Wau: The Most Amazing, Ancient, and Singular Number

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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?