User blog:QuasarBooster/Googology in Geometry

Euclidean geometry is a well formulated system with quite a bit of power. It can describe the entire field of quadratically closed rational numbers, so it definitely has the expressiveness to yield some precise numbers. This made me wonder if the same power could be used to create reasonably large numbers as well. So I devised a function to study the googological abilities of Euclidean geometry. Here are my current results.

Let Geom(n) be the maximum distance between any two points yielded in at most n constructions in Euclidean geometry starting with two points at unit distance.

After some thought, it's easy to come up with 2^(n-1) as a lower bound for Geom(n): draw a line through the two starting points, let one of those points be A, then repeatedly conduct a circle passing through A whose center is the point furthest from A.



But that's absolutely puny! Can we do any better than exponential? Well, we can also construct squares of lengths in geometry. If you start out with vertical and horizontal axes with length 'a' marked on one and the unit length on the other, then a^2 can be constructed in 10 steps (actually you can do it in 9) like so:



It takes five constructions to create such axes and mark off the unit length from just the two starting points. Since doubling a given length requires just one construction and squaring requires nine, it doesn't provide an advantage until n becomes a bit larger (around 25). But this does mean we can do better than exponential, kind of.

'''[Here will be a table showing some lower bounds for Geom(n). Stay tuned.]'''

Thanks to squaring, it seems that Geom(n) is at least as powerful as (very roughly) 2^(2^n). That's... kinda underwhelming. And my intuition says we can't get much better than that either. Maybe it can, but it doesn't look like geometry has a lot of googological potential after all. Oh well. This was a fun investigation anyway, and hey, a double-exponential growth rate is pretty fast according to the average person at least!