Interactive Mathematics

Euler Circles, Hilbert Space & the Three-Body Problem

From the simple rotation $e^{i\theta}$ to infinite-dimensional Hilbert spaces, the same circle builds Fourier analysis, tori, and the chaos of gravitating bodies. Each periodic orbit lives on a torus $T^3$ — built from three Euler circles. Chaotic orbits need infinitely many circles: a point in $L^2(T^3)$.

Euler: $e^{i\theta} = \cos\theta + i\sin\theta$ Torus: $T^n = S^1 \times \cdots \times S^1$ Hilbert: $L^2 = \{\sum c_k e^{ik\cdot\theta}\}$

1Euler's Circle — The Building Block

Every complex phase is a point on the unit circle. Real and imaginary parts are orthogonal projections. Fourier series are just sums of these spinning phasors.

$\cos\theta$ = 0.540

$\sin\theta$ = 0.841

$e^{i\theta}$ = 0.54 + 0.84i

Orthogonal basis: $\langle e^{im\theta}, e^{in\theta}\rangle = \delta_{mn}$

2Three Euler Circles = 3-Torus $T^3$

A 3-torus is the product of three circles: $(e^{i\theta_1}, e^{i\theta_2}, e^{i\theta_3})$. Each $\theta_k = \omega_k t$ spins at its own frequency. Fourier modes on $T^3$ are $e^{i(l\theta_1+m\theta_2+n\theta_3)}$.

Fourier mode

$e^{i(l\theta_1+m\theta_2+n\theta_3)}$ = 0.00 + 0.00i

3Three-Body Problem — Newton Meets Fourier

Three masses with $F = Gm_im_j/r^2$. Periodic solutions (Lagrange, Euler, figure-8) live on a low-dimensional torus — finite Fourier series. Chaotic ones fill $L^2(T^3)$ requiring infinite modes.

0.000%
Energy drift
0.00
Lyapunov λ
~3
Fourier modes

Stable orbits: 3 frequencies. Chaos: spectrum → continuous.

4Hilbert Space $L^2(T^2)$ — Infinite Circles

Functions on a torus expand as $f(x,y)=\sum_{l,m} c_{lm} e^{2\pi i(lx+my)}$. The coefficients $c_{lm}$ are coordinates in infinite-dimensional Hilbert space. Truncating at $|l|,|m|\le N$ projects onto a finite torus.

Original $f(x,y)$

Reconstruction $\sum_{|l|,|m|\le N}$

$|c_{lm}|$ heatmap (log scale)

Parseval: $\|f\|_{L^2}^2 = \sum |c_{lm}|^2 = $ 1.000   •   Truncation error: 0.0%