Thread by @TivadarDanka

Math Twitter!

I have an annoyingly simple problem that has been bugging me for years. It is about the 3-regular infinite tree graph with a root.

Can we collaboratively solve this problem? I'll explain below.

(Retweet so this reaches as many smart people as possible.)

The 3-regular infinite tree with a root (3RT) is very simple to define.

The root vertex has two children, and besides that, every vertex has two more. This goes on infinitely.

You get the pattern. This is illustrated below.

It is easy to see that 3RT is a planar graph; that is, you can draw it on the plane without any edges intersecting.

I am particularly interested in drawing the 3RT inside a bounded set of the plane without any edges intersecting.

Of course, if you are allowed to draw the edges shorter and shorter, it is possible.

Can you still do it if the edges have to be at least unit length?

This requires some thinking, but the answer is still yes. You can wrap the 3RT around a triangle by utilizing a convergent series.

It is better to draw a picture instead of a formal proof, so here you go.

What I can't solve, and has been bothering me for years: is it possible to draw the 3RT inside a bounded set without crossing edged if the length of the edges has to be exactly one?

I suspect that it is not possible can't, but I don't have a proof. The solution might be a very simple one that I have missed so far.

Let's solve this together!