Odd Dyadic Continuum
Consider this fractal image, “Octaves S”:
This image illustrates binary notation on the unit interval. To find the binary expansion of a real number, locate the number on the horizonal line, then note, for each wave, whether it is above or below the line. For instance, for the point at 2/3, the ½ wave is above the line; the ¼ wave is below the line; the 1/8 wave is above the line; and so on, corresponding to the binary expansion 2/3 = .10101010… Conversely, with this image one could find the point, given the expansion.
Consider the expansion ½ = .10000 = .01111… This is a form of Zeno’s Paradox: ½ = ¼ + 1/8 + 1/16 + 1/32 + … This double-expansion is at the center of the Octaves image. Just left of the ½ point, the biggest wave is below the line, and all the others are above; and just right of the ½ point, it is the other way around. At every dyadic point (i.e. equal to k/2^n, for some integers k and n), infinitely many bits flip.
I say that Zeno’s Paradox explains Cantor’s Paradox. Any enumeration of the continuum has an antidiagonal with a bit equal to its own negation. The usual reading of that buzzing bit is that the continuum is not denumerable, and there must be infinities beyond infinity. Was more ever made from less? I say that Cantorian bit-flip detects Zenonian bit-flip at a dyadic; so the antidiagonal is a dyadic, the continuum is countable (as the Lowenheim-Skolem theorem allows) and there is only one infinity, and it is paradoxical, as Zeno demands.
A subtler interpretation yields transfinitely simpler mathematics. That is called elegance; sign of a correct theory. But questions remain. Why must infinitely many bits flip at a dyadic? Is it possible to denote the continuum so that at most only one bit flips at any point?
Yes, it is possible! Consider this image; “Odd Dyadic Waves”:
The 1/2 wave passes through the line at 1/2; the 1/4 wave passes through the line at 1/4 and 3/4; the 1/8 wave passes through the line at 1/8, 3/8, 5/8, and 7/8; and the 1/2^k wave passes through the line at n/2^k, where n is an odd number. Therefore the name “odd dyadic”, and therefore at any point, at most one bit flips.
Just as with Octaves, this image illustrates a system of binary expansions for any real number on the unit interval. For instance, at the 2/3 point, the 1/2 wave is above the line, as are the 1/4, 1/8, 1/16, … waves; yielding the odd-dyadic expansion 2/3 = .11111… For the 1/3 point we get the odd-dyadic expansion 1/3 = .01111… The 1/2 point has two odd-dyadic expansions: ½ = .00000… = .10000…
Odd-dyadic notation has a bit-flip at every dyadic, but with only 1 bit-flip at each dyadic.
Odd-dyadic notation extends to the integers. Here is a list of the integers from -15 to +15:
- 1000.
- 1001.
- 1011.
- 1010.
- 1110.
- 1111.
- 1101.
- 1100.
- 0100.
- 0101.
- 0111.
- 0110.
- 0010.
- 0011.
- 0001.
- 0000. = + 0000.
+ 0001.
+ 0011.
+ 0010.
+ 0110.
+ 0111.
+ 0101.
+ 0100.
+ 1100.
+ 1101.
+ 1111.
+ 1110.
+ 1010.
+ 1011.
+ 1001.
+ 1000.
Only one place changes between successive integers. The +/- sign is a sort of infinity-place for this dyadic-binary notation, which flips at zero, where positive meets negative.
You can transform between binary and odd dyadic notation:
Binary to odd dyadic:
. a1, a2, a3, a4, … in binary is, in odd dyadic:
. a1, a1+a2, a2+a3, a3+a4, …
Odd dyadic to binary:
. a1, a2, a3, a4, … in odd dyadic is, in binary:
. a1, a1+a2, a1+a2+a3, a1+a2+a3+a4, …
where + is addition modulo two.
Addition mod 2 is the same as subtraction mod 2 because x+x=0 mod 2 for all x. The binary to odd-dyadic map is the difference between successive places, and the odd-dyadic to binary map is summation of places. Therefore I call odd dyadic the “place difference of binary” and binary the “place summation of odd dyadic”.
Here are more images of how binary and odd-dyadic notation divide up the line.
Horizon:
Odd Dyadic Horizon Edge:
Odd Dyadic Horizon:
Odd Dyadic Octaves S:
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