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FRACTURED CONNECTION TENSION OVERWHELM GRIEF GROWTH CYCLES CONFUSION SHAME SOLITUDE NOSTALGIA DESIRE MELANCHOLY SURRENDER WONDER PEACE AWE ENVY RAGE JOY RESILIENCE TRUST EUPHORIA PRIDE ANTICIPATION LONGING

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$Bifurcation — x_{n+1} = rx_n(1 - x_n)$

Bifurcation

x_{n+1} = rx_n(1 - x_n)

Catastrophe Fold — V(x) = x⁴ + ax² + bx

Catastrophe Fold

V(x) = x⁴ + ax² + bx

Glass Fracture — K_I = σ√(πa)

Glass Fracture

K_I = σ√(πa)

Seismic Fault — log₁₀(N) = a - bM

Seismic Fault

log₁₀(N) = a - bM

Voronoi Shatter — V(p) = {x : d(x,p) ≤ d(x,q) ∀q}

Voronoi Shatter

V(p) = {x : d(x,p) ≤ d(x,q) ∀q}

Coupled Oscillators — ẍ₁ = -ω₁²x₁ + κ(x₂ - x₁)

Coupled Oscillators

ẍ₁ = -ω₁²x₁ + κ(x₂ - x₁)

Double Helix — r(t) = (cos t, sin t, t/2π) ± d/2

Double Helix

r(t) = (cos t, sin t, t/2π) ± d/2

Phase Sync — dθ/dt = ω + K sin(θ₂ - θ₁)

Phase Sync

dθ/dt = ω + K sin(θ₂ - θ₁)

Torus Knot — r(t) = ((R + r cos qt) cos pt, ...)

Torus Knot

r(t) = ((R + r cos qt) cos pt, ...)

Two Body — F = -Gm₁m₂/r² · r̂

Two Body

F = -Gm₁m₂/r² · r̂

Buckling — P_cr = π²EI / (KL)²

Buckling

P_cr = π²EI / (KL)²

Fracture — σ = Eε (until σ > σ_y)

Fracture

σ = Eε (until σ > σ_y)

Interference — A(x) = A₁sin(k₁x) + A₂sin(k₂x)

Interference

A(x) = A₁sin(k₁x) + A₂sin(k₂x)

Opposition — F_net = F₁ - F₂ = 0, |F₁| > 0

Opposition

F_net = F₁ - F₂ = 0, |F₁| > 0

Torsion — τ = Tr/J

Torsion

τ = Tr/J

Attractors — dx/dt = σ(y-x), dy/dt = x(ρ-z)-y

Attractors

dx/dt = σ(y-x), dy/dt = x(ρ-z)-y

Kuramoto — dθᵢ/dt = ωᵢ + (K/N)Σsin(θⱼ - θᵢ)

Kuramoto

dθᵢ/dt = ωᵢ + (K/N)Σsin(θⱼ - θᵢ)

$Levy Swarm — P(x) ~ |x|^{-1-α}, 0 < α < 2$

Levy Swarm

P(x) ~ |x|^{-1-α}, 0 < α < 2

$Phase Flood — ψ(x,t) = Σ Aₙ e^{i(kₙx - ωₙt + φₙ)}$

Phase Flood

ψ(x,t) = Σ Aₙ e^{i(kₙx - ωₙt + φₙ)}

Turbulence — Re = ρvL/μ >> Re_cr

Turbulence

Re = ρvL/μ >> Re_cr

Heat Diffusion — ∂u/∂t = α∇²u

Heat Diffusion

∂u/∂t = α∇²u

Heaviside Cascade — H(t - t₀) = {0, t < t₀; 1, t ≥ t₀}

Heaviside Cascade

H(t - t₀) = {0, t < t₀; 1, t ≥ t₀}

$Spectral Erosion — S(f,t) = S₀(f) · e^{-γ(f)t}$

Spectral Erosion

S(f,t) = S₀(f) · e^{-γ(f)t}

Void — ∫∫ ρ(x,y) dA → 0

Void

∫∫ ρ(x,y) dA → 0

$Weight — F = mg, m(t) = m₀(1 - e^{-t/τ})$

Weight

F = mg, m(t) = m₀(1 - e^{-t/τ})

