CYCLES

Phase portraits closing on themselves. Möbius strips with no beginning. Orbits that always return to where they started.

Closed orbits, phase portraits, Möbius topology, and recurrence plots. Mathematics that loops — systems that always return, never quite the same way twice.

55 renders. 5 survived.

Loom — x(t) = x(t + T), T = 2π/ω

Loom

x(t) = x(t + T), T = 2π/ω

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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))

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Orbit — r(θ) = a(1-e²)/(1 + e cos θ)

Orbit

r(θ) = a(1-e²)/(1 + e cos θ)

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

Phase Portrait

ẋ = f(x,y), ẏ = g(x,y)

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

Recurrence

R(i,j) = Θ(ε - ||x_i - x_j||)

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Complete CYCLES Series

All 5 pieces as high-resolution PDFs. Print-ready at 300 DPI.

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▶The Story Behind This Series

The universe is built on cycles. Seasons, tides, heartbeats, orbits — patterns that return without being asked. This series renders the geometry of return: phase portraits that close, orbits that repeat, and Möbius strips that have no beginning or end.

Mathematical primitive: closed orbits, phase portraits, Möbius topology, recurrence