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The evolving lemniscates of a time-choreographed complex polynomial. *DM me if you would like to see the math framework...bots are killing me.

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Mathelirium

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18,318 views • 10 months ago •via X (Twitter)

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The roots of a polynomial aren't just numbers, but a moving geometric field that reacts like a resonant surface in the complex plane. See the description for the animation below
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The roots of a polynomial aren't just numbers, but a moving geometric field that reacts like a resonant surface in the complex plane. See the description for the animation below

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You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.
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You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

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Harmonics of Roots of Complex Polynomials polynomial equation by Simone Conradi
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What happens when you take a polynomial and keep changing its coefficients every frame? You get a resonant root field in the complex plane where clusters, arcs and outer flares form, dissolve, and reappear as the system evolves in time.
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What happens when you take a polynomial and keep changing its coefficients every frame? You get a resonant root field in the complex plane where clusters, arcs and outer flares form, dissolve, and reappear as the system evolves in time.

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“i would like to see it” the bisexual panic at the end is killing me
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Finding the roots of a polynomial is one of the oldest and most central problems in Algebra. Once the coefficients are allowed to move through the complex plane, the roots start revealing a much richer geometry. Here is the Mathematics, try it yourself and play around with the coefficients and polynomial structure: P(x) = x¹² + a x⁸ + b x⁶ + c x⁴ + d x + 1 a = (sin(t₁) + i cos(sin(t₂) + 0.5 sin(2t₂))) M b = 0.3 (t₁ + t₂) Mᶜ c = (exp(i t₁) - cos(t₂)) M d = (cos(t₁) - 0.5 i sin(sin(t₂) + 0.5 sin(2t₂))) Mᶜ M = 1 + ∑ₖ₌₁ᵐ Aₖ exp(i 2π sₖ n / N) Mᶜ = 1 + ∑ₖ₌₁ᵐ Aₖ exp(-i 2π sₖ n / N) Here t₁ and t₂ lie on the unit circle, n is the frame index, N is the total number of frames, and Aₖ, sₖ are the modulation amplitudes and speeds.
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🥟: stay's do it list: stay has to think of me. before bed, you have to think of me. you have to like me. no matter what i do, you have to like it. got it? if you see me on the street, just confess to me ㅎ if you see me on the street, you'll go "it's hyunjin! i love you!" please say it
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