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# Wau: The most amazing, ancient, and singular number

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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 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?