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∀ x ∈ ℂ — An Interactive Mathematics Laboratory

The Beauty of Mathematics

Where equations breathe, patterns emerge, and infinity unfolds before your eyes.

$$e^{i\pi} + 1 = 0 \qquad \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
Begin Exploration ↓
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01

Interactive Fractals

Infinite complexity from simple rules. WebGL-powered. Zoom into infinity.

🖱 Drag to pan · Scroll to zoom · Double-click to zoom in

Center: (0, 0) | Zoom: 1×

02

Equation Explorer

Type any equation. Watch mathematics come alive in real time.

Quick Presets:

03

Mathematical Icons

The most beautiful results in history — click any card to explore in full.

Click to expand ↗
Euler's Identity
$$e^{i\pi} + 1 = 0$$

Five fundamental constants united in one sublime equation.

Click to expand ↗
Golden Ratio & Fibonacci
$$\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618$$

Nature's blueprint — the golden ratio hidden in every spiral.

Click to expand ↗
Ulam Spiral
$$\pi(x) \sim \frac{x}{\ln x}$$

Primes in a spiral reveal mysterious diagonal patterns.

Click to expand ↗
Fourier Series
$$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$$

Any periodic signal decomposed into pure harmonics.

Click to expand ↗
Mandelbrot Set
$$z_{n+1} = z_n^2 + c,\quad z_0=0$$

Infinite complexity from a single quadratic iteration.

Click to expand ↗
Pythagorean Theorem
$$a^2 + b^2 = c^2$$

The foundation of Euclidean geometry — over 370 known proofs.

Click to expand ↗
Normal Distribution
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

The bell curve — governing randomness in nature and science.

Click to expand ↗
Basel Problem
$$\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$$

Why do the reciprocal squares of integers sum to π²/6?

04

Riemann Zeta Function

The deepest unsolved mystery in mathematics — visualized.

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p\text{ prime}} \frac{1}{1-p^{-s}}$$

The Riemann Hypothesis: all non-trivial zeros lie on the critical line Re(s) = ½.

First few zeros: 14.135, 21.022, 25.011...

05

Complex Function Explorer

Domain coloring makes the invisible visible. Every color tells a story.

$$f: \mathbb{C} \to \mathbb{C}, \quad \arg(f(z)) \mapsto \text{hue}, \; |f(z)| \mapsto \text{brightness}$$

Presets:

Hue = argument · Brightness = magnitude

06

Dynamical Systems

Chaos, order, and everything in between.

07

Reaction-Diffusion Patterns

Turing's morphogenesis — how nature generates patterns from chemistry.

$$\frac{\partial A}{\partial t} = D_A \nabla^2 A - AB^2 + f(1-A) \qquad \frac{\partial B}{\partial t} = D_B \nabla^2 B + AB^2 - (k+f)B$$

🖱 Click/drag to add chemical B

08

Fluid Dynamics

Navier-Stokes equations made tactile. Paint with mathematics.

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu\nabla^2\mathbf{u}$$

🖱 Move mouse to stir the fluid

09

Hyperbolic Geometry

Where parallel lines diverge and infinity lives at the boundary.

$$ds^2 = \frac{dx^2 + dy^2}{(1-x^2-y^2)^2}$$

🖱 Drag to navigate Poincaré disk

10

4D Geometry

Projections of four-dimensional objects into our three-dimensional world.

11

Number Theory Visual Art

The hidden music of integers revealed through geometry.

12

Mathematics → Sound

Hear the shape of equations. Every function has a voice.

Sound Presets:

13

Function Laboratory

Manipulate parameters. Discover patterns. Your space to experiment.

$$y = \sin(ax + b)\,e^{-cx}$$
14

3D Mathematical Surfaces

Geometry that transcends three dimensions.

ADV 01

Mandelbulb — 3D Fractal

Ray-marched 3D fractal sculpture — distance-estimated on the GPU.

$$|z_{n+1}| = |z_n|^n + c, \quad z,c \in \mathbb{R}^3$$
$$\text{DE}(p) = \frac{|p|\ln|p|}{2|p'|}$$

🖱 Drag to rotate · Scroll to zoom

ADV 02

Quaternion Julia Set

A 4D fractal rendered as a 3D cross-section via GPU ray marching.

$$q_{n+1} = q_n^2 + c, \quad q,c \in \mathbb{H}$$
$$\mathbb{H} = \{a+bi+cj+dk\}$$
ADV 03

Minimal Surfaces

Zero mean curvature — nature's most efficient geometries.

$$H = \frac{\kappa_1 + \kappa_2}{2} = 0$$
ADV 04

Hopf Fibration

The beautiful fiber bundle S¹ → S³ → S² discovered by Heinz Hopf in 1931.

$$\pi: S^3 \to S^2, \quad \pi^{-1}(p) \cong S^1$$
ADV 05

Geodesic Flow

Shortest paths on curved surfaces — the generalisation of straight lines.

$$\ddot{\gamma}^k + \Gamma^k_{ij}\dot{\gamma}^i\dot{\gamma}^j = 0$$
ADV 06

Riemann Surface Visualizer

Multi-sheeted surfaces that resolve branch cuts of complex functions.

$$w = f(z), \quad z \in \mathbb{C}$$
ADV 07

Calabi-Yau Manifold

Compact Kähler manifolds with vanishing first Chern class — the extra dimensions of string theory.

$$\Omega_{ij} = \frac{\partial^2 K}{\partial z^i \partial \bar{z}^j}$$
ADV 08

Schrödinger Wave Function

Quantum wave packet evolution — probability density, interference, tunneling.

