This, THIS! I just got a result that rather shocked me. I wanted to understand the relationship between areas of circles of various bases... which of course is equivalent to what we call circumference, so I set about investigating that with a system of equations, the old familiar circle formulas C = 2pi*r and A = pi*rSalmoneus wrote:The area of a circle might be a basic unit, at least for mathematicians

^{2}. The first of those can be restated as r = C/2pi, which then plugs into the other: A = pi(C/2pi)(C/2pi), and the pi on top cancels out one of the lower ones, leaving A = C

^{2}/ 4pi.

Then, defining a circle of base=1 to be the basic unit of area - let's name this unit to help us keep track, we can call it the "smurf" (sorry, running out of smart ideas here) - I want to know how many smurfs there are in a circle of base=2. I got slightly lost at this point and pharazon rescued me, but here it is: the relationship of Area

_{2}to Area

_{1}is the ratio A

_{2}/ A

_{1}, which means (C

_{2}

^{2}/ 4pi) / (C

_{1}

^{2}/ 4pi), the 4pi terms cancel and we get C

_{2}

^{2}/ C

_{1}

^{2}.

Still with me? A

_{2}/ A

_{1}= C

_{2}

^{2}/ C

_{1}

^{2}. Now C

_{1}and A

_{1}were set equal to one as the circle we started from, and 1 squared equals 1, so the denominators go away too and we can rewrite the entire thing as

**A = C**. That means that, even within this alien circular cyclical modular math,

^{2}*area is still the "square" of distance*. So if we have a circle where the circumference is 10, its area is 100 smurfs. And we arrived at that relationship without ever having to think about any four-sided figures, and it doesn't even have to involve pi! That Area = Circumference

^{2}is just as true in circles as Area = Length

^{2}is in rectangles is surely something plenty of mathematicians are already aware of, but I wasn't.

Aside from surprising me, why is it important? First - because it could be this area relationship rather than that of rectangles that is first hit on by our hypothetical con-people, so it shows that they can arrive at an understanding of the squaring relationship by a totally different path, even if I don't yet know what the details should look like. (Whats above is proof of its possibility, not the path itself.) And second - because it suggests that quite a lot else might be gotten to from the cycle system. And third, though I have no interest in putting in the work to verify it, it also suggests that sphere volume varies simply by C

^{3}and so on up the dimensions, so our conpeople can probably have a fully fleshed system for dimensions and exponentiation based on this too. And lastly, it implies that pi only enters the picture as a

*conversion factor*between line-based and circle-based math (which is apparently true according to pharazon), so our conpeople don't actually need it at all just because they're counting around circles instead of in lines, they'll only need it for converting, just like we do.

So thank you, thank you, for providing the big hint I needed for how to approach area in this cyclic math.