\(f(z)=\frac{z^{10}-a}{z^{12}+\frac{z}{a}}\)
with \(a\) on the unit circle.
Perfect loop
#Animation #Math #Mathematics #Mathart #Complex #ComplexAnalysis #ComplexPlane #GLSL #Shadertoy #PerfectLoop
#ComplexPlane
Sort of \(f(z)=\frac{(z^2-1)(z-2-i)^2}{z^2+2+2i}\)
Shadertoy: https://www.shadertoy.com/view/wcyGRw
#Animation #Math #Mathematics #Mathart #Complex #ComplexAnalysis #ComplexPlane #GLSL #Shadertoy
I asked ChatGPT to extend the Collatz Conjecture to the complex plane (and it made sense)
I was at a loose end, so I thought I would see if ChatGPT could extend Collatz into complex numbers. This was the reply.
Extending the Collatz Conjecture to the complex plane is a fascinating idea, though it leads into uncharted and nontrivial mathematical territory. Here’s a breakdown of the options and some of the considerations involved:
🔢 Original Collatz: Defined on ℕ
The original Collatz function is a piecewise function on the positive integers:
This is based on parity (odd or even), which doesn’t naturally extend to the complex plane ℂ.
🧩 Challenges in Extending to ℂ
To generalize to complex numbers, we must:
- Replace parity-based rules with something continuous or analytically defined.
- Define a continuous analogue or complex dynamical system resembling Collatz behavior.
🧮 Option 1: Complex Analogue Using Modular Extension
One approach is to construct an analytic function that smoothly interpolates between the two piecewise rules.
Lagarias & Weiss (1992) considered functions like:
I was surprised that ChatGPT cited a real paper.
This simplifies to:
I’d hardly claim this was much of a simplification.
Which behaves like:
Thus, this function smoothly mimics the parity behavior of integers, but it is now defined for all complex numbers.
🎨 Option 2: Visualizing Collatz-like Dynamics in ℂ
Instead of defining a “true” complex Collatz, some researchers explore dynamical systems that generalize the iterative structure:
- Iterate a function
- Use color plots or escape-time plots (similar to Mandelbrot sets) to visualize the behavior.
⚙️ Option 3: Integer Sequence Extended to ℂ Numerically
Alternatively, you could treat the original Collatz steps as a discrete complex function:
This would define a heuristic parity rule based on the real part of zzz, though it’s less elegant and doesn’t retain the analytic properties you’d want for complex dynamics.
Summary
ApproachDescriptionAnalytic extensionUse functions involvingto interpolate Collatz behavior
Dynamical systems viewStudy iterations in ℂ for generalized maps (e.g., Collatz-like Julia sets)Discrete parity heuristicExtend the piecewise definition to ℂ via heuristicsNGL, some of that was lost on me. So I asked Chat to explain the Complex Analogue Using Modular Extension like I was a GCSE student.
That kinda makes sense. I might write some bad Python and play with this. (later)
Decagon (fractal version)
\(z_{n+1}=fold(z_n)^2+c\)
where fold is a generalized absolute value function. A complex number has two components: a real and an imaginary part.
If we take the absolute value of one of these parts, we can interpret this as a fold in the complex plane. For example, |re(z)| causes a fold of the complex plane around the imaginary axis, which means that the left half ends up on the right half. If we do this for the imaginary component |im(z)|, we fold the complex plane around the real axis which means that the bottom half ends up on the top half.
These two operations are quite similar, because the imaginary fold is just like the real fold of the plane, except that it was previously rotated 90 degrees (z * i). But what if we rotate the plane by an arbitrary number of degrees?
An arbitrary rotation of the complex plane can be expressed as rot(z, radians) = z * (cos(radians) + sin(radians) * i), where radians encodes the rotation.
The image here is produced, by rotating the plane exactly five times, and folding the imaginary part each time.
I found this algorithm in the Fractal Formus under the name “Correction for the Infinite Burning Ship Fractal Algorithm”.
It can be seen as a generalization of the burning ship obtained by folding the complex plane twice with a rotation of 90 degrees, i.e. folding both the real and the imaginary part.
#fractalfriday #fractal #burningship #mandelbrot #complexplane #complexnumbers #mathart #math #escapetimefractals
#mathartmarch II: On a square grid
A third degree Newton fractal zoom in.
Domain coloring technique adapted from this beauty: https://www.shadertoy.com/view/wld3zl
Sorry for the not-so-high quality, I'm too lazy to export it to a direct video at the moment.
#grid #newton #fractal #complexnumbers #complexplane #glsl #shader
The #Zhukovsky #Aerofoil (sometimes transliterated as #Joukowsky from #Russian), is a 2D model of #streamlined #Airflow past a #wing. It uses #ComplexVariable and is an #AnalyticFunction (i.e. #Differentiable everywhere, save at isolated #Singularities). Take a circle in the #ComplexPlane which is not quite centred at the #origin but passes through the #coordinate (1,0) or (z=1+0i).
#MyWork #CCBYSA #AppliedMathematics #WxMaxima #FreeSoftware #Aeronautics #Aerodynamics #LaminarFlow
I'm trying to work out how to find a prime in the complex plane, and per chance in quads and above.
A lot of the language and techniques used in the boring 1D space don't work in higher dimensions, and I don't have the prerequisite knowledge to find thesepotential primes in multidimensional spaces.
I wish I could go to school again.
#SabineHossenfelder - Do #ComplexNumbers Exist?
"Do complex #Numbers exist or are they just a convenient, #MathematicalTool that we use in science? With the exception of #QuantumMechanics, it is easy to get rid of complex numbers. But can you do quantum mechanics without complex numbers? A recent paper says no, you can't."
https://www.youtube.com/watch?v=ALc8CBYOfkw&ab_channel=SabineHossenfelder
#Math #Maths #Mathematics #Science #Physics #Equations #ComplexPlane #TheComplexPlane #ComplexNumber #e #i #Pi #QM #QuantumPhysics
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