CNB2 Necking, Cycling and Squiggling
Hi! How did you get on with the cycle-number triangle T? Did you check out the L-and R-border diagonals? By the way, I define the 0th cycle-number to be 0100000… with the last 0 cycling onwards indefinitely. And I designate it by 0 .
Did you get the next line in T: 010101010 ? And the next: 0110110110 ? I won’t tell you the secret of how to proceed, line by line, just yet. I will however give a couple of clues.
(i) You will have seen that the L and R borders are the same, as they are in Pascal’s triangle. And I can tell you that after that 1 in row R2, they continue with zeros for ever.
(ii) To get the next row, I use a device (pnemonic if you like) which I call a neck-tie, because it looks like one thus:
(iii) What you have to is to place the (0,1)-string which is in T, at the current row … e.g. R8 …, then complete the neck-tie shape by cycling the Row-string down both the arrowed-diagonals(legs of the neck-tie), as far as you need (one more step for each row). That sounds horribly complex, so experiment a bit with what there is already in the T(7) triangle until you see what I mean by ‘cycling’.
Now, look at the Blog title. What do I mean by ‘squiggling‘. I am using the symbols (or squiggles) in the alphabet set {0,1} to fill up T; then from T I will show you (after a few Blogs) how to get an infinite sequence of what I call (0,1)-strings; and each of those strings will be infinite in length. I shall then proceed to call the strings cycle-numbers: and go on to tell you lots about them. You might think then, since I use the ‘binary digits 0,1′ as my alphabet, that these should be called binary numbers or something like that. But you already know about binary numbers; and my numbers are not all like them. I have merely used the alphabet {0,1} because they are convenient squiggles for me to proceed with. You may know that most people in Western Europe thought (until the 15th centuryand more) that the squiggles 0,1,2,3,4,5,6,7,8,9 which now all Western Europeans use to write their numbers, were to be called Hindu/Arabic numerals, and everyone should learn how to use them. Just imagine the opposition there would be to memorizing a dozen multiplication tables, using strange squiggles!
Just to make my point on squiggling, I will show you what I might have done if I had written T using two different squiggle alphabets, such as {#,@} or {x,y}. How long would you stick with my Blog story, if I had?
That’s enough for today. Try to sort out your neck-tie exercises, before the next posting.
Two Cycle-Number Triangles T(6), using different ‘squiggle-sets’