conformal_mapping_posts.txt

=== SOCIAL MEDIA POSTS: CONFORMAL MAPPING ===

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1. TWITTER/X (280 characters max)
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Just visualized conformal mappings in complex analysis! 🎯 Functions like w = e^z preserve angles while transforming grids into beautiful curved patterns. Perfect for fluid dynamics & electrostatics.

#Python #Mathematics #ComplexAnalysis #ScienceCoding

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2. BLUESKY (300 characters max)
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Explored conformal mapping: transformations that preserve local angles. The exponential function e^z turns rectangular grids into polar patterns, while Möbius transformations map the upper half-plane to a unit disk. Fundamental for solving PDEs in physics & engineering.

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3. THREADS (500 characters max)
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Ever wonder how mathematicians transform complex shapes while preserving angles? Just coded up conformal mappings in Python!

These special functions (like w = e^z and w = z²) take grid patterns and warp them into circles, spirals, and curves - but angles between lines stay the same. It's not just pretty math; engineers use this to solve fluid flow problems and design airfoils.

The visualizations show how rectangles become polar grids under exponential mapping. Mind-bending stuff! 📐

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4. MASTODON (500 characters max)
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Implemented conformal mapping visualizations in Python using complex analysis. Key findings:

• Functions analytic at z₀ with f'(z₀) ≠ 0 preserve angles locally
• Exponential mapping e^z converts horizontal strips to annular sectors
• Möbius transformations (az+b)/(cz+d) map upper half-plane to unit disk
• Numerical verification shows angle preservation to machine precision

Applications: 2D potential flow, electrostatics, Riemann mapping theorem. The Jacobian has rotation matrix form, confirming geometric interpretation.

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5. REDDIT (Title + Body)
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TITLE:
[OC] Visualizing Conformal Mappings: How Complex Functions Preserve Angles While Warping Space

BODY:
I created an interactive exploration of conformal mapping in complex analysis using Python and NumPy. Here's what I learned:

**What is Conformal Mapping?**

A conformal mapping is a transformation of the complex plane that preserves angles between curves. If you have two lines intersecting at 45°, after the transformation they'll still intersect at 45° - even though the lines themselves might be curved into circles or spirals.

**The Math (ELI5 version):**

For a function w = f(z) to be conformal at a point z₀, two things must be true:
1. The function must be analytic (smooth in a complex sense) at z₀
2. The derivative f'(z₀) cannot be zero

The derivative's magnitude |f'(z₀)| tells you how much the mapping stretches or shrinks distances, but the key insight is that it stretches equally in all directions - that's why angles are preserved.

**Three Classic Examples I Visualized:**

1. **Exponential Mapping (w = e^z)**: Horizontal lines become circles, vertical lines become rays from the origin. It's like converting rectangular coordinates to polar coordinates, but in a continuous, smooth way.

2. **Square Mapping (w = z²)**: Doubles angles and squares distances. A grid gets warped into a curved mesh, but perpendicular lines stay perpendicular.

3. **Möbius Transformation (w = (z-i)/(z+i))**: This one's wild - it maps the entire upper half of the complex plane into a unit disk. The real axis wraps around to form the circle's boundary.

**Why This Matters:**

Conformal mappings are used in:
- **Fluid dynamics**: Solving flow around airfoils (Joukowsky transformation)
- **Electrostatics**: Finding electric fields in complex geometries
- **Cartography**: Map projections that preserve angles (Mercator projection)
- **Pure math**: The Riemann Mapping Theorem says any simply connected domain can be mapped to a unit disk

**Numerical Verification:**

I tested angle preservation numerically by computing angles between horizontal and vertical directions before and after transformation. The angles matched to within 10⁻⁹ radians - basically machine precision.

**View the interactive notebook:**
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/conformal_mapping.ipynb

The code is self-contained with matplotlib visualizations showing grid transformations. The beauty of conformal mapping is that it connects pure mathematics with practical applications in physics and engineering.

What other complex analysis topics would you like to see explored?

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6. FACEBOOK (500 characters, general audience)
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🔬 Just explored one of the coolest concepts in mathematics: conformal mapping!

Imagine drawing a grid on a rubber sheet, then stretching and bending it - but with one special rule: angles between lines must stay the same. That's what conformal mappings do in the complex number plane.

I visualized three classic transformations: exponential functions that turn rectangles into spirals, squaring that warps grids, and Möbius transformations that bend entire half-planes into circles.

Used in: airplane wing design, map projections, and solving physics equations!

🔗 View the interactive notebook:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/conformal_mapping.ipynb

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7. LINKEDIN (1000 characters, professional tone)
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Conformal Mapping: Angle-Preserving Transformations in Complex Analysis

I recently implemented a computational exploration of conformal mappings using Python, demonstrating key concepts in complex analysis with practical applications in engineering and physics.

**Technical Overview:**

Conformal mappings are complex functions w = f(z) that preserve angles locally. For a mapping to be conformal at z₀:
• f must be analytic at z₀
• f'(z₀) ≠ 0

The Cauchy-Riemann equations ensure the Jacobian matrix has the form of a scaled rotation, confirming angle preservation.

**Implementations:**

1. Exponential mapping (w = e^z): Converts rectangular grids to polar patterns
2. Square mapping (w = z²): Doubles angles while preserving orthogonality
3. Möbius transformation: Maps upper half-plane to unit disk

**Real-World Applications:**

• Aerodynamics: Joukowsky transformation for airfoil design
• Electrostatics: Solving Laplace's equation in complex geometries
• Cartography: Angle-preserving map projections
• Riemann Mapping Theorem: Fundamental result in geometric function theory

**Methodology:**

Numerical verification confirms angle preservation to 10⁻⁹ radian precision. Grid visualization demonstrates geometric properties of analytic functions.

🔗 Full computational notebook:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/conformal_mapping.ipynb

#Mathematics #Python #ComplexAnalysis #ComputationalScience #Engineering

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8. INSTAGRAM (500 characters, visual-focused)
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🎨 CONFORMAL MAPPING IN ACTION

[Image shows three grid transformations side-by-side]

Ever seen math turn straight lines into perfect circles while keeping all angles intact? That's conformal mapping ✨

✦ LEFT: Exponential function e^z
Rectangles → polar spirals

✦ MIDDLE: Square function z²
Grids → curved meshes

✦ RIGHT: Möbius transformation
Half-plane → unit disk

These aren't just pretty patterns - engineers use them to design airplane wings and solve physics equations 🛩️

The secret? Functions that preserve angles even while warping space.

#Mathematics #Python #ComplexAnalysis #DataVisualization #STEM #ScienceArt #Coding #MathArt

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