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Pythagorean theorem, written on the unit circle: sin²θ + cos²θ = 1.

The Math Flow

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63,394 views • 3 days ago •via X (Twitter)

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Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.

Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.

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A radius turns at a steady rate inside a regular hexagon, a decagon and a circle, and the height y = r(θ) sin θ of its tip is traced against time. For an n-gon of circumradius R with a vertex at θ = 0, r(θ) = R cos(π/n) / cos((θ mod 2π/n) − π/n). Each edge is one piece of the wave; the flat top edge is a plateau at R cos(π/n), 0.866R for n = 6, 0.951R for n = 10. As n grows the plateaus narrow and rise to R, and the wave tends to R sin θ.

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A disk of radius a rolls on a table at tilt θ, its contact point circling at rate Ω. Gravity against the gyroscopic torque fixes Ω² = 4g/(a sin θ), with energy (3/2)Mga sin θ. Rolling resistance μ drains it as dθ/dt = −(2/3)μΩ, so near the end θ√θ falls linearly to zero: a finite-time stop. With a = 0.40 m and μ = 0.0086, θ drops from 70° to 1.4° in 15 s while Ω climbs from 1.63 to 10.08 Hz. The disk itself turns ever slower, at Ω sin θ.

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3D rotation matrices about coordinate axes. Rₓ(θ) fixes x and rotates yz-plane [[1 0 0] [0 cosθ −sinθ] [0 sinθ cosθ]]. Rᵧ(θ) fixes y and rotates xz-plane [[cosθ 0 sinθ] [0 1 0] [−sinθ 0 cosθ]]. Rz(θ) fixes z and rotates xy-plane [[cosθ −sinθ 0] [sinθ cosθ 0] [0 0 1]].

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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)

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my parasite pen finally came in the mail ϵ( ♡Θ♡ )϶

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An electric dipole p(t) = p₀ sin(ωt) ẑ oscillating at a point radiates a field that, far away, falls off only as 1/r. Its radiation term is E = (μ₀/4π) p̈(t − r/c) sin θ / r along θ̂, with B = r̂ × E / c along φ̂, so E, B and the direction of travel r̂ are mutually perpendicular. The intensity goes as sin²θ / r²: zero along the axis, strongest at the equator. Every sphere around the source carries the same mean power, p₀²ω⁴ / (12πε₀c³).

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Ok, this is amazing. I asked Claude 3 to generate an animation of the Pythagorean Theorem and this is what it created:

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The Hidden Geometry of Trigonometric Functions and the Unit Circle.

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You don't like math because you don't follow me. Check out the beautiful, intricate pattern generated by this equation: sin (sin (x) + cos (y)) = cos (sin (x*y) + cos (x))

You don't like math because you don't follow me. Check out the beautiful, intricate pattern generated by this equation: sin (sin (x) + cos (y)) = cos (sin (x*y) + cos (x))

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