geometry
strand1 = [R cos(4πy + 2t), y], strand2 = [R cos(4πy + π + 2t), y], base_rung = line(s1, s2) Hodgkin-Huxley Action Potential Transmission
spike = (timer % 2.5) < 0.25, pulse(px, py) = soma_i + (soma_j - soma_i) * (2t % 1) Electrocardiogram Fourier Waveform & Vascular Rhythm
ecg(t) = P(0.18) + Q(-0.15) + R(1.0) + S(-0.3) + T(0.35), heart_scale = 1 + 0.22*ecg Differential Growth Curve & Radial Polyp Bloom
r_ridge = R0 + 12 sin(6θ + 0.5t) + 6 cos(12θ - t), tentacle = quadCurve(mouth, ctrlWave, tip) Trochoidal Deep Water Wave Superposition
x = x0 - (dx/k)*Q*sin(phase), y = y0 - A*cos(phase), phase = k*(dx*x0 + dy*y0) - ωt Iterated Function System Botanical Attractor
nextX = 0.85x + 0.04y, nextY = -0.04x + 0.85y + 1.6 (p = 85%) Recursive Lindenmayer Botanical Branching
branch(x, y, len * ratio, angle ± Δθ + wind, depth - 1) Equiangular Logarithmic Chamber Geometry
r = a * exp(0.1759 * θ), septum_arch = quadCurve(inPoint, outPoint) Traveling Wave Pectoral Wing Hydrodynamics
wing_z = A(x)*sin(kx - ωt)*exp(-y²/σ²), tail = sin(3t - 4s)*20s Segmented Pleon Arching & Pleopod Propulsion
antenna = -s*L - s^1.4*70 + sin(3t - 5s)*18s, pleopod = sin(6t - 0.8s)*0.6 Bilateral Crustacean Gait Kinematics
knee = coxa + [cos(θ)*45 + reach, -20 + lift], pinch = 0.35 + 0.3*sin(3t) Harmonic Umbrella Kinematics & Phosphor Pulse
r_dome = R*(1 + 0.22*sin(3t)), y_tentacle = y0 + s + sin(3t - 6s + k)*18s Complex Dynamic Polynomial Iteration
z = z² + (-0.8 + 0.15 e^(iωt)), iter_escape = min(k | |z_k| > 2) SmoothLife Multi-Kernel Continuous Automata
avg = (1/8) * Σ neighbors, s_{t+1} = (0.25 <= avg <= 0.45) ? s + 0.08 : s - 0.05 Lin-Shu Astrophysical Density Wave Theory
armPhase = (1/pitch) * ln(r/10) - 0.4t, r_eff = r * (1 - 0.12 * sin(m*(θ - armPhase))) Conic Section Gravitational Ephemerides
E_{k+1} = E_k - (E_k - e*sin(E_k) - M) / (1 - e*cos(E_k)), x = a*cos(E) - ae Schwarzschild Metric Relativistic Deflection
v_orb = 2.5 / √r, y_warp = y - (1.5*rs / d)*22, doppler_hue = 190 + 30*sin(θ) Bessel Multi-Source Wave Interference
w = Σ (sin(dist_i * freq - 2t) / (sqrt(dist_i)*0.1 + 1)), r_point = 0.5*(w + 1) Korteweg-de Vries Non-Linear Wave Dynamics
u(x, t) = 2k² * sech²(k(x - 4k² t)), total_u = u1(x, t) + u2(x, -t) 2D Chladni Resonance Nodal Lines
w(x, y) = sin(nπx)*sin(mπy) - sin(mπx)*sin(nπy), p -= ∇|w| * dt Epicycle Series Waveform Synthesis
r_k = (4 / kπ) * R, x += r_k cos(k t), y += r_k sin(k t), trace(y) Recursive Golden Triangle Aperiodic Tiling
p4 = p1 + (p2 - p1) / φ, subdivide(RobinsonTriangle(depth - 1)) Superimposed Spatial Frequency Grating
fringe = arc(c1, r) + arc(c2(t), r), beat_frequency = |k1 - k2| Polar Symmetry Modulo Geometry
rotate(2πk / N), drawEllipse(0, r, w * (1 + 0.2 sin(2t + r)), h, rotOffset) Conformal Disk Hyperbolic Geometry
r_hyp = R * tanh(l * 0.25 + 0.1 sin(t)), θ = 2πi / p*l + t/l Craig Reynolds Flocking Algorithm
v += align * 0.05 + cohere * 0.005 + separate * 0.15, p += clamp(v, maxSpeed) Dynamic Euclidean Distance Graph
d = dist(p_i, p_j), if (d < R) drawLine(p_i, p_j, alpha = 1 - d/R) 3D Projected Harmonic Knot Curves
x = sin(3φ + δ), y = sin(4φ), z = cos(5φ + t), px = cx + rotX * depth Binary Core N-Body Gravity
a_x = G * dx / (dx² + dy² + ε²)^(3/2), a_y = G * dy / (dx² + dy² + ε²)^(3/2) Stommel-Munk Oceanic Stream Function
u = 2π sin(πx) cos(2πy), v = -π cos(πx) sin(2πy) Discrete Eulerian Advection
ρ_{x,y}^{t+1} = ρ_{x,y}^t * 0.94 + sin(r*0.4 - 3t)*cos(4θ + 2t)*exp(-0.1r)*0.15 Divergence-Free Solenoidal Flow
vx = (N(x, y+ε) - N(x, y-ε)) / 2ε, vy = -(N(x+ε, y) - N(x-ε, y)) / 2ε Biot-Savart Point Vortex Dynamics
v_θ(r) = (Γ / 2πr) * (1 - exp(-r² / r_core²)) 2D Fractal Brownian Flow
angle = fbm(x * 0.003, y * 0.003 + t * 0.15) * 4π, v = [cos(angle), sin(angle)] Gielis Morphological Symmetry
r = (|cos(m*φ/4)/a|^n2 + |sin(m*φ/4)/b|^n3)^(-1/n1) Golden Ratio Plant Morphogenesis
θ = n * 137.508°, r = c * √n, x = r * cos(θ + t), y = r * sin(θ + t) Sinusoidal Deformation Field
y = sin(x * 0.02 + t) * cos(y * 0.01 + t * 0.5) * 35 + sin(dist(x, y) * 0.05 - t * 2) * 20