Making things with Maths

Steven Wittens

unconed
http:// acko.net

$$ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t) $$

“I don’t see how it’s doing society any good
to have its members walking around with
vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.”

Paul Lockhart, “A Mathematician's Lament”

$$ Area = \frac{b \cdot h}{2} $$

“The power to understand and predict
the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.”

Bret Victor, “Kill Math”

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Background

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Tip: Hold shift to slow down animations.

$$ y = 0.5 - 0.5 \cdot \cos x $$

$$ r = 0.5 - 0.5 \cdot \cos \theta $$

$$ r = 0.5 + 0.5 \cdot \cos 8 \theta $$

$$ \begin{array}{rl} r = & \arcsin(0.5 + 0.5 \cos 8 \theta) \\ & ( 0.5 + 0.5 \cos \theta) \\ \end{array} $$

$$ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{v} + \mathbf{f} $$

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$$ y = \frac{x+1}{x-3} \frac{1}{x} \frac{1}{x+2} $$

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Math Machines

Bezier Curves

Paul de Casteljau

1959

Pierre Bézier

1962

Vectors

$$ \class{mj-blue}{\vec a} $$

$$ \class{mj-blue}{\vec a} , \class{mj-red}{\vec b} $$

$$ \class{mj-blue}{\vec a} = \class{mj-red}{\vec b} $$

$$ \class{mj-blue}{\vec a} \neq \class{mj-red}{\vec b} $$

$$ \class{mj-blue}{\vec a} + \class{mj-red}{\vec b} = \class{mj-green}{\vec c} $$

$$ \class{mj-green}{\vec c} - \class{mj-red}{\vec b} = \class{mj-blue}{\vec a} $$

$$ \class{mj-blue}{\vec a} + \class{mj-blue}{\vec a} + \class{mj-blue}{\vec a} = 3 \cdot \class{mj-blue}{\vec a} = \class{mj-green}{\vec b} $$

$$ \class{mj-blue}{\vec a} = \frac{\class{mj-green}{\vec b}}{3} = \frac{1}{3} \cdot \class{mj-green}{\vec b} $$

$$ \class{mj-red}{\vec b} - \class{mj-blue}{\vec a} = \class{mj-green}{\vec c} $$

$$ \class{mj-purple}{\vec d} = \class{mj-blue}{\vec a} + \frac{1}{3} \cdot (\class{mj-red}{\vec b} - \class{mj-blue}{\vec a}) $$

Linear interpolation

$$ lerp({\vec a}, {\vec b}, t) = \vec a + t \cdot (\vec b - \vec a) $$

Quadratic Bezier Curve

Cubic Bezier Curve

Bicubic Bezier Surface

Procedural Generation

Iñigo Quilez, Pixar

“To make a film, we have to direct
almost 200 billion pixels.
That's a lot of mathematics.”

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Longitude

Latitude

longitude = cos(time + sin(time * 0.31)) * 2
          + sin(time * 0.83) * 3 + time * 0.02
latitude  = sin(time * 0.7 + 1)
          - cos(3 + time * 0.43 + sin(time) * 0.13) * 2.3

$$ x $$

$$ x =\hspace{3 pt}? $$

function (x) { return x }

$$ x = $$

$$ \ldots $$

$$ -4 $$

$$ -3 $$

$$ -2 $$

$$ -1 $$

$$ 0 $$

$$ 1 $$

$$ 2 $$

$$ 3 $$

$$ 4 $$

$$ \ldots $$

$$ f(x) = x $$

$$ f(x) = 1 $$

$$ f(x) = 2x $$

$$ f(x) = 2x + 1 $$

$$ f(x) = \left|2x + 1\right| $$

$$ f(x) = \left| 2x + 1 \right| - 4 $$

$$ f(x) = \left| \left| 2x + 1 \right| - 4 \right| $$

$$ f(x) = \left| \left| 2x + 1 \right| - 4 \right| - 2.5 $$

$$ \class{mj-blue}{f(x) = 2.5 \cdot \arctan x} $$

$$ \class{mj-green}{g(x) = x} $$

$$ \class{mj-blue}{f(x) = 2.5 \cdot \arctan x} $$ $$ \class{mj-green}{g(x) = \sin 6x} $$

$$ \class{mj-red}{f(x) + g(x)} $$

$$ \class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot \cos x } $$

$$ \class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot \cos ( max(-\pi, min(\pi, x))) } $$

$$ \class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot \cos ( max(-\pi, min(\pi, x))) } $$ $$ \class{mj-green}{ g(x) = thing(x, t) } $$

$$ \class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot \cos ( max(-\pi, min(\pi, x))) } $$ $$ \class{mj-red}{ g(x) = f(x) \cdot thing(x, t) } $$

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Stacked Value Noise

Physics

$$ \class{mj-blue}{\vec p} $$

$$ \class{mj-blue}{\vec p}, \class{mj-green}{\vec v} $$

$$ \class{mj-blue}{\vec p_1} = \class{mj-blue}{\vec p_0} + \class{mj-green}{\vec v_0} \cdot 1 s $$

$$ \class{mj-blue}{\vec p_{1.5}} = \class{mj-blue}{\vec p_1} + \class{mj-green}{\vec v_1} \cdot 0.5 s $$

$$ \class{mj-blue}{\vec p_{i+1}} = \class{mj-blue}{\vec p_i} + \class{mj-green}{\vec v_i} \cdot \Delta t $$

$$ \vec f = \ldots $$ $$ \class{mj-red}{\vec a} = \frac{\vec f}{m} $$

$$ \class{mj-blue}{\vec p_{i+1}} = \class{mj-blue}{\vec p_i} + \class{mj-green}{\vec v_i} \cdot \Delta t $$

$$ \class{mj-blue}{\vec p_{i+1}} = \class{mj-blue}{\vec p_i} + \class{mj-green}{\vec v_i} \cdot \Delta t $$ $$ \class{mj-green}{\vec v_{i+1}} = \class{mj-green}{\vec v_i} + \class{mj-red}{\vec a_i} \cdot \Delta t $$ Euler integration

$$ f = -k \cdot x $$

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$$ \class{mj-blue}{\vec p_{i+1}} = \class{mj-blue}{\vec p_i} + \class{mj-green}{\vec v_i} \cdot \Delta t \\ \class{mj-green}{\vec v_{i+1}} = \class{mj-green}{\vec v_i} + \class{mj-red}{\vec a_i} \cdot \Delta t $$

$$ \vec p_{i+1} = \vec p_{i} + \int_T^{T + \Delta t}{\vec v(t) \cdot dt} $$

$$ ????? $$

Verlet Integration
Runge-Kutta Methods

$$ O(\Delta t) \\ O(\Delta t^2) \\ O(\Delta t^3) $$