Wau: The most amazing, ancient, and singular number

I'm going to tell you about a number that's cooler than pi, more mysterious than the golden ratio, weirder than e or i. Somehow in recent times, we seem to have forgotten about just how incredible this number is, yet many ancient cultures knew of it. Some even worshipped it. It was known to Pythagoras, Ptolemy, and Zeno of Elea. It was independently discovered in East Asia and there's even evidence that the Aztecs were aware of it. It's pretty different from most other numbers you've heard of, but today's mathematicians are becoming increasingly fascinated by this ancient concept. And I want to bring back into the common knowledge how cool, how amazing, this number really is. A fitting name for this number is the obsolete Greek letter digamma, which originally represented a w sound was called wau. Trying to write wau in decimal notation is, I think, a misleading exercise, which distracts from the nature of the number itself. One difficulty is that there's actually more than one way to represent it as an infinite decimal. But wau can be defined in some unconventional ways. Take this unusual fractal fraction. How would you find out what this equals? Let's see if it converges to something. If you had just one layer, it would be 2 over 4 or 1/2. Take two layers and it's 3/4 plus 1/4, that's 4/4. So this is 2 over 1, or just 2. Add another layer, that's 3 over 1 plus 1 over 1, 2 over 4 again. And for any finite number of layers, it will be one of those. But it turns out if you continue this fraction on to infinity you get neither 2 over 1, nor 1 over 2, but the curious number wau. Here's another way to write wau. Wau equals 5/6 plus all over 6, 5/6 plus all over 6 5/6 plus all over 6, 5/6, and so on. This infinite fractal quality of wau lets you do some really crazy stuff. Take this. Wau to the wau to the wau to do way, but this is wau times wau. It's all wau times wau-- I don't even know how to pronounce this, but infinite fractal exponentiation of wau loops back and equals wau. I mean, that's just wau. And wau does have an intimate relationship with other special numbers. Check this out. Wau to the pi, to the wau to the 2 pi, to the wau to the 4 pi, to the wau to the 8 pi, and so on, equals wau times square root wau, times cube root wau times, and so on. I mean, this isn't much more mind boggling than thinking about what might happen if you try to raise a number to an imaginary power. Speaking of which e to the 2 i pi equals wau. Relatedly, you can find wau in calculus. The derivative of e to the wau equals wau e. And e to i, to the e i o is e to the wau to the tau wau wau. You might be tempted to try and solve these things using logarithms, but when you try to take log base wau, it kind of doesn't work. It's like dividing by 0. Cool, right? People talk about the geometry of the golden ratio as a bit special, but it's actually pretty normal. Normal numbers you can make a rectangle with that proportion just fine. Make the proportion wau, and well, you get something that most people wouldn't call a rectangle unless you're a mathematician and being technical. But if x and y are the size of a wau rectangle, meaning if x over y is wau, then x plus x to the y, over y plus y to the x, equals wau. And now if I take the previous two, x to the x to the y, and down here, y to the y to the x. This equals wau. And if I make the next term take just the previous two, x to the y to x to the x the y, and y to x to the y to the y to the x. This also equals wau. And you can keep going and get these non repeating patterns related to the Fibonacci sequence. And this will all work out to wau for any numbers x and y, where x over y is wau. Yeah, wau is awesome. You can make an equiangular spiral when you make the angle phi as you go around and get a golden spiral, pretty straightforward. But wau is such a weird number, if you try and make the angle wau, then the spiral curls up infinitely on itself, collapsing and entangling like a quantum string. Wau, in fact, shows up in physics, too. E to the wau divided by m c squares equals wau squared. And wau shows up everywhere in nature. I mean everywhere, every single flower or tree you see embodies wau. But instead of going on about that, I'll just give you one more crazy concept. Just imagine that you took a number to the power of a number to the power of a number all the way to infinity and then beyond infinity, so far that it came back down and became the number's root of the number's root of the number's root of, all the way past infinity until it gets back around to the beginning. It's not a notion that even makes sense according to standard mathematics. But if you make this number wau, it actually becomes possible to argue that all this equals 1. What amazing properties of wau can you come up with?