# Making things with Maths

## Steven Wittens

### github / twitter  unconedhttp:// acko.net

“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”

“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”

f=Math; e=document.body.children[$=0]; G="globalCompositeOperation"; Q=.43; P=.05; with(e){ with(style)width=(w=innerWidth-9)+"px", height=(h=innerHeight-25)+"px"; W=(width=w/=2)/2; H=(height=h/=2)/2; g=getContext("2d"); t=w/h} with(g){ scale(W/t,H); translate(t,1); setInterval(function(){ with(E=e.cloneNode(0)) width=height=H, c=getContext("2d"); c.fillRect(0,0,h,h); g[G]=c[G]="lighter"; C=f.cos; S=f.sin; L=f.atan2; q=C($); r=S(q-$*.7)+Q; u=C(r-$*Q)+Q; a=L(q,-u*2); b=L(r,u*u+q*q); n=C(a); o=S(a); N=C(b); O=S(b); $+=P; clearRect(-t,-1,2*t,2); for(i=14; i>4; --i){ v=0; for(j=25; j; ){ M=f.log(j+.2)*Q; j--; _=$-j*.07-i*4; A=C(_+S(_*.8))*2+_*P; B=S(_*.7-C(_*Q))*3; x=C(A)*C(B)*M-q; y=S(A)*C(B)*M-r; z=S(B)*M-u; k=x*n+z*o; _=z*n-x*o; l=y*N+_*O; z=_*N-y*O; lineTo(k/=z,l/=z); lineWidth=P/z; strokeStyle="hsl("+60*S($-z)+",60%,"+~~(40-j)*(Q+!j+(.1>($-j*P)%1))+"%)"; if(z>.1)v++&&stroke(); else{ v=0} beginPath(); moveTo(k,l)} } A="drawImage"; N=H/2; c.globalAlpha=Q; c[A](e,0,0,H,H); X=k*N+N; Y=l*N+N; K=1.1; c.translate(X,Y); while(i--)c.scale(K,K),c[A](E,-X,-Y,H,H); g[A](E,-t,-1,2*t,2)} ,33)}

WebGL

Framework

Live Graphing

Slideshow

Typesetting

# Background

Tip: Hold shift to slow down animations.
$$y = 0.5 - 0.5 \cdot \cos x$$
$$r = 0.5 - 0.5 \cdot \cos \theta$$
$$r = 0.5 + 0.5 \cdot \cos 8 \theta$$
$$\begin{array}{rl} r = & \arcsin(0.5 + 0.5 \cos 8 \theta) \\ & ( 0.5 + 0.5 \cos \theta) \\ \end{array}$$
$$r = 0.4 + 0.4 \cdot \cos ( 1.1 \cdot \theta ^{2} )$$

# Sandboxes

Grapher.app
$$y = \frac{x+1}{x-3} \frac{1}{x} \frac{1}{x+2}$$
$$map(x) = \left\{ \begin{array}{l} 2 - \frac{1}{x} & x > 1 \\ x & |x| \le 1 \\ -2 - \frac{1}{x} & x < -1 \end{array} \right.$$

GLSL Sandbox

Processing(.js)

Web Audio

Games

3D Prints

Arduino

# 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
$$lerp({\vec a}, {\vec b}, t) = \vec a + t \cdot (\vec b - \vec a)$$
Cubic Bezier Curve
Bicubic Bezier Surface

# Procedural Generation

Textures

3D Models

Maps / Terrains

Fractals
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

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

function (x) { return x }

# $$x =$$

$$\ldots$$
$$-4$$
$$-3$$
$$-2$$
$$-1$$
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$\ldots$$
$$f(x) = x$$
$$f(x) = \frac{x}{2}$$
$$f(x) = \frac{x}{2} + 1$$
$$\class{mj-blue}{f(x) = 2 \cdot \arctan x}$$
$$\class{mj-green}{g(x) = \sin 6x}$$
$$\class{mj-red}{f(x) + g(x)}$$
$$\class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot cos x }$$
$$\class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot cos ( min(-\pi, max(\pi, x))) }$$
$$\class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot cos ( min(-\pi, max(\pi, x))) }$$ $$\class{mj-green}{ g(x) = fractal(x, t); }$$
$$\class{mj-blue}{ f(x) = 0.5 + 0.5 \cdot cos ( min(-\pi, max(\pi, x))) }$$ $$\class{mj-red}{ g(x) = f(x) * fractal(x, t); }$$

# 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
$$f = -k \cdot x$$
$$\class{mj-blue}{\vec p_{i+1}} = \class{mj-blue}{\vec p_i} + \class{mj-green}{\vec v_i} \cdot \Delta t \\ \class{mj-green}{\vec v_{i+1}} = \class{mj-green}{\vec v_i} + \class{mj-red}{\vec a_i} \cdot \Delta t$$
$$\vec p_{i+1} = \vec p_{i} + \int_t^{t + \Delta t}{\vec v(t) \cdot dt}$$

$$?????$$