# Math

## ThreeJS Rhombic Dodecahedron

I’ve been working with threejs a bunch lately. Here’s a pretty shiny tessellation of a space-filling polyhedron called the rhombic dodecahedron:

## UPDATE: Delaunay Triangulation

Make your delaunay triangulation squirm in the updated version of this project, here.

This version also allows you to toggle between wallpaper and wireframe mode.

I decided to build in this functionality because I wanted to make my polygonal background wallpaper be constantly moving slightly, *but* it’s way too computationally expensive to do that atm. In other words, be warned: your fan will spin up.

## Delaunay Triangulation in JS

My initial goal was to make a dynamically generated triangle pattern for a site, like so:

### The pattern.

That site will generate a new pattern on each visit.

Here you see a set of points, randomly generated, that has been triangulated. In other words, every generated point has become the vertex of a triangle and all triangles are non-overlapping.

### The demo.

In the demo, drag points to see the triangulation recalculate.

## Random Normally Distributed Values in JS

I want random data! But…I want them to be distributed normally!

The ability to generate random data that are normally distributed is very useful. Here’s how you do it in Javascript:

## Math of Bézier Curves

If you are at all interested in SVG or Bézier curves, you’ve probably seen something like Jason Davies’ animation. I found that those animations are an excellent way of intuitively grasping how Bézier curves work. However, the math behind it all is less intuitive.

I just read this really illuminating article.

Something I hadn’t realized before reading the article is that, mathematically, Bézier curves are not defined as run-of-the-mill functions. Whereas generally one would plug an `x`

value into a function to determine a `y`

value, à la `f(x) = y = ax + b`

, Bézier curves are defined *parametrically*. The values of `x`

and `y`

are determined independently, according to a third parameter, dubbed `t`

.

This is the general formula for a cubic Bézier:

B(t) = (1-t)^{3}·P0 + 3·(1-t)^{2}·t·P1 + 3·(1-t)·t^{2}·P2 + t^{3}·P3

where `P0`

and `P3`

are the start and end points, and `P1`

and `P2`

are the first and second control points.

## Pentagram ≅ Pentagon

animation javascript math svg graphs

Pentagrams and pentagons are actually a bit spooky…

## Graph Theory

animation javascript math svg graphs

I’m currently reading a book on graph theory.

A graph, in the mathematical sense, is a completely abstract object made up of sets. However, it definitely lends itself to visual representation, so I’m having a bit of fun making visualizations of some of the concepts.

## Circle w/ Cubic Bézier

This paper demonstrates an extremely accurate approximation of a circle using cubic bezier curves.

It takes 4 curves, one for each 90° circle arc.
As you can see above, I’ve implemented that approximation. The black dots represent the start (`P_0`

) and end (`P_3`

) of each segment, and the red dots represent the control points (`P_1`

& `P_2`

).