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To celebrate Tau Day, I've decided to write my first ever Rithmatic theory. This one, appropriately enough, will involve circles, circular functions, and τ. There will also be ellipses, but we'll be talking about their relationship to circles. (A slight pause to explain what τ is.) Before I continue, if you haven't seen it already, you guys really need to read KalynaAnne's awesome series on Rithmatics first. She has a guide on how to construct different Lines of Warding, including a couple that were not mentioned in the book: the five-point and the eight-point circles, both of which have been confirmed by Brandon. She even has theories about Lines of Vigor that have also been confirmed, which is pretty cool. KalynaAnne also speculated about elliptic Lines of Warding, including the mysterious Blad Defense (I think her version of that looks very promising). But while most of her work had been quite brilliant, ellipses is where she had one tiny problem. Since elliptical Lines of Warding are stronger where the curvature is greater and weaker where the curvature is less, she initially theorized that smaller circles (which have greater curvature) must be stronger than larger circles. But this theory had one obvious flaw: If larger circles are weaker, then the Great Circle of Nebrask would be pathetically weak, and that's not very likely. Confronted with this, she gave a list of possible alternatives. The only one in her list that actually solves the problem of the Great Circle is this: She proposed that each Line of Warding has a total amount of "charge" directly proportional to its circumference (i.e. the bigger the Line of Warding, the more charge it has). This "charge", which determines the strength of the Line at a given point, is distributed along the points of the line based on curvature, so points on the line with more curvature attract more of the "charge" and become stronger. Since circles have constant curvature, the charge is evenly distributed all throughout. This solution works perfectly well, but I'd like to analyze this problem using a different approach, create a solution that is essentially compatible with KalynaAnne's proposal, and then discuss its implications. Here is what the book says about the strength of an elliptic Line of Warding: The question an observant reader might ask is "Which circle?" This is because, as KalynaAnne correctly pointed out in her discussion on curvature, the curvature of a circle depends on its radius. This means it varies depending on the circle's size, so there isn't a single curvature for all circles that can be compared to that of a point on an ellipse. So, which circle? The simplest interpretation is that each Line of Warding has its own reference circle. This reference circle would of course have a constant curvature (hence a constant strength at each point), and by comparing the curvature of a point on an ellipse to the curvature of that circle, we can determine the Warding strength at that point through this ratio: σP = (κP / κC) * w Where σP is the strength at point P of a particular Line of Warding, κP is the curvature at point P, and κC is the curvature of the reference circle. As you can see, points of greater curvature with respect to the reference circle would be stronger than the reference circle, and points of less curvature are weaker. Finally, w is the coefficient of Warding, which has a constant value for the unit of "Warding strength" being used. We don't know any Warding strength units, so we'll just set w to 1 and ignore it. Note that this equation should also apply to perfectly circular Lines of Warding because circles are also ellipses. In this case, it would make sense to assume that a circular Line of Warding is its own reference circle, meaning κP=κC, so σP=1 for all circles. This solves the Great Circle problem. (Edit: I've been reminded by ccstat of this recent WoB showing that there is in fact some sort of weakening involved when creating large circles, but not as fast as the curvature would indicate. So in fact σP is not 1 for all circles. I formulated a possible explanation, but please read the rest of this post first.) Either way, we still need to define what this reference circle is for non-circular ellipses. I could think of three natural candidates: The inscribed circle of the ellipse ("incircle"), a circle whose radius is equal to the ellipse's semi-minor axis The circumscribed circle of the ellipse ("circumcircle"), a circle whose radius is equal to the ellipse's semi-major axis The circle whose circumference is equal to that of the ellipse (let's call this the perimeter circle or the "pericircle" of the ellipse) (Conveniently, when the ellipse in question is actually a circle, then all four circles are equal to each other. This jives with our assumption that the reference circle of a circular Line of Warding is itself.) Here's a badly drawn diagram of an ellipse and its incircle, circumcircle, and pericircle: In one of her diagrams, KalynaAnne called the incircle of an ellipse the reference circle of that ellipse. Later I'll explain why I don't think the incircle's curvature is the best candidate for the basis of Warding strength. The "pericircle" is an appealing choice because it shares something quite fundamental with the ellipse: the circumference. It also jives with KalynaAnne's "charge" theory which uses the circumference as a basis of total strength charge. However, the equation