ape-visualize

Visualize Skill: Interactive 3D Math & Science Visualizer

Takes any concept from mathematics or science and creates an interactive, visually stunning, browser-based 3D visualization using Three.js, packaged as a single self-contained HTML file.

The visualization is grounded in real mathematical formulations and physical laws—not just decorative graphics, but a live, interactive simulation or geometric construction that users can rotate, zoom, tweak with sliders, and inspect in real time.

---

Inputs

The user provides:

1. Concept: Any mathematics or science concept. For example:
   • Math: Eigenvalues & Eigenvectors, 3D Linear Transformations, Gradient Descent on Surfaces, Curl & Divergence of Vector Fields, Riemann Surfaces, Möbius Strip & Topology, Fourier Epicycles in 3D, Lorenz Attractor & Chaos, Spherical Harmonics, Minimal Surfaces (Helicoid/Catenoid).
   • Science: Lorentz Force, Gravitational Orbits & Precession, Double-Slit Wave Interference, Maxwell-Boltzmann Distribution, VSEPR Molecular Geometry, Quantum Harmonic Oscillator, DNA Replication Fork.
2. Depth: One of three simple depth levels (default to Intermediate if not specified):
   • Simple (Level 1 — Intuition & Big Picture)
   • Intermediate (Level 2 — Mechanics & Equations)
   • Advanced (Level 3 — Deep Dive & Rigorous Internals)

Regardless of the starting depth, every generated HTML document embeds an in-browser Depth Selector (Simple | Intermediate | Advanced) so the user can fluidly toggle between levels live.

---

The Three Depth Levels (Simple Words)

The depth determines both the visual representation and the level of mathematical/physical rigor exposed:

Depth	Target Audience	Core Focus	Visual Elements (Math & Science)	Interactive Controls	Telemetry & HUD
Simple	Beginners, general audience, visual learners	Big-picture intuition & visual metaphors. What does it look like? What is the main effect?	Math: Deforming shapes, flowing streamlines, animated paths, color gradients.Science: Macro objects, smooth orbits, glowing wavefronts. No overwhelming formulas.	Play / Pause, Speed slider, 1–2 intuitive dials ("Morph", "Stretch", "Strength", "More / Less", scenario presets).	"What's happening?" card in plain English explaining the core takeaway and visual intuition.
Intermediate	Undergraduates, engineers, technical learners	How it works mechanically — governing rules, vectors, coordinates, and equations.	Math: Basis vectors ($\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}$), tangent planes, 3D vector arrow grids, coordinate axes, parameter curves.Science: Force/velocity arrows, field lines, particle trails, component breakdowns.	Sliders with real variables and units (matrix components $a_{ij}$, surface curvature $k$, mass $m$, charge $q$, frequency $\omega$).	Live numerical telemetry (current coordinates, matrix determinant $\det(A)$, energy, flux) + live 2D canvas sparkline. Typeset formula.
Advanced	Researchers, math/physics majors, deep divers	Rigorous math & deep mechanics — differential equations, phase space, edge cases.	Math: Invariant eigenvector axes, differential forms, phase portraits ($\dot{x}$ vs $x$ or complex plane), branch cuts, geodesic curvature.Science: Runge-Kutta trajectories, Hamiltonian drift, quantum probability clouds, perturbation indicators.	Precision mathematical dials (eigenvalue parameter $\lambda$, perturbation $\epsilon$, integration step $\Delta t$, non-linear coupling, boundary conditions).	Real-time phase-space canvas, conservation/drift monitor ($\Delta E / E_0$, $\det(A)$ conservation), matrix/tensor readout, raycaster probe on click.