Sometimes, I think it would be great if I could turn my brain off at will. You know, just stop thinking and float for a bit. This usually comes after I’ve spent the night tossing and turning trying to visualize the fourth dimension. Note: while I’m actually doing that, no way in the world do I want to turn my brain off! No, it’s only after I get up in the morning and drag my carcass off to work that the benefits of being able to turn my brain off exert themselves.

Oh, and if you’re trying to read between the lines, this did not occur last night, although it has in the past. Instead, it happened after I picked up my lunch at Subway this afternoon and started back to work. So it’s not an inconvenient time at least.

Here’s what I thought about. You remember the math stuff (I can see you roll your eyes, you know) about how zero = infinity and stuff like that? And you remember the cool nifty squares I made of numbers in various bases? Well, I think I’ve found a way to combine them.

As way of example, let me give you a base-12 square, where A = 10 and B = 11. Oh, and ou might want to look at this and this before you continue if you need a refresher as to what I am doing:

1 2 3 4 5 6 7 8 9 A B   Sum = 66
2 4 6 8 A 0 2 4 6 8 A   Sum = 60 (1 zero)
3 6 9 0 3 6 9 0 3 6 9   Sum = 54 (2 zeros)
4 8 0 4 8 0 4 8 0 4 8   Sum = 48 (3 zeros)
5 A 3 8 1 6 B 4 9 2 7   Sum = 66
6 0 6 0 6 0 6 0 6 0 6   Sum = 36 (5 zeros)
7 2 9 4 B 6 1 8 3 A 5   Sum = 66
8 4 0 8 4 0 8 4 0 8 4   Sum = 48 (3 zeros)
9 6 3 0 9 6 3 0 9 6 3   Sum = 54 (2 zeros)
A 8 6 4 2 0 A 8 6 4 2   Sum = 60 (1 zero)
B A 9 8 7 6 5 4 3 2 1   Sum = 66

Now, as before with base-10 math, if you give the zero an actual value (in this case, 6) then the sums add up to 66 per row in base-12. Oh, and if you’re wondering, the pattern seems to be this:

Take any even base, b. Then take the midpoint, as m = b/2. Then find m2 + [(m – 1)m]. That will be what each row adds up to, if you assume a zero value = your m value.

So we show this in base 14. In base 14, b = 14, so m = 7. 72 = 49, while [(7 – 1)(7)] = 6 x 7 = 42. 42 + 49 = 91. We can test it by seeing that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 +13 = 91 (this would be your first row if you constructed a box like the above, although it would look like 1 2 3 4 5 6 7 8 9 A B C D).

In any case, you can convert

m2 + [(m-1)m] = 2m2 – m

which is easier to use on a calculator.

In any case, look at what would happen if we included a zero:

1 2 3 4 5 6 7 8 9 A B 0  Sum = 66 (1 zero)
2 4 6 8 A 0 2 4 6 8 A 0  Sum = 60 (2 zeros)
3 6 9 0 3 6 9 0 3 6 9 0  Sum = 54 (3 zeros)
4 8 0 4 8 0 4 8 0 4 8 0  Sum = 48 (4 zeros)
5 A 3 8 1 6 B 4 9 2 7 0  Sum = 66 (1 zero)
6 0 6 0 6 0 6 0 6 0 6 0  Sum = 36 (6 zeros)
7 2 9 4 B 6 1 8 3 A 5 0  Sum = 66 (1 zero)
8 4 0 8 4 0 8 4 0 8 4 0  Sum = 48 (4 zeros)
9 6 3 0 9 6 3 0 9 6 3 0  Sum = 54 (3 zeros)
A 8 6 4 2 0 A 8 6 4 2 0  Sum = 60 (2 zeros)
B A 9 8 7 6 5 4 3 2 1 0  Sum = 66 (1 zero)
0 0 0 0 0 0 0 0 0 0 0 0  Sum = 0  (12 zeros)

Now, let’s add the value of 0 = 6 to the equation above:

1 2 3 4 5 6 7 8 9 A B 0  Sum = 72 (1 zero)
2 4 6 8 A 0 2 4 6 8 A 0  Sum = 72 (2 zeros)
3 6 9 0 3 6 9 0 3 6 9 0  Sum = 72 (3 zeros)
4 8 0 4 8 0 4 8 0 4 8 0  Sum = 72 (4 zeros)
5 A 3 8 1 6 B 4 9 2 7 0  Sum = 72 (1 zero)
6 0 6 0 6 0 6 0 6 0 6 0  Sum = 72 (6 zeros)
7 2 9 4 B 6 1 8 3 A 5 0  Sum = 72 (1 zero)
8 4 0 8 4 0 8 4 0 8 4 0  Sum = 72 (4 zeros)
9 6 3 0 9 6 3 0 9 6 3 0  Sum = 72 (3 zeros)
A 8 6 4 2 0 A 8 6 4 2 0  Sum = 72 (2 zeros)
B A 9 8 7 6 5 4 3 2 1 0  Sum = 72 (1 zero)
0 0 0 0 0 0 0 0 0 0 0 0  Sum = 72 (12 zeros)

And what is the relationship of m = 6 to the result of 72?

2m2
2(62 = 2(36) = 72.

In some way, it appears that because of the way the pattern works, the zero (which is where the pattern repeats) seems to take on the value of the midpoint of the pattern. In other words, zero doesn’t act like “nothing” in these patterns. Zero acts as if it has value, but the value comes not because of the digit itself, but because of the patterns that are formed.

Now here’s the thing. You can extend this out to infinity, at least as long as you’ve got an even infinity for your pattern. Of course, we’d quickly run out of digits to express our pattern, so we couldn’t make a physical box like we’ve done above. But if you have an infinite number base that’s even (when I have time later I may see how this works in odd bases too, but at the moment my lunch break is almost over), then the zero value in the patterns would seem to take on 1/2 infinity as its value.

Of course, this begs the question: does it really take on this value? Obviously in terms of pure math, we’d say, “of course not.” On the other hand, looking at the patterns above, it seems so elegant to get the rows all to the same value that it feels like there’s something to this notion.

In any case, I think it’s safe to agree that zero isn’t what we usually think it is.