Randomness is a Halo About Symmetry

S.Roof,

June 11, 2003

As per my paper "The Implicate Order of Randomness", I decided to demonstrate my idea that randomness is a halo about symmetry. Recall that I mention two types of order in my paper - symmetry and grouping order. I now have demonstrable proof that the 'randomness factor' given by my theory does indeed peak on either side of symmetry between symmetry order and grouping order. The image above is an edited clip from the real plot shown below.

The most symmetrical ordering of bits is ...01010101010101..... so I take that as my starting conditions. I arbitrarily constructed a bit string 4000 bits long using maximum symmetry as the template. Then I began randomly moving bits from the right half and placing them in random positions in the left half of the string. I wanted to keep the overall bitmass (number of 'one' bits) constant so that the bitmass density was always 0.5. This entailed searching for on and off bits that I could swap. This process gradually moved all the on bits to the left half of the string. I then reinitialized to symmetry and repeated the whole proceedure moving bits from left to right. This gave me the full range of possible variations between maximum symmetry and maximum grouping on either side. Actually it wasn't all possible variations but it does give a fair random sampling of the whole spectrum of possible strings given the 50/50 balance of ones/zeros i.e. 0.5 bitmass density.

After each shuffling of bits, I measured the randomness. This would be the y-coordinate on my plot. I also measured the relative center of mass and used that as the x-coordinate. Both coordinates of each point are thus measurements made on the string as it randomly moves between symmetry and grouping order.

The graphic below the x-axis is just a visualization of the string bits to confirm the center of mass measurements. After each point on the graph proper is plotted, a vertical line is drawn below with red being 'on' bits and gray being 'off' bits. The graphic thus gives a running depiction of the string as it changes. As expected, when the clumping is at a maximum, say on the left, then the center of mass of that clump is 0.25 because the clump is bits/2 long.

The center deep cleavage is the maximum symmetry situation at bitmass density 0.5. One can easily see that my randomness measure reflects that.

Q.E.D!