$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi$$
▬ |ψ|²   ▬ Re(ψ)   ▬ Im(ψ)
ADV 09

Spherical Harmonics

The eigenfunctions of the angular Laplacian — atomic orbitals, gravitational fields.

$$Y_l^m(\theta,\phi) = \sqrt{\frac{(2l+1)(l-|m|)!}{4\pi(l+|m|)!}}P_l^m(\cos\theta)e^{im\phi}$$
ADV 10

Monte Carlo π Estimation

Stochastic geometry — estimating π through random sampling.

$$\pi \approx 4 \cdot \frac{\text{points inside circle}}{\text{total points}}$$
Total Points0
Inside Circle0
π ≈—
Error—
ADV 11

Lyapunov Fractal

Stability and chaos revealed by the Lyapunov exponent heatmap.

$$\lambda = \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\ln\left|f'(x_n)\right|$$
ADV 12

Kuramoto Oscillator Network

Spontaneous synchronization — how fireflies, neurons, and clocks align.

$$\dot{\theta}_i = \omega_i + \frac{K}{N}\sum_{j=1}^{N}\sin(\theta_j - \theta_i)$$
Order Parameter r0.00
Sync StateIncoherent
ADV 13

Boids + Chaotic Attractors

Emergent flocking shaped by underlying strange attractors.

$$\mathbf{v}_i \leftarrow \mathbf{v}_i + \alpha\mathbf{s}_i + \beta\mathbf{a}_i + \gamma\mathbf{c}_i + \delta\mathbf{f}_{\text{attractor}}$$
ADV 14

Cellular Automata Universe

Complex behavior from brutally simple rules — Conway, Langton, Wireworld.

🖱 Click/drag to draw on grid

ADV 15

Mean Curvature Flow

Shapes evolving toward spheres — the geometric heat equation.

$$\frac{\partial X}{\partial t} = H\mathbf{n}, \quad H = -\nabla \cdot \hat{n}$$

Watch as any shape smoothly deforms toward a circle (sphere in 3D)

🎨

Mathematical Generative Art

Where pure mathematics becomes digital poetry

ART 01

Cosmic Fractal Nebula

Fractal Brownian Motion sculpting an infinite mathematical galaxy.

$$ ext{fBm}(x) = \sum_{k=0}^{n} rac{1}{2^k}\, ext{noise}(2^k x)$$

🖱 Move mouse to warp space

ART 02

Fourier Flow Field

Thousands of luminous particles surfing a Fourier-series vector field.

$$\mathbf{v}(x,y) = \sum_{k=1}^{n} A_k\sin(a_k x + b_k y + \phi_k)$$

🖱 Click to seed particles

ART 03

Topological Morphing Sculpture

A 3D manifold continuously transforming between topological identities.

$$\gamma(t) = (1-t)\,\mathbf{S}^2 + t\,\mathbf{T}^2 o \mathbf{K}_{3,3}$$
ART 04

Chaotic Attractor Particle Cloud

Millions of luminous particles accumulating on strange attractors.

$$\dot{x}=\sigma(y-x),\quad \dot{y}=x( ho-z)-y,\quad \dot{z}=xy-eta z$$

🖱 Drag to orbit · Scroll to zoom

ART 05

Recursive Geometric Kaleidoscope

Complex-number rotations and reflection groups generating infinite symmetry.

$$z \mapsto rac{az+b}{cz+d}, \quad egin{pmatrix}a&b\c&d\end{pmatrix}\in ext{PSL}(2,\mathbb{Z})$$

🖱 Move mouse to rotate axis

ART 06

Complex Domain Coloring Art

Phase portraits of complex functions rendered as psychedelic living art.

$$f(z) = rac{z^6 - 1}{z^4 + 0.5}, \quad rg(f(z)) o ext{hue}$$

🖱 Click to recenter · Scroll to zoom

ART 07

Fractal Tree Ecosystem

L-system forests growing, branching, and breathing with mathematical wind.

$$F o FF\!-\!\!\left[\!-F\!+F\!+F\! ight]\!+\!\left[\!+F\!-F\!-F\! ight]$$
ART 08

Voronoi Galaxy Generator

Animated Voronoi cells orbiting like a living cosmic diagram.

$$\mathcal{V}(p_i) = \{x \in \mathbb{R}^2 : |x-p_i| \le |x-p_j|\; orall j eq i\}$$

🖱 Click to add seed points

ART 09

Strange Loop Visualizer

Self-referential recursive geometry — Hofstadter's mathematical mirror.

$$f(f(\ldots f(x)\ldots)) = x, \quad \dim_H > \dim_T$$

🖱 Scroll to dive into recursion

ART 10

Mathematical Light Interference

Wave superposition rendered as luminous interference patterns.

$$\Psi(x,y,t) = \sum_i A_i\sin\!\left(k_i r_i - \omega t + \phi_i ight)$$

🖱 Click to add wave sources