## Voronoi diagram of people enjoying a park

Starting with an overhead shot of people sitting out in the sun in NYC’s Bryant Park, Rod Bogart laid what’s called a Voronoi diagram on top of it. A Voronoi diagram is a way of mapping out areas where any point in a given area is closer to a seed point than it is to any other seed point. You can think of it as a sphere of influence…and in this case, you can see how the park-goers have organized themselves into having their own personal space. As Bogart says:

It’s fascinating to see the real world optimization problem of wanting to get a nice large patch of grass.

I often think about Voronoi diagrams when I get into an elevator.

I stand alone in the elevator, right in the middle, equidistant from the four walls. Before the doors close, a woman enters. Unconsciously, I move over to make room for her. We stand side by side with equal amounts of space between the two of us and between each of us and the walls of the elevator. On the 12th floor, a man gets on and the woman and I slide slightly to the side and to the back, maximizing the space that each of us occupies in the elevator. At the 14th floor, another man gets on. The man in front steps to the back center and the woman and I move slightly toward the front, forming a diamond shape that again maximizes each person’s distance from the elevator walls and the people next to them.

## Elliptical pool table

In a perfect world, if you place a cue ball at the focal point of an elliptical pool table, you can hit it in any direction you want and it will go into a pocket located at the other focal point. Geometry! Of course, in the real world, you need to worry about things like hitting it too hard, variations in the table, spin on the ball, etc., but it still works pretty well.

How would you play an actual game on an elliptical table though? Like this. (Hint: to sink the intended ball on the table, hit it as though it came from the opposite focal point.)

## The fractal and geometric beauty of plants

When you look at some plants, you can just see the mathematics behind how the leaves, petals, and veins are organized.

## Heliocentrism vs geocentrism

With hindsight, it seems bloody obvious the Sun and not the Earth is the center of the solar system. Occam’s razor and all that. (via @somniumprojec)

## Always buy the bigger pizza

Planet Money: always buy the bigger pizza because geometry.

The math of why bigger pizzas are such a good deal is simple. A pizza is a circle, and the area of a circle increases with the square of the radius.

So, for example, a 16-inch pizza is actually four times as big as an 8-inch pizza.

And when you look at thousands of pizza prices from around the U.S., you see that you almost always get a much, much better deal when you buy a bigger pizza.

## The Sierpinski triangle

More than you’ve ever wanted to know about the Sierpinski triangle.

Throughout my years playing around with fractals, the Sierpinski triangle has been a consistent staple. The triangle is named after Wacław Sierpiński and as fractals are wont the pattern appears in many places, so there are many different ways of constructing the triangle on a computer.

All of the methods are fundamentally iterative. The most obvious method is probably the triangle-in-triangle approach. We start with one triangle, and at every step we replace each triangle with 3 subtriangles:

The discussion even veers into cows at some point…but zero mentions of the Menger sponge though? (via hacker news)

## Have you seen this fractal?

Unknown fractal. It’s sort of like a Sierpinski gasket but with circles. (via migurski)

Update: Turns out that this fractal is “the orbit of a circle under a Kleinian group generated by two Mobius transformations”. (thx, david)

## Using a geometric shape called a Reuleaux

Using a geometric shape called a Reuleaux triangle, it’s possible to drill square holes. Click through for all the exciting math!

Update: A video of a Reuleaux triangle rotating in a square. (thx, will)

Update: More on the Reuleaux triangle at MathWorld. (thx, nevan)

Update: The Reuleaux triangle is also the basis for the Wankel engine.. (thx, brian & adam)