Nature’s Hidden Network
This is how soap bubbles divide space.
🧿 Complex Function Visualizer
This shows how each point in the complex plane gets transformed by your selected function.
Hermetic Interpretation
Solve represents harmonic dissolution, where f(t) is decomposed into oscillatory modes.
θ marks rotational phase — the boundary where decomposition shifts into recomposition.
Coagula is harmonic condensation: the frequency domain re-emerges into structured form.
Witch of Agnesi
Geometric Construction
📈 Curve Equation
\[ y = \frac{8a^3}{x^2 + 4a^2} \]
Parametric:
\[ x = 2a\tan\theta \]
\[ y = 2a\cos^2\theta \]
📐 Geometric Construction
- Circle with diameter 2a centered at (0, a)
- Line through origin with slope \(t\)
- Intersection point P with circle
- Project P horizontally to y-axis → Q
- The locus of Q is the Witch curve
📊 Mathematical Properties
- Asymptote: \(\lim_{x\to\pm\infty}y=0\)
- Max point: \(y_{\max}=2a\)
- Area: \(A=4\pi a^2\)
- Volume: \(V=4\pi^2 a^3\)
📜 Historical Background
Named after Maria Gaetana Agnesi (1718–1799).
“Witch” is a mistranslation of the Italian versiera.
\[ y = a\sqrt{\frac{a-x}{x}} \]
The Cartan Tartan
The Cartan Tartan blends two worlds:
- Élie Cartan’s structural geometry — recursion, symmetry, and decomposition of spaces.
- Tartan’s woven pattern language — nested squares, recursive subdivision, and self-similar crosshatching.
Here, the Fourier harmonic backdrop acts as a shifting manifold, while the recursive quadtree subdivides itself over time — echoing Cartan’s ideas of structure-from-structure.
This name conveys:
- Recursion — fractal quadtree depth
- Harmony — Fourier interference field
- Geometry — Cartan’s influence across modern mathematics
- Weaving — the grid’s tartan-like appearance
It is playful, symbolic, and mathematically accurate — a signature blend of harmonic analysis and recursive geometry.
You must be logged in to post a comment.