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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 görüntüleme • 10 ay önce •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
0:33

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

Mathelirium

23,885 görüntüleme • 2 ay önce

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.
0:32

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.

Mathelirium

15,566 görüntüleme • 5 ay önce

Harmonics of Roots of Complex Polynomials polynomial equation by Simone Conradi
0:33

Harmonics of Roots of Complex Polynomials polynomial equation by Simone Conradi

Mathelirium

18,496 görüntüleme • 1 yıl önce

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.
0:33

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.

Mathelirium

17,351 görüntüleme • 2 ay önce

What you're seeing here is the root dynamics of a smoothly time-modulated complex polynomial. A living membrane breathes while four satellites orbit, dock, fuse with the core, then split again. From the simplest task in algebra...finding roots...emerges natural-feeling, sophisticated motion. #ComplexPolynomials #ComplexAnalysis #MathArt #Mathematics
0:50

What you're seeing here is the root dynamics of a smoothly time-modulated complex polynomial. A living membrane breathes while four satellites orbit, dock, fuse with the core, then split again. From the simplest task in algebra...finding roots...emerges natural-feeling, sophisticated motion. #ComplexPolynomials #ComplexAnalysis #MathArt #Mathematics

Mathelirium

63,140 görüntüleme • 8 ay önce

What would you do if you saw me a few lockers down? Who wants to see the whole video?? Dm me!!
0:21

What would you do if you saw me a few lockers down? Who wants to see the whole video?? Dm me!!

Tallscuzz | 7’3” 221cm

37,015 görüntüleme • 5 ay önce

“i would like to see it” the bisexual panic at the end is killing me
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Sensitive content

“i would like to see it” the bisexual panic at the end is killing me

ceo of evan buckley

73,395 görüntüleme • 2 ay önce

DM if you would like to support me
0:57

DM if you would like to support me

Gianni Poli

16,678 görüntüleme • 10 ay önce

Hey fam 🧡 I'm going to go shopping tomorrow if anyone from out of state would like me to send them Dolphin's merch. You would obviously be paying for the items you choose and the shipping fee. Please DM me if you are interested. SERIOUS INQUIRIES ONLY!!
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Hey fam 🧡 I'm going to go shopping tomorrow if anyone from out of state would like me to send them Dolphin's merch. You would obviously be paying for the items you choose and the shipping fee. Please DM me if you are interested. SERIOUS INQUIRIES ONLY!!

Lesley Diaz

49,001 görüntüleme • 10 ay önce

I’m doing a series on TikTok where I show people how to draw seemingly complex things in perspective Give me some ideas of furniture or other objects you would like to see the process of me drawing
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I’m doing a series on TikTok where I show people how to draw seemingly complex things in perspective Give me some ideas of furniture or other objects you would like to see the process of me drawing

Britt Ortega

533,875 görüntüleme • 1 yıl önce

Roots of complex polynomials with time-modulated geodesic coefficients. What are the roots telling us?
0:45

Roots of complex polynomials with time-modulated geodesic coefficients. What are the roots telling us?

Mathelirium

15,166 görüntüleme • 6 ay önce

Hi everyone! I’ve decided to formally open my commissions for the first time! If you enjoy my work and would like to create something together, you can DM me or email mimarymi11@gmail.com. Here are a couple of visuals and some of my favorite cover projects I have been a part of:
0:09

Hi everyone! I’ve decided to formally open my commissions for the first time! If you enjoy my work and would like to create something together, you can DM me or email [email protected]. Here are a couple of visuals and some of my favorite cover projects I have been a part of:

mary

16,051 görüntüleme • 6 ay önce

One of the first polynomial root plots I made 2 years ago after learning the idea from Simone Conradi
0:31

One of the first polynomial root plots I made 2 years ago after learning the idea from Simone Conradi

Mathelirium

15,640 görüntüleme • 2 ay önce

Next time you stop me in the street to Rap, I expect to hear math & science like this…
4:01

Next time you stop me in the street to Rap, I expect to hear math & science like this…

DJSemtex

271,588 görüntüleme • 3 yıl önce

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍
1:32

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

Mathelirium

29,231 görüntüleme • 9 ay önce

🥟: 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
0:42

🥟: 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

비니

101,890 görüntüleme • 8 ay önce

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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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.

Mathelirium

23,042 görüntüleme • 3 ay önce

Shoot me a dm if you like this guy would love to hear your opinion about him
0:30

Shoot me a dm if you like this guy would love to hear your opinion about him

h2hypnotize

24,854 görüntüleme • 1 yıl önce