Well, what I got from this was this:
He tried to find a way to represent any positive integer in a wave packet.
Now, the easiest way to represent a number as a wave packet would simply be to go:
0 = 0Hz
1 = 1Hz
2 = 2Hz
...
but he apparently wanted to find a way that is more compact? - His representation allows up to two frequencies to be used. Why? No idea, but that's what he did.
So apparently he worked on an appropriate representation for a whopping 40 years.
What he came up with was, that you could use frequencies of powers of two for this purpose, using the following construction:
0 would simply be 0Hz.
1 would be 1Hz.
2 would be 2Hz.
3 and beyond would be different though:
for
you'd first half the number (rounding down to a full integer each time) until you reach 1 and count how many steps that took.
This happens to be equal to
 \right \rceil)
, if I'm not mistaken. That's the variable
he uses.
With this he finds the two frequencies he chose for his representation:
and
.
More conventionally you might want to name those
and
or something.
From here you must construct how often to repeat a wave (e.g. how often to go through
) to uniquely represent your given integer.
He uses an additional constraint here: Every integer must be exactly a second long. i.e. precisely a full 1Hz wave must happen.
So this restricts how many repetitions of the lower- and higher-frequency wave you need. For instance, this restriction automatically means that you can only ever have an even number of repetitions of the higher frequency since else the intervals wouldn't add up to be precisely one second long.
Next, to get to the number of high frequencies necessary, you take the original number
and subtract from it the low frequency
and, as anticipated to be necessary, you make sure it is even by multiplying that by 2:
)
.
From there you then find the number of repetitions for the low frequency to be your original number minus the just found high frequency repetitions:
. - This gives rise to a pattern you can see in this notation, namely that the sum of repetitions of the high and low frequency (the numerators of the fractions of his weird fractional notation) is equal to the original integer.
This construction also causes all integer powers of 2 to only feature a single frequency. (i.e. the numerator of the left fraction, which denotes the repetitions of the the lower-frequency wave is 0 for those values, so the lower frequency doesn't actually happen and the corresponding representations coincide with the more straightforward representation given in the beginning.)
Here is his notation and what may be a more usual notation side by side wherever I actually get his notation:
... by this he means, that he can write the wave representation of a given number as the lower frequency wave plus/followed by the higher frequency wave.
\to1\Leftrightarrow)
"count how often you need to halve T to make it equal to 1. Disregard the digits after the decimal point"
So basically, this is a substitution/iteration.
)
actually doesn't mean "
", it means "
" as in "replace
by
".
Meanwhile, the arrow doesnt mean a limit in the usual sense. It's not "This is what this will go against" per se, nor is it "replace the previous expression with the following", both of which would be more usual interpretations for it. No,
here means "stop once you reach the following".
So something like
\to c)
really means
s=0;
while(a!=c)
{
a=b;
s++;
}
What I'm not entirely sure about is what the logic behind the varying prime notation (i.e.
would mean.
Since only one of the various symbols even uses
I'd almost think it's a typo if it didn't consistently happen throughout the entire... err... work.
The various instances of
and
are just what gotta be the most confusing way to denote units as of yet. You can completely ignore them. They literally just mean "this is a wavelength" or "frequency" respectively. Mentally set them to 1.
And as for the "Jewelz-Set" I have a strong hunch that this is basically the same idea iterated upon.
My guess is that this "Jewelz-Set" for 9 would be the following:
=2_4\left(\left(4_32\right)_31\right))
Where now, 9 is represented solely by powers of 2. (I simply iterated the same algorithm for all values that aren't a power of 2. If you continue from here, you'll just end up with the same thing repeated over and over.)
In his own notation this would be something like:
It's basically just confusing and pointless. It does make for some neat fractal patterns of smaller and smaller numbers in more and more towering fractions though, so at least it has that going for it which some may find nice. (As said you could totally continue this ad infinum if you don't stop at powers of 2. You'd just get more of the same powers of 2 and a bunch of 0s though, which doesn't seem particularly useful (even relative to any of this) but it does maximize fractalness.)
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May 14, 2008, 12:23:13 PM
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