Yes, that's really another good thus find, especially with MDawg, and, MrV, going at it. Ha.Originally Posted by DGenBen
Now to make the connection between the ET (Einstein Tile), and, my theory of everything.
The tile's Achilles' heel is the long side between two short sides, which has two corresponding parallel sides. Each of the thirteen sides has a corresponding parallel side, but, that side has two corresponding parallel sides. So, an asymmetry in the tile. Incidentally, the blurb wasn't written nearly so well as with the one for the Penrose tiling, back in 1993. Specifically, in more precise terminology, a thus tiling must be "non-random non-periodic", which, loosely speaking, is akin to symmetrically asymmetrical, but, there's a deep reason for the non-non definition of thus things.
My theory of everything, in general, has to do with the numerals growing out of themselves, alongside the same process for dimensions, which then grow together with the numerals (extensions) as they sort themselves out. Which leads to the simple underlying numerical progression: from 0, to 1, and, next, from 0/1, to 2, and 3, and, next, from 0/1//2/3, to 4, 5, 6, and 7. By which the only other true symmetries involve that of the 5, and 7, -fold. Note that 2, and 3, involve 4, or 8, and 6, -fold symmetries, respectively, with the other symmetries of numerals/dimensions beyond 16 entirely based equally on those two. The digits reduce to two sets of digits, and, their inter-symmetries in their overall groupings from 1, to 16. Namely, the sets, {0, 3, 5, 6, 9}, and {1, 2, 4, 7, 8}, from which the neutral digits may, at times, be grouped separately, as {0, 1, 4, 5}, with the 10 going back to 5, and, the 4 going to 8.
So, it's not really a surprise that the essentially self- sufficient and necessary, ET tile has to do with 4, or 8, and 6, -fold symmetries. But, the give-away is the number of its sides, at thirteen, along with the aforementioned asymmetry in each tile. What gives is that my theory, in specific, has to do with runs of digits of lengths nine, and seventeen, which overlap where the runs reverse to add together, and, next, add together without reversing, respectively. Ie, the runs of length nine digits add together by reversing from, and overlapping at, their last digits, to form the runs of length seventeen digits, which add together, as are, by overlapping at their first/last digits. Note that the runs of length nine digits form the runs of length seventeen digits, for an average run-length, [(17 + 9) / 2] = 13. And, that the tiles go together in the same way if one of those two corresponding parallel sides above overlaps a side of the next tile, so that each side has, in theory, really only one parallel side. Doing so restores the symmetry in each tile, so that each side has only one corresponding parallel side. The loosely defined paradoxical, symmetrical asymmetry, but, which still lies between (regular) crystalline, and, (amorphous) glassy, chemical atomic structures.
This really makes a lot of numerical sense, going by the successive lengths of the overall run of length seventeen digits, and, of the runs of thirteen sides of the tiles. The thus overlapped sets of seventeen digits add by the sums: 17, (17 + 16) = 33, (33 + 16) = 49, and so on. And, the thus overlapped sets of thirteen sides, by the sums: 13, (13 + 12) = 25, (25 + 12) = 37, (37 + 12) = 49, and so on. Which meet up at multiples of sums of 49 = 2^7 ---> 2/7, and 49 = (50 - 1) ---> 51, which have to do with the digits of the related sums, and products, of the fine-structure constants, within their calculations. I have yet to add the bits of connective explanations in my main write-up, in the blog, of which this bit will be the first, in terms of how the thus constant for 142 was calculated exactly to eight decimal places, for the first half of its overall run of seventeen digits. For some reason, things go the route of the digits of 15 = (6 + 9), and, 51 = (10*6 - 9*1) ---> 1961, instead of the 6's, and 9's, directly, as with the 2's, and 7's. As well, if label the quadrants clockwise from quadrant-0 at the bottom left, to quadrant-2, at top right, which involves the straight-up Fibonacci sequence, at dimension - (1/)5, things work back to quadrant-2 by, (3/)7, (5/)9, (7/)11, and, (9/)13, with the ET, at 13, to coincide with the straight-up Fibonacci sequence. Ie, the ET is quite based on it, even though the basic angles involved are of the hexagon instead of pentagon.
Something a bit far out, which I noticed only a few minutes ago, is that, say, in the three blue tiles at the bottom of the image above, are three 7's with three 2's on top of the 7's. Do they occur only thus together?
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