for getting the circumference of a non-circular ellipse (which involves a factorial, a double factorial, and the sum of an infinite series) is so complex that I doubt Brandon would bother with it. (Although there is a nifty rough approximation of the elliptic circumference involving tau: τ * sqrt((a2 + b2) / 2)) I'm going with the circumcircle as the reference circle, mainly because it is the largest of the three (for non-circular ellipses), and therefore has the lowest curvature. If the front and back of an elliptical Line of Warding really are "much stronger" than a circular Warding, then the κC ought to be pretty low compared to the ellipse's greatest κP. Using the circumcircle of the ellipse as the reference circle creates stronger elliptic Lines of Warding than using the incircle or the pericircle. I am open to counterarguments, though. Feel free to defend either the incircle or the pericircle. Even as I type this, I am growing more fond of the pericircle. Perhaps it's because I invented the term; there is no widely used name for the circle whose circumference is equal to that of an ellipse, so I had to make a name up as I developed this theory. Alas, I need a stronger argument in favor of the pericircle before I actually switch. (Edit: I now believe that the reference circle is the one whose area is equal to that of the ellipse, as I explain later on in the thread.) (More talk about curvature and its relation to circles.) Now for the fun part. Let us play with extreme values for the numerator κP (or the denominator rO, if you prefer the equation I gave in the spoilered note on osculating circles) and see the results σP. If κP is negative, then you're looking at a non-convex part in your Line of Warding (either that, or you managed to draw a hyperbola, which stretches to infinity, so... no). Well-drawn circles or ellipses are convex all throughout. If κP is zero, you've got a line segment (another solution is two parallel lines stretching infinitely in both directions, but that's impossible). P is somewhere on the line segment other than the endpoints, and σP is also zero. A line segment is a degenerate ellipse whose foci are on its endpoints, meaning it's so squashed that the foci have moved as far apart as they possibly can. If κP is ∞, then you are on an endpoint of a line segment. If you used the incircle as the reference circle, you get a weird strength value: σP=∞/∞. Otherwise, you only get the slightly less weird σP=∞. What does it mean that a Line of Warding that looks like a line segment has infinite strength at its bind points while having no strength at all anywhere else along the line? In real life, things get really weird when they start involving infinities; for example, a portion of spacetime with infinite curvature becomes a black hole. I believe something similar happens in the 2D universe of Rithmatic lines when Rithmatic Lines involve infinities in their construction: the very fabric of the "chalk space" is affected, and the effects leak towards "people space". You already know where I'm going with this. I think a Line of Forbiddance is actually a Line of Warding whose infinite curvature at its endpoints has transformed it into something that warps "chalk space", and this is what causes the electromagnetic-like force field that affects both the world of chalk and the world of people. A Line of Forbiddance is a degenerate Line of Warding. Can this theory of degenerate Rithmatics be generalized to other Rithmatic lines? I believe it can. Let us look at two more pairs of Lines: the Line of Vigor and the Line of Revocation. Vigors are basically sinusoidal waves of varying frequency and amplitude. They are used to either move or destroy other Lines. Sine waves are generated through the following function of time: f(t) = A * sin(τft + p) Where A is the amplitude, f is the frequency, and p is the phase of oscillation. The "sin" is, of course, the circular function sine. In Rithmatics, the amplitude is defined by how large your Vigors are. Frequency is probably defined by how many individual waves are drawn. The phase is probably just zero. You could add different sine waves together to form different-looking periodic waveforms, some of which can look quite weird. Such waves are difficult to draw (remember, you need at least two repetitions of a waveform for the Line of Vigor to work) and would be impractical when used as Lines of Vigor. But what if we add an infinite number of different sine waves? Well, that could be interesting. Consider the following summation of an infinite series: f(t) = 2A * (Σ(-1n * sin(nτtf) / n) from n=1 to n=∞) / τ When graphed, that function looks like a sawtooth wave, which looks exactly like Lines of Revocation. The Line of Revocation is just a degenerate Line of Vigor, which is how it can affect things beyond the chalk universe. (More talk on Lines of Revocation in the second part of this post.) Alas, the remaining Lines aren't circle-related. Let's just discuss them briefly. Lines of Making are able to actually harm people once given the Glyph of Rending as an instruction. This Glyph must be a degenerate form of other Glyphs, which seem to all involve straight lines. This makes me think the Glyph of Rending is just a dot. The Line of Silencing, which is four spiral loops combined, is already degenerate because it can affect the outside world. I just don't know what it is a degenerate form of. It kinda looks like this, though. I think I need to read up on fractals.