Friday, December 1 2006
23:42
When writing numbers in base seven, the only valid digits are 0 through 6. Moving digits to the left means multiplying by seven; to the right, dividing. So, writing "125" in base seven means "one times seven squared, plus two times seven, plus 5": the number sixty-three. And, writing 3.4 means "three and four sevenths", or 3.57142857 etc written in normal base 10 (decimal).
In decimal, the ratio "one seventh" has an interesting property. It is written as 0.142857142857... a repeating decimal, meaning that it just keeps repeating 142857 over and over again. That's not so uncommon, but this is: if you multiply the result by 2, 3, 4, 5, or 6, the results look very similar, only shifted a bit:
1/7 = 0.14285714285714...
2/7 = 0.28571428571428...
3/7 = 0.42857142857142...
4/7 = 0.57142857142857...
5/7 = 0.71428571428571...
6/7 = 0.85714285714285...
Oh, and 7 times 1/7 = 0.99999999999999.... which is basically a funny way of saying "one". But an interesting one.
Seven is known as a "
cyclic number" because of these properties. In decimal, there are more cyclic numbers, however seven is special: it is the only one where the set of repeated digits, 142857, does not have any duplicates. I'll somewhat arbitrarily call this a "pure" cyclic number. You may also have noticed that there are four decimal digits that do not appear in the set: 0, 3, 6, and 9. If we were to, say, make some repeated decimals out of these digits, we'd get:
0/3 = 0.00000000000000...
1/3 = 0.33333333333333...
2/3 = 0.66666666666666...
3/3 = 0.99999999999999...
And there's that funny way of writing one again. And really, since one can also be 1.0000000000..., there's a certain something shared by 0/3 and 3/3. What we end up with putting all this together is a peculiar repeating sequence using six digits, plus four left over digits that divide the space up between the whole numbers into three parts. If we were to put these digits around a circle (with 0 and 9 on the same spot), connecting each set, we might end up with something known as the
Enneagram, a symbol which has taken on various meanings and uses in history, but that's another subject.
Moving on... these interesting properties depend heavily on the fact that we're writing the numbers in decimal. In another base – say, base seven, things change drastically. Because of course, one seventh is just written 0.1 – nothing repeating there. I'm not sure if there are cyclic numbers in all bases, however I know that not all bases have "pure" cyclic numbers. (see the Wikipedia link above, look at "Other numeric bases" – none of the examples there are "pure").
I went looking for some other pure cyclic numbers a while back, and found one pretty quickly: in base seven, the value "one fifth". In base seven, you have to write that as 0.12541254... that's "one seventh plus 2 forty-ninths plus five three-hundred-forty-thirds plus ... " you get the idea. So, we have:
1/5 = 0.1254125412...
2/5 = 0.2541254125...
3/5 = 0.4125412541...
4/5 = 0.5412541254...
Oh, and 5 times 1/5 = 0.6666666666... which is a funny way of saying "one" in base seven. If you're thinking ahead, you may have noticed that the digits 0, 3, and 6 never appear in that cyclic number. Making some repeating decimals out of those digits in base seven, we have:
0/2 = 0.000000000000...
1/2 = 0.333333333333...
2/2 = 0.666666666666...
What we end up with putting all this together is a peculiar repeating sequence using four digits, plus three leftover digits that divide the space up between the whole numbers into two parts. If we were to put these digits around a circle (with 0 and 6 on the same spot), connecting each set, we might come up with something called the . . . I dunno. Technically, it's a hexagram, but I've never seen a hexagram that looked exactly like this. It'd look something like this:

Has this been, or will it be, of any meaning or use to anybody?