Simple Harmonic Motion visualized from the formula x(t) =... A cos(ωt + φ)show more

This video looks like a rotating wheel. But watch... just one single sphere. It never rotates. Every ball only moves in a perfectly straight line back and forth. This is simple harmonic motion where linear position follows: x = R cos(ωt) By perfectly offsetting the timing of each straight path, the collective motion creates the illusion of rotation.show more

A turnstile antenna is two crossed dipoles driven 90°... apart, one dipole p turning at ω about its mast. Its far field E ∝ r̂ × (r̂ × p̈(t − r/c)) has phase ωt − kr − φ at every latitude, so each wavefront is the spiral r = (ωt − φ − 2πn)/k. Along the mast E ∝ (1/r)[cos(kr − ωt) ŷ + sin(kr − ωt) x̂] keeps its length and turns once per period, and B = (1/c) r̂ × E turns with it. In the plane of the cross the same wave is linearly polarized.show more

This fascinating experiment reveals how simple harmonic motion can... be generated from uniform circular motionshow more

Richard Feynman called Euler's formula "the most remarkable formula... in mathematics." And there's a fascinating engineering reason why this equation is so important. e^(iθ) = cos(θ) + i·sin(θ) It connects exponential functions, imaginary numbers, trigonometry, and rotation through a single mathematical relationship. But its usefulness goes much further. Consider a sinusoidal signal: x(t) = A cos(ωt + φ) Using Euler's formula, we can represent it as the real part of a complex exponential: A e^(i(ωt + φ)) Why would engineers use this representation? Because differentiating e^(iωt) simply multiplies it by iω. Phase shifts can be represented through complex multiplication. For linear, time-invariant electrical circuits in sinusoidal steady state, this allows differential equations to be reduced to algebraic equations. Engineers use this mathematics in phasor analysis, AC circuit calculations, impedance, and frequency-response analysis. The same complex-exponential representation is fundamental to Fourier analysis, signal processing, communication systems, and the study of mechanical vibrations. Feynman also described the complex-plane representation of numbers as the unification of algebra and geometry. And that's what makes Euler's formula so remarkable. An equation connecting imaginary numbers and circular motion provides a practical way to analyze real electrical signals, oscillations, and physical systems.show more

Euler's formula (eⁱᶿ = cos θ + i sin... θ) is fundamentally a dynamic statement about rotation, growth, and circular motion in the complex plane.show more

2/11 20:08 Φ∞∞∞∞∞∞Φ MV OUT Φ∞∞∞∞∞∞Φ

Newton's first law of motion, visualized.

1. Spatial Constraints & Magnitude: K(u) = A₁ *... cos(u / λ₁) E(v) = (v / λ₂) - δ_offset Distance Driver: D(u,v) = √( K(u)² + E(v)² ) 2. Pulsation Kinematics: Q(u,v,t) = αsin(2K) + (β/K) + sin(v/λ₃) * K * [γ + εsin(ω₁E - ω₂D + ω₃t)] ↳ The (β/K) term creates the asymptotic stretch of the tentacles, while the nested sine waves act as a damped harmonic oscillator for the contraction of the bell. pause - push - pause - push (action) 3. Polar-to-Cartesian Mapping Rotational Phase: θ(u,v,t) = D(u,v) - t X(u,v,t) = Q * cos(θ) * S_x + C_x Y(u,v,t) = Q * sin(θ) + S_y * D - C_y 4. Particle Sizing Filter: Radius = K(u)² > τ ? 2 : 1 Alpha-Blended Buffer: C_final = (1 - ρ) * C_new + ρ * C_buffer (u,v = particle indices, t = continuous time, ρ = retention rate for trails)show more

a=(x,y,d=5*cos(o=mag(k=x/8-25,e=y/8-25)/3.5))=>[(k*atan(3*cos(d*9))+x/2)*cos(c=(d+o)/4-t/8)+200,(k*d*cos(o-d+t)+y/2)*sin(c)+200] t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(255,96);for(t+=PI/60,y=99;y<300;y+=2)for(x=99;++x<300;)point(...a(x,y))} #つぶやきProcessing

a=(x,y,o=mag(k=x/4-12.5,e=y/9)/9)=>point((q=x+99+cos(9/k)+o*k*(cos(e*9)/3+cos(y/9)/.7)*sin(o*4-t))*.7*sin(c=o*e/30-t/8)+200,200+y/9*cos(c*4-t/2)-q/2*cos(c)) t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(w,46);for(t+=PI/60,i=3e4;i--;)a(i%100,i/150)}#つぶやきProcessing

a=(x,y,d=mag(k=(4+cos(x/9-t))*cos(x/30),e=y/7-13)+sin(y/99+t/2)-4)=>point((q=3*sin(k*2)+sin(y/29)*k*(9+2*sin(cos(e)*9-d*4+t)))+40*cos(c=d-t)+200,q*sin(c)+d*35) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,96);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}#つぶやきProcessing

a=(x,y,k=x/12-12.5,d=abs(cos(k/2)+sin(y/2)*cos(k/5)/2+1))=>point((q=x/4+99+(d+0)*k*(.5+cos(d*2-t+y/8)))*.8*cos(c=y/69-t/8+d/6)+200,q/2*cos(c+d/3+y/69)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,46);for(t+=PI/30,i=4e4;i--;)a(i%300,i/300)}#つぶやきProcessing

a=(x,y,k=x/8-12.5,d=abs(cos(k/2)+sin(y/4)))=> point((q=x/4+90+(d+.1)*k*cos(d-t*2+y/24))*.7*cos(c=y/99-t/8+d*d/32)+200,(q+y)*.7*sin(c)+(q-y)/2*cos(c*2)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%200,i/300)}#つぶやきProcessing

a=(x,y,k=x/12-12.5,d=2*abs(cos(k/2)+sin(y/4)+cos(k/8)))=>point((q=x/4+90+(d)*k*cos(d-t*2+y/24))*.7*cos(c=y/79-t/8+d/8)+q/8*sin(c*4)+200,(q)*.7*sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%300,i/300)}//#つぶやきProcessing

This Lissajous Machine is straight-up sorcery. Two cranks at... 33:37 ratio, linked arms, and x=3cos(33t), y=3sin(37t) weave a perfect lattice from thin air—no pixels, just raw harmonic motion. Trig functions don’t just describe beauty. They are the beauty.show more

a=(x,y,o=mag(k=x/4-12.5,e=y/9+9)/9)=>point((q=x+39+y+1/k+o*k*(cos(y)/4+cos(e))*sin(o*4-t))*.7*sin(c=o+e/99-t/8+i%2*3)+200,200+y/2*cos(c*4-o)-q/2*cos(c)) t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(w,46);for(t+=PI/60,i=4e4;i--;)a(i/2%100,i/350)}//#つぶやきProcessing

a=(x,y,d=mag(k=(4+cos(y))*cos(x/4),e=y/8-20))=>point((q=sin(k*3)+sin(y/19+9)*k*(6+sin(e*14-d)))*cos(d/8+t/4)+50*cos(c=d-t)+200,q*sin(c)+d*7*sin(c/4)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,116);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}//#つぶやきProcessing

a=(x,y,k=x/8-12.5,d=cos(k/2)+sin(y/3)-.5)=>point((q=x/4+60+d*k*(1+cos(d*4-t*2+y/14)))*.7*cos(c=y*d/169-t/8+d/9)+200+60*sin(t*3/32+c/4),(q+59)*.7*sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/30,i=4e4;i--;)a(i%200,i/400)}#つぶやきProcessing