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Rithmatics: Part 5 - Curvature, Ellipses and a Guess at the Blad Defense

Let's start by considering a snippit from one of the diagrams in The Rithmatist:

(Note: full diagram can be found at http://brandonsanderson.com/books/the-rithmatist/the-rithmatist/rithmatist-maps-and-illustrations/ )

Now we have a problem, namely, circles don't all have the same curvature. In fact (a slight simplification of) the idea of calculating curvature is to determine what radius circle would best approximate the curvature of the line at that point. A circle of radius r has constant curvature 1/r.

The basic idea here is reasonable though – apparently, lines of warding are stronger when they have a higher curvature. You can think of an ellipse as a circle that has been stretched along one axis. This means that if you start with a circle and then stretch it, we can talk about the resulting ellipse being stronger than the original circle where it curves more and weaker where it curves less. Here is what that diagram might look like if we add in the relevant reference circle:

Assuming this interpretation is correct, there are some important implications. The biggest is probably that the size of the circle used to form a defense matters. If you have two otherwise completely equivalent defenses and one of them is a scaled up version of the other, every point in the wall of the smaller defense will be stronger than the equivalent point in the wall of the larger.

Note: There are at least two potential underlying explanations for what is going on here. One option is that there is a certain strength inherent in a portion of a curve of a given curvature. This is the assumption that I am going to work from here. There is also the possibility that there is a fixed total strength for any closed curve of warding and that this strength distributes itself based on curvature. If we stick to circles and assume that strength and curvature are proportional, the two notions are equivalent. The second option is intriguing, but leads to rather messy calculations when we start looking at more interesting constructions. If I stick with this long enough we may eventually get there. I have no idea which option is correct or whether there is a third one I haven't considered.

One way to think about this (and this is almost certainly an oversimplification of things) is that it might take approximately the same amount of chalkling effort to destroy the entire dark blue segment as to destroy the entire dark green segment in the figure below:

 The important take away is that it should be easier to break a small hole in a large circle than it is to break an equally sized hole in a smaller circle. This means that when you are drawing your initial circle for your defense, you should be actively thinking about how large you really need it to be. It also means that even the weakest point on an ellipse could still be stronger than the wall of a much larger circle.

From an offensive standpoint, this means that the small circles and Mark's crosses added to your opponent's main circle are going to be much harder to affect than their main line of warding. They aren't just in the way – they are actually stronger.

We will talk about ellipses in more depth in future installments, but for now let's close with a guess at what the Blad Defense might look like. All we know about it is that it combines four ellipsoid segments in a non-traditional manner and that it is strong enough that some people think it should be banned from competitions.