Non Euclidean Geometry

Rodrigo Vilanova asked,

"Please confirm that non-Euclidean geometries are in fact Fractal geometries."

While I have an answer for that, I don't think it's very funny, so I'm not going to put it in the show. However, I thought that you, the enterprising Lovecraft fan, might be interested in my guess at how Non-Euclidean Geometry works. So, here goes:



Well, yes and no. You see, Fractal geometry is an expedient to explain repeating complex patterns in nature by reducing them to simple repeating formulas, and as far as that goes, yeah.

The problem is that non-Euclidean geometry is based on a whole different number system. A basic ruleset for Euclidean geometry is that the shortest distance between two points is a straight line. Based on that, you can tell the distance between two points, you can define an object as a series of points, and you can map out the world. Unfortunately, in non-Euclidean geometry, there is no shortest distance between two points. Not a straight line, not a Bezier curve, not even a definable parabola.

Imagine that you treated the distance between two points as a four-dimensional value, where it is not only based on the length, depth, and breadth of the points, but also based on where they are at a given time. This assumes that all points move in a constant, but unpredictable manner. Now, despite the fact that distance is a four-dimensional value, it is still a single value (for the undefinable determinant of time).

But it gets worse. Now, instead of a four-dimensional line, you make it an n-dimensional line (where n can be a freaking huge number). There's almost no way of knowing from one moment to the next where a point is going to be, using your static form of math.

Now, add to that the fact that these almost undefinable values are just the simple functions used as a foundation for the more complicated fractal, and you add an order of magnitude more difficult than you had before, just to figure out the distance between two points.

Hope that helps clear up the issue.