skaa

Theory: Circles, Lines, and Degenerate Rithmatics

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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.)

 

You might have heard of the math constant π ("pi"), the so-called fundamental circle constant, which is defined as the circumference of a circle divided by its diameter. Some people have recently decided that since the circle's radius is more fundamental than its diameter, the proper fundamental circle constant should instead be the circumference divided by the radius. This is equivalent to 2π, and is called τ ("tau") by many of its proponents. Check the Tau Day link above for more info.



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:
 

Quote

[An elliptic Line of Warding] will be stronger where it curves more than a circle, and be weaker where it curves less than one.

 

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 κPC, 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:

post-6598-0-85027800-1435488161_thumb.pn

 

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.)

 

To make curvatures easier to visualize, you can imagine them as sort of like the opposite of a circle's radius. The smaller the radius, the more curved a circle is (i.e. greater curvature). This property means that even in non-circular curves (e.g. ellipse, parabola, etc.), the curvature at a given point can be visualized as a circle, called an osculating circle, whose curvature matches that of the given point and is equal to the reciprocal of its radius. So, the equation above can be replaced with the following:

σP = (rC / rO) * w

Where rC is the radius of the reference circle/circumcircle, and rO is the radius of the osculating circle at a particular point on the Line of Warding.

Here is a (badly-drawn) diagram showing an ellipse, its circumcircle, and the osculating circle at a point of maximum curvature.

 

post-6598-0-56260700-1435488986_thumb.pn

You can see how much bigger the circumcircle is compared to the osculating circle at that point, meaning that point really is pretty strong. Compare that to this (badly-drawn) diagram showing an ellipse, its circumcircle, and the osculating circle at a point of minimum curvature.

 

post-6598-0-40632900-1435489052_thumb.pn
 


 


 


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.

Edited by skaa
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This is amazing.

 

Thanks! And welcome to the forums! :)

 

Hmmm... I should probably post a shorter theory next time. I think writing all that gave me RSI. :(

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Good work! This is the first rigorous attempt I've seen to unify the 2D-only and 3D-affecting lines. I really like the way you apply this to various different Rithmatic lines, not just Warding and Forbiddance. That last Julia set reference actually is the piece that most convinced me there is some real merit to this approach.

A few points to bring up:

In the recent AMA on Reddit, we got this:

With a circle, there is an innate structural strength that does weaken with larger sizes, but it isn't as fast as the curvature would indicate.

That doesn't change any of the degenerate infinities above, but it does mean that the conclusion that "σP=1 for all circles" is not correct, so there is another factor at play here.

I want to say that i particularly like the way you explicitly approach line segments, including their endpoints. The way to apply infinite conics has so far eluded me. However, the weakest connection here for me is how the treatment of a LoF line segment requires all of the strength to be concentrated at the bind points. IIRC the chalklings preferentially attack the ends of LoF, such as the corners of squares, because they are weaker there than on the flat face, which is clearly still strong enough to present a barrier.

I think it is also important to note that the line of Revocation is not simply a triangle wave, but also has a straight line superimposed on it along the axis, so something else is happening there besides an infinite series. I do think it would be instructive to ask Brandon whether another summed waveform, e.g. a square wave, has Rithmatic functions. That might tell us if we are on the right track with this thought process.

Edited by ccstat
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Ah, that WoB puts a damper on things a bit. Oh well. I wonder what we're missing here. Either there is something that partly counteracts a curvature-related weakening of larger circles (thereby lessening the expected weakening), or the thing that weakens larger circles has nothing to do with curvature per se.

The city-sized circle at Nebrask is apparently not large enough to be rendered useless by this structural weakening, so I'm not going to give up on my equation just yet. I'm just going to assume that the w term in the equation is not really a coefficient, but is rather a variable that shrinks very slowly as the circumference of the circle increases. Perhaps it has something to do with the Earth's own curvature affecting very large circles?

On 6/29/2015 at 8:22 PM, ccstat said:

I want to say that i particularly like the way you explicitly approach line segments, including their endpoints. The way to apply infinite conics has so far eluded me. However, the weakest connection here for me is how the treatment of a LoF line segment requires all of the strength to be concentrated at the bind points. IIRC the chalklings preferentially attack the ends of LoF, such as the corners of squares, because they are weaker there than on the flat face, which is clearly still strong enough to present a barrier.

Yes, Joel even explicitly say that Lines of Forbiddance have no bind points. Also, the curvature rule of ellipses would dictate that the body of the line segment should have no strength