Take this series of numbers; 1,2,3,4,5,6,7,8,9,10.
Each of these numbers is different from the previous number by the same amount: 1.
So the series of first differences for this series is 1,1,1,1,1,1,1,1,1.
And the series of second differences is 0,0,0,0,0,0,0,0,0
Its boring. While manipulation of differences can be useful - the Difference Engine after all was not a machine designed to generate social alternatives - its seems to be a rarely used tool in the mathematics toolbox. However its a simple enough tool for my poor brain to understand. Lets try a better series.
The first ten thousand prime numbers:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 ...
Bear with me, I'm coming to a point.
The first differences:
1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4 2 6 4 6 8 4 2
4 2 4 14 4 6 2 10 2 6 6 4 6 6 2 10 2 4 2 12 12 4 2 4 6
2 10 6 6 6 2 6 4 2 10 14 4 2 4 14 6 10 2 4 6 8 6 6 4 6
8 4 8 10 2 10 2 6 4 6 8 4 2 4 12 8 4 8 4 6 12 2 18 6
10 6 6 2 6 10 6 6 2 6 6 4 2 12 10 2 4 6 6 2 12 4 6 8
10 8 10 8 6 6 4 8 6 4 8 4 14 10 12 2 10 2 4 2 10 14 4 2
4 14 4 2 4 20 4 8 10 8 4 6 6 14 4 6 6 8 6 12 4 6 2 10
2 6 10 2 10 2 6 18 4 2 4 6 6 8 6 6 22 2 10 8 10 6 6 8
12 4 6 6 2 6 12 10 18 2 4 6 2 6 4 2 4 12 2 6 34 6 6 8
18 10 14 4 2 4 6 8 4 2 6 12 10 2 4 2 4 6 12 12 8 12 6 4
6 8 4 8 4 14 4 6 2 4 6 2 6 10 20 6 4 2 24 4 2 10 12 ...
Now this seems to be patterned. The pattern reminds me of a wave form with increasing amplitude and decreasing frequency. Also note the new series ..... remove the redundancies and you end up with
One could also ignore every number in the series that is less than the previous number. This generates
2,4,6,8,10,12....
The set of primes having been proved infinite, I wonder if this means that this subset of differences is also infinite, containing within itself all integers that are a multiple of 2...
I would like to graph this, but tonight I've managed to crash Excel three times, and Quattro wasn't any better at handling it. I would think that a 2.2 Ghz processor with 512 MB of RAM could graph a measly 9,999 data points, even with Micro$oft bloatware...
I'll have to find a better program , and one that preferably does not require two hours of study in order to produce a stinkin' graph. If I was going to bother to learn something, I'd figure out how to do this in Mathematica.
I was going to be a bastard and post all 10,000 primes and 9,999 differences, but that seemed anal. I've uploaded both and here are the links:
First Differences
Second Differences
And , of course the primes:
(Ever notice that there is not a word for 'all three'?
I hereby coin 'Throce' to fill the void...)
Now for the second differences:
1 0 2 -2 2 -2 2 2 -4 4 -2 -2 2 2 0 -4 4 -2 -2 4 -2 2 2 -4 -2 2 ...
Note the negative numbers. This causes a fork in the inquiry. It was the negatives that led me to visualize waves in the first place. If I calculate the differences using absolute values, I get:
1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 4 2 2 4 2 2 2 4 2 2 ...
and third differences:
1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 0 2 0 2 2 0 0 2 2 0 0 ...
Amused myself in using the first differences to create a bitmap. Copied a BMP header onto the textfile using DOS EDIT in binary mode; used Find/Replace to deal with the pesky CR/LF characters (that's Carriage Return/Line Feed, a beautiful vestige of the teletype, btw ) ; and then took a peek with photoshop:
The aspect ratio is completely arbitrary. Its a little gray.
I tweaked it a bit, boosted the contrast as high it would go for the pleasant polychrome effect:
Now, if I was feeling progressive, I'd get the computer to generate several thousand copies at various aspect ratios that I could page through and see if any patterns emerged. But not tonight - I'm tired, see?
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