$Bifurcation — x_{n+1} = rx_n(1 - x_n), r ∈ [2.5, 4]$

Bifurcation

x_{n+1} = rx_n(1 - x_n), r ∈ [2.5, 4]

Dendrite — L → F[+L][-L]FL

Dendrite

L → F[+L][-L]FL

Lissajous Bloom — x = A sin(at + δ), y = B sin(bt)

Lissajous Bloom

x = A sin(at + δ), y = B sin(bt)

Logistic Cascade — dN/dt = rN(1 - N/K)

Logistic Cascade

dN/dt = rN(1 - N/K)

Reaction Diffusion — ∂u/∂t = Dᵤ∇²u + f(u,v)

Reaction Diffusion

∂u/∂t = Dᵤ∇²u + f(u,v)

Loom — x(t) = x(t + T), T = 2π/ω

Loom

x(t) = x(t + T), T = 2π/ω

Moebius — r(t) = ((2 + cos(t/2))cos t, (2 + cos(t/2))sin t, sin(t/2))

Moebius

r(t) = ((2 + cos(t/2))cos t, (2 + cos(t/2))sin t, sin(t/2))

Orbit — r(θ) = a(1-e²)/(1 + e cos θ)

Orbit

r(θ) = a(1-e²)/(1 + e cos θ)

Phase Portrait — ẋ = f(x,y), ẏ = g(x,y)

Phase Portrait

ẋ = f(x,y), ẏ = g(x,y)

Recurrence — R(i,j) = Θ(ε - ||x_i - x_j||)

Recurrence

R(i,j) = Θ(ε - ||x_i - x_j||)

Aliased — f_alias = |f - n·f_s|, n = round(f/f_s)

Aliased

f_alias = |f - n·f_s|, n = round(f/f_s)

Knot — K: S¹ → S³

Knot

K: S¹ → S³

Labyrinth — maze(x,y) = {0,1} | ∃! path(start, end)

Labyrinth

maze(x,y) = {0,1} | ∃! path(start, end)

Tangle — γ(t): [0,1] → R³, self-intersecting

Tangle

γ(t): [0,1] → R³, self-intersecting

$Vertigo — r(t) = e^{-at}(cos ωt, sin ωt, t)$

Vertigo

r(t) = e^{-at}(cos ωt, sin ωt, t)

Contraction — T(x) : ||T(x)-T(y)|| < ||x-y||

Contraction

T(x) : ||T(x)-T(y)|| < ||x-y||

Crumple — κ(s) → ∞ at fold lines

Crumple

κ(s) → ∞ at fold lines

Fold — f(x) = f(−x), x → 0

Fold

f(x) = f(−x), x → 0

$Shrink — A(t) = A₀ · e^{-λt}$

Shrink

A(t) = A₀ · e^{-λt}

Veil — f(x) = f(x) · (1 - g(x))

Veil

f(x) = f(x) · (1 - g(x))

Echo — f(t - τ) · α^n, α < 1

Echo

f(t - τ) · α^n, α < 1

Footprints — δ(x - x_n), n = 1,2,...,N

Footprints

δ(x - x_n), n = 1,2,...,N

Island — Ω = {x : f(x) > 0} ⊂ R², |∂Ω| < ∞

Island

Ω = {x : f(x) > 0} ⊂ R², |∂Ω| < ∞

Lighthouse — I(r) = I₀/r² · rect(θ/Δθ)

Lighthouse

I(r) = I₀/r² · rect(θ/Δθ)

