Wythoff construction

Okay kiddo, have you ever played the game of pick-up sticks? It's a game where you have a bunch of sticks and you have to pick them up one by one without moving any of the other sticks. Well, the Wythoff construction is kind of like that game but with a bunch of lines instead of sticks.

First, we start with two lines that cross each other like this:

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Now, we imagine that each line represents a number line, and there's a dot at the point where the two lines cross. This dot represents the number 0.

Next, we have to pick a number to start with. Let's say we pick the number 1. We draw a line starting at the dot and going out to the right like this:

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This new line represents the number 1.

Now, we have to pick another number, let's say we pick the number 2. We draw a line starting at the dot and going out to the right, but we make it twice as long as the last line:

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This new line represents the number 2.

We keep doing this, picking bigger and bigger numbers and making the lines longer and longer, until we can't fit any more lines on the page.

The cool thing is that if we keep doing this, all of the numbers between 0 and 1 will be represented by lines that are shorter than the line representing 1, and all of the numbers between 1 and 2 will be represented by lines that are shorter than the line representing 2, and so on.

That's pretty much it! The Wythoff construction is a way to visualize all of the numbers on the number line between 0 and infinity using a bunch of lines that get longer and longer. It's kind of like playing pick-up sticks, but with lines instead of sticks!