Category Archives: geometry

The chromatic number of the plane, part 1

About a week ago, Aubrey de Grey published a paper titled “The chromatic number of the plane is at least 5”, which is a really cool result. It’s been widely reported already, so I’m actually a bit late to the … Continue reading

Posted in geometry, proof | Tagged , , , , | 1 Comment

Why drawing orthogons is hard

We’re nearing the end of this little diversion on orthogons. We now know that orthogons are in 1-1 correspondence with orthobraces, and we can efficiently generate orthobraces. The only thing left is to find a way to turn orthobraces into … Continue reading

Posted in computation, geometry | Tagged , , , , , , | 2 Comments

Orthogons and orthobraces

One of these days soon I will get back to writing about primality tests, but for now I am having fun getting sidetracked on orthogons! In a previous post I gave rules for when two orthogons will be considered the … Continue reading

Posted in combinatorics, geometry | Tagged , , , | 4 Comments

Properties of orthogons II

In my previous post I proved three out of the four properties of orthogons I originally stated. Now let’s prove the final property: Every sequence of an even number of X’s and V’s, with exactly four more X’s than V’s, … Continue reading

Posted in combinatorics, geometry | Tagged , , , | 3 Comments

Properties of orthogons I

First things first: from now on, when talking about polygons with only right angles, instead of calling them “orthogonal polygons” I’m going to start calling them “orthogons”, which sounds cool, is much less clunky than “orthogonal polygons”, and doesn’t seem … Continue reading

Posted in combinatorics, geometry, proof | Tagged , , , , , , , | 10 Comments

Orthogonal polygons

It’s time to say more about PWW #21, in which I exhibited things like this: Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to … Continue reading

Posted in combinatorics, geometry | Tagged , , | 5 Comments

Post without words #21

Posted in combinatorics, geometry, posts without words | Tagged , , | 17 Comments