Wanderer — x(t) = x₀ + ∫₀ᵗ v(s)ds, v random

Wanderer

x(t) = x₀ + ∫₀ᵗ v(s)ds, v random

$Carousel — r(t) = R · e^{iωt}, ω → 0$

Carousel

r(t) = R · e^{iωt}, ω → 0

$Daguerreotype — I(x,y,t) = I₀(x,y) · e^{-t/τ} + η(x,y)$

Daguerreotype

I(x,y,t) = I₀(x,y) · e^{-t/τ} + η(x,y)

Locket — f(x) = f₀(x) * g(σ), σ → ∞

Locket

f(x) = f₀(x) * g(σ), σ → ∞

$Music Box — x(t) = Σ Aₙe^{-γₙt}sin(nωt)$

Music Box

x(t) = Σ Aₙe^{-γₙt}sin(nωt)

$Remnant — R(x) = lim_{t→∞} f(x,t)$

Remnant

R(x) = lim_{t→∞} f(x,t)

Eclipse — I(r) = I₀ · (1 - disk(r/R))

Eclipse

I(r) = I₀ · (1 - disk(r/R))

Flame — ∂T/∂t = α∇²T + Q(x,y)

Flame

∂T/∂t = α∇²T + Q(x,y)

Hunger — dN/dt = rN(1 - N/K) - aN/(1+bN)

Hunger

dN/dt = rN(1 - N/K) - aN/(1+bN)

$Inspiral — r(t) = r₀(1 - t/t_c)^{1/4}$

Inspiral

r(t) = r₀(1 - t/t_c)^{1/4}

Pursuit — dr/dt = v₁ · (r_target - r)/||r_target - r||

Pursuit

dr/dt = v₁ · (r_target - r)/||r_target - r||

$Dissolve — c(t) = c₀ · e^{-kt}$

Dissolve

c(t) = c₀ · e^{-kt}

Drift — dx = μdt + σdW

Drift

dx = μdt + σdW

$Elegy — f(t) = A · t^{-α}, α ∈ (0,1)$

Elegy

f(t) = A · t^{-α}, α ∈ (0,1)

Entropy — S = -k Σ pᵢ ln pᵢ, dS/dt ≥ 0

Entropy

S = -k Σ pᵢ ln pᵢ, dS/dt ≥ 0

Lethe — ∂c/∂t + v·∇c = D∇²c

Lethe

∂c/∂t + v·∇c = D∇²c

Dissolution — dm/dt = -kA(c_s - c)

Dissolution

dm/dt = -kA(c_s - c)

Flow — ∇·v = 0, Re << 1

Flow

∇·v = 0, Re << 1

Melt — ρL(ds/dt) = k(∂T/∂n)

Melt

ρL(ds/dt) = k(∂T/∂n)

Settle — v_t = (2r²(ρ_p - ρ_f)g)/(9μ)

Settle

v_t = (2r²(ρ_p - ρ_f)g)/(9μ)

Shed — f(t) = f₀ · H(t₀ - t)

Shed

f(t) = f₀ · H(t₀ - t)

Apollonian Gasket — (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

Apollonian Gasket

(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

$Harmonograph — x = A₁sin(f₁t+φ₁)e^{-d₁t} + A₂sin(f₂t+φ₂)e^{-d₂t}$

Harmonograph

x = A₁sin(f₁t+φ₁)e^{-d₁t} + A₂sin(f₂t+φ₂)e^{-d₂t}

$Mandelbrot Orbit — z_{n+1} = z_n² + c$

Mandelbrot Orbit

z_{n+1} = z_n² + c

Recursion — f(n) = f(f(n-1))

Recursion

f(n) = f(f(n-1))

Strange Attractor — dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz

Strange Attractor

dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz

Cloud — ∂ρ/∂t = 0, ∇·(ρv) = 0

Cloud

∂ρ/∂t = 0, ∇·(ρv) = 0

Resolved — ∇²f = 0

Resolved

∇²f = 0

Sand — ∂h/∂t = ν∇²h - λ(∇h)² + η

Sand

∂h/∂t = ν∇²h - λ(∇h)² + η

Settled — v(t) → 0, x(t) → x_eq

Settled

v(t) → 0, x(t) → x_eq

Zen Garden — ψ = A · sin(kx) · sin(ky)

