#complexplane — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #complexplane, aggregated by home.social.
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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)
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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
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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
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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
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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
-
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
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#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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#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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#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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#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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#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