Interactive three-dimensional simulations & visualizations

Visualizing the beauty in physics and mathematics

Project maintained by zhendrikse Hosted on GitHub Pages — Theme by mattgraham

Somehow it’s okay for people to chuckle about not being good at math. Yet, if I said “I never learned to read,” they’d say I was an illiterate dolt. — Neil deGrasse Tyson


Mathematics


Plots for $f(x, y) \rightarrow \mathbb{R}$


The application below let’s one render functions in two real variables $x$ and $y$. The color-coding below is associated with the height of the object and is added purely for aesthetical purposes.

Multivariate functions    
Surface plot for $f(x, y) = \sin(\pi x)\cos(\pi y)$.
Complex functions
Surface plot for $f(z) = \exp(-z^2)$.

Scalar fields $f(x, y, z) \rightarrow \mathbb{R}$ and vector fields $f(x, y, z) \rightarrow \mathbb{R}^3$


A field is an algebraic structure that is defined as a non-empty collection with two (binary) operations: addition, $a+b$, and multiplication, $a\cdot b$.

These operations are accurately defined by the conditions they must suffice, but roughly speaking they should behave similarly as we know them already from the rational numbers $\mathbb{Q}$ and real numbers $\mathbb{R}$.

The applications below render two such fields, that are abundant in physics, namely scalar fields and vector fields. The former assigns a value (scalar) to every point in space (e.g. the temperature in a room), the latter a vector (e.g. the force and direction of the wind).

Vector field
Rendering of three-dimensional vector field and implied flow.
Divergence and curl
Visualization of divergence and curl.

Scalar field
Rendering a temperature distribution as a 3D scalar field.

Differential geometry


Twisted torus
Surface plot of twisted torus. For more surfaces, visit the Math Art Gallery.
Self-intersecting disk
Contour plot of self-intersecting disk. For more surfaces, visit the Math Art Gallery.

Cellular automata


Conway's game of life
Conway's game of life.
Ising spin model

Mandelbrot & Julia sets


Mandelbrot set
Check out this page and learn about the difference between Mandelbrot and Julia sets!

Fractals


Dragon curve fractal
Generation of two-dimensional fractal shapes.
Vicsek fractal
The Vicsek fractal generated using a chaos game.

Sierpiński's pyramid & Menger's sponge


Sierpinski pyramid
Sierpiński pyramid is a 3D analogue of the Sierpiński triangle
Menger sponge
Menger Sponge is another famous three-dimensional fractal curve.

Lorenz & Rössler attractors


Lorenz attractor
VPython program that renders the Lorenz attractor.
Rössler attractor
VPython program that renders the Rössler attractor.

Fractal terrains


Fractal terrain    
Fractal terrain surface.
Fractal terrain
Fractal terrain contour plot.

Dalton board and harmonograph


Galton board
A Galton board that demonstrates the binomial distribution.
Harmonograph simulator
A three-dimensional harmonograph simulator.

Spherical harmonics


Spherical harmonics
Spherical harmonics play an important role in both physics and mathematics.
Spherical harmonics
Spherical harmonics play an important role in both physics and mathematics.

⇓ Python code snippet for plotting spherical harmonics ⇑ The spherical harmonic function is given by $$\begin{cases} \rho & = 4 \cos^2(2\theta)\sin^2(\phi) \\ \theta & = [0, 2\pi] \\ \phi & = [0, \pi] \end{cases}$$ This can then easily be translated to the graphing software, that can also be seen in the mathematics section on this page: 
def sphere_harmonic():
    theta = np.linspace(-1.1 * pi, pi, 100)
    phi = np.linspace(0, pi, 100)
    U, V = np.meshgrid(theta, phi)

    R1 = np.cos(U.multiply(2)).multiply(np.cos(U.multiply(2)))
    R2 = np.sin(V).multiply(np.sin(V))
    R = R1.multiply(R2).multiply(4)

    X = np.sin(U).multiply(np.cos(V)).multiply(R)
    Y = np.sin(U).multiply(np.sin(V)).multiply(R)
    Z = np.cos(U).multiply(R)
    return X, Y, Z, None, None

Numerical methods


Numerical methods
Comparison between an exact solution and those from various numerical methods, such as Euler method and Runge-Kutta methods (2 and 4).
Numeric integration
Illustration of using polar coordinates when numerically integrating spherically symmetric functions.

References


Computational Physics

Mathematics


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