Zen Garden

ψ = A · sin(kx) · sin(ky)

Cathedral — h(x) = Σ Aₙ sin(nπx/L), n → ∞

Cathedral

h(x) = Σ Aₙ sin(nπx/L), n → ∞

Chladni — ∇⁴w - k⁴w = 0

Chladni

∇⁴w - k⁴w = 0

Monolith — f(x,y) = H(x-a)H(b-x)H(y-c)H(d-y)

Monolith

f(x,y) = H(x-a)H(b-x)H(y-c)H(d-y)

Radiance — L(x,ω) = Lₑ + ∫ f_r L_i cos θ dω

Radiance

L(x,ω) = Lₑ + ∫ f_r L_i cos θ dω

Singularity — f(z) = 1/(z-z₀)ⁿ, n ≥ 1

Singularity

f(z) = 1/(z-z₀)ⁿ, n ≥ 1

Covet — d(x, S) = inf{||x-s|| : s ∈ S}

Covet

d(x, S) = inf{||x-s|| : s ∈ S}

Glass Ceiling — f(x) = min(g(x), c)

Glass Ceiling

f(x) = min(g(x), c)

Mirror — f(-x) = f(x)

Mirror

f(-x) = f(x)

Shadow — P(x) = x - (x·n̂)n̂

Shadow

P(x) = x - (x·n̂)n̂

Watch — θ(t) = arctan(y(t)/x(t))

Watch

θ(t) = arctan(y(t)/x(t))

$Chaos — x_{n+1} = 4x_n(1-x_n)$

Chaos

x_{n+1} = 4x_n(1-x_n)

Detonation — D = √(2(γ²-1)q)

Detonation

D = √(2(γ²-1)q)

Eruption — p(z) = ρgz + p₀, p > p_yield

Eruption

p(z) = ρgz + p₀, p > p_yield

Shatter — E_release > Σ G_c · A_crack

Shatter

E_release > Σ G_c · A_crack

Shockwave — v > c, M = v/c >> 1

Shockwave

v > c, M = v/c >> 1

Bloom — r(θ) = a + b·cos(nθ), a > b

Bloom

r(θ) = a + b·cos(nθ), a > b

Confetti — P(x,y) ~ Uniform(Ω), N >> 1

Confetti

P(x,y) ~ Uniform(Ω), N >> 1

Lissajous — x = sin(3t), y = sin(4t)

Lissajous

x = sin(3t), y = sin(4t)

Pinwheel — f(r,θ) = cos(nθ + αr)

Pinwheel

f(r,θ) = cos(nθ + αr)

Sunburst — I(r,θ) = I₀ · (1/r) · Σ δ(θ - 2πn/N)

Sunburst

I(r,θ) = I₀ · (1/r) · Σ δ(θ - 2πn/N)

Forged — σ_y(ε) = σ₀ + Kε^n

Forged

σ_y(ε) = σ₀ + Kε^n

Growth — dN/dt = r(K' - N), K' > K

Growth

dN/dt = r(K' - N), K' > K

$Phoenix — f(t) = A(1 - e^{-t/τ₁})e^{t/τ₂}, τ₂ > τ₁$

Phoenix

f(t) = A(1 - e^{-t/τ₁})e^{t/τ₂}, τ₂ > τ₁

$Recovery — x(t) = x_eq + (x₀-x_eq)e^{-t/τ} + overshoot$

Recovery

x(t) = x_eq + (x₀-x_eq)e^{-t/τ} + overshoot

$Repair — G(t) = G₀(1 - e^{-t/τ_r}) · H(t-t_d)$

Repair

G(t) = G₀(1 - e^{-t/τ_r}) · H(t-t_d)

