Thread: 2-d binary representation View Single Post
 2009-02-21, 20:01 #9 only_human     "Gang aft agley" Sep 2002 2×1,877 Posts Balanced ternary seems frustratingly close to a useful alternative to binary in these two dimensional patterns that I am examining. One problem is symbols; if I use either a vinculum (overscore) or an underscore to represent -1, the mark is close to vertically adjacent symbols. The first 5 positive numbers squared using underscored '1's: Code:  111 11 10 11 111 1 11 00 11 111 Adjacent underscore symbols without accompanying '1's are adequate on grid paper scratchings but run together horizontally in computer fonts. Using the minus sign collides with its' ordinary usage in subtraction. The factors of a product are on the pattern edges. Looking at the second square above, the top and left edges represent the value two, but the bottom and right edges hold the value negative two. The basic situation is that the corners share a symbol between the two factors being multiplied. So for these 2D patterns, one factor may end in a symbol that is used for the start of the other factor; also viable is that both numbers share a most significant symbol in a corner as in the representation of 2 squared above. In that number two is "11" and the factors are found on the top and left sides (note negative two is found on the bottom and right sides). There is no representation for some negative products. If you look at 52 above, the sides represent 5 (top or left) or -5 (bottom or right). Negating the entire pattern does make a pattern that sums correctly to -25 but it is an invalid pattern for a product (the center symbol is not correctly a product of the corresponding edge factor symbols. This is expected as no squared integer results in a negative product. Using red for the -1 valued symbols: Code: 111 This invalid pattern but was made by negating the symbols in the 52 pattern. 111 111 Here is a cheat sheet of all the 3 symbol values: Code: -13,-12,-11,-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 111,110,111,101,100,101,111,110,111, 11, 10, 11, 1, 0, 1, 11, 10, 11,111,110,111,101,100,101,111,110,111 One problem with using colored symbols is difficulty with casual cut & paste editing. Lets try to examine the 2D pattern for 5 x -8. Laying out these patterns I choose to place the negative 8 at the top and the 5 on the right side: Code: 101 ?01 101 The center symbol could be determined by the single digit multiply of the middle symbols on the top and right edges. However it can also be determined (as can the bottom edge middle symbol) by knowing that any symbol multiplied by 0 is 0; thus zeros on edges make 0 filled rows or columns. The bottom left corner symbol is determined by multiplying the top left and bottom right symbols. Knowing that the 5 x -8 pattern must total -40, and that the defined locations total -67, I know that the left edge middle symbol must be a '1' to represent 27. Code: 101 101 101 The end result is that the top and right side represent 5 x -8 as can be seen by direct inspection of the edge. The left and bottom represent -5 x 8. This is consistent mathematically and each interior symbol of the pattern is determined by using the edges as if they were the borders of a multiplication table. The crucial thing is that a corner represents the most significant bit in one of the factors and also the least significant bit in the other factor or its' negation. Does this restriction of sharing a digit between factors or their negation on the corners prevent some products from being drawn in these 2D patterns? I haven't checked yet, it should be easy enough to examine but this message is already long and I am taking a break. Other people are welcome to post in this thread. In general I interpret silence as either kindness, a lack of interest, or that I haven't said much worthy of a reply. In all these cases I do appreciate not having been taken too hard to task for my possibly vacuous musings. Last fiddled with by only_human on 2009-02-21 at 20:30