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# Wau: The Most Amazing, Ancient, and Singular Number: What other amazing properties of Wau can you think of? Leave them in the comments.

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- 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 seemed to have forgotten 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 Illyria.
- 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 and 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 do you find out what this equals?
- Let's see if it converges to something
- If you had just one layer, it would be two over four
- or, one half
- Take two layers and it's three-fourths plus one-fourth
- so that's four-fourths, so this is two over one
- or just two.
- Add another layer that's three over one, plus one over one
- two over four 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 two over one, nor one over two.
- but the curious number wau.
- Here's another way to write wau.
- Wau equals five-sixths plus, all over six, five-sixths
- plus, all over six, five-sixths, plus, all over six
- five-sixths, 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 the wau,
- 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 two pi, to the wau,
- to the four pi, to the wau, to the eight 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 tried raising
- a number to an imaginary power.
- Speaking of which, e to the two i pi equals wau.
- Relatedly, you can find wau in calculus.
- The derivative of e to the wau equals
- wau e!
- And e to the 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 if it's 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 sides 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 the x to the x to the y,
- and y to the 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 you get a golden spiral
- Pretty straightforward. But wau is such a weird number
- if you tried to 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 mc squared, is equal to
- 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 that number
- to the power of that number, all the way to infinity
- and then beyond infinity. So far that it came back
- and became that 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 one.
- What amazing properties of wau can you come up with?

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