Breath Together — φ₁(t) - φ₂(t) → 0 as t → ∞

Breath Together

φ₁(t) - φ₂(t) → 0 as t → ∞

Handshake — SYN → SYN-ACK → ACK

Handshake

SYN → SYN-ACK → ACK

Mirror — y₁(t) = αy₂(t-τ) + (1-α)y₁(t-τ)

Mirror

y₁(t) = αy₂(t-τ) + (1-α)y₁(t-τ)

Sync — dφ/dt = Δω - K sin(φ)

Sync

dφ/dt = Δω - K sin(φ)

Weave — f(x,y) = sin(x)sin(y) + sin(x)cos(y)

Weave

f(x,y) = sin(x)sin(y) + sin(x)cos(y)

$Crown — f(θ) = |cos(nθ/2)|^{1/n}$

Crown

f(θ) = |cos(nθ/2)|^{1/n}

Firework — r(t) = v₀t - ½gt², θ ~ Uniform(0,2π)

Firework

r(t) = v₀t - ½gt², θ ~ Uniform(0,2π)

Kaleidoscope — f(r,θ) = f(r, θ + 2π/n), n = 6

Kaleidoscope

f(r,θ) = f(r, θ + 2π/n), n = 6

Prismatic — n(λ) = A + B/λ² + C/λ⁴

Prismatic

n(λ) = A + B/λ² + C/λ⁴

$Stained Glass — T(λ,x,y) = Σ cᵢ · χ_{Ωᵢ}(x,y) · S(λ)$

Stained Glass

T(λ,x,y) = Σ cᵢ · χ_{Ωᵢ}(x,y) · S(λ)

Crown — r(θ) = 1 + ε·cos(nθ), n ≥ 5

Crown

r(θ) = 1 + ε·cos(nθ), n ≥ 5

Flourish — γ(t) = (t·cos(t), t·sin(t), t²)

Flourish

γ(t) = (t·cos(t), t·sin(t), t²)

Golden Ratio — φ = (1 + √5)/2 ≈ 1.618...

Golden Ratio

φ = (1 + √5)/2 ≈ 1.618...

$Spire — h(x) = H · (1 - |x/w|^p)^{1/p}$

Spire

h(x) = H · (1 - |x/w|^p)^{1/p}

$Unfurl — θ(t) = θ₀ + (π - θ₀)(1 - e^{-t/τ})$

Unfurl

θ(t) = θ₀ + (π - θ₀)(1 - e^{-t/τ})

$Charge — V(t) = V₀(1 - e^{-t/RC})$

Charge

V(t) = V₀(1 - e^{-t/RC})

Convergence — aₙ → L as n → ∞, |aₙ - L| < ε

Convergence

aₙ → L as n → ∞, |aₙ - L| < ε

Countdown — f(t) = N - ⌊t/Δt⌋

Countdown

f(t) = N - ⌊t/Δt⌋

$Kindling — T(t) = T_ign - ΔT·e^{-t/τ}$

Kindling

T(t) = T_ign - ΔT·e^{-t/τ}

Potential — U(x) = mgh, h = h_max

Potential

U(x) = mgh, h = h_max

$Harmonic Decay — f(t) = A₀ · cos(ωt) · e^{-γt}$

Harmonic Decay

f(t) = A₀ · cos(ωt) · e^{-γt}

Magnetic — F = μ₀m₁m₂/(4πr²)

Magnetic

F = μ₀m₁m₂/(4πr²)

$Tantalus — lim_{x→a} f(x) = L, f(a) undefined$

Tantalus

lim_{x→a} f(x) = L, f(a) undefined

Vanishing — x' = x·f/(f+d), y' = y·f/(f+d)

Vanishing

x' = x·f/(f+d), y' = y·f/(f+d)

Zeno — S = Σ (½)ⁿ = 1, but no finite step reaches 1

Zeno

S = Σ (½)ⁿ = 1, but no finite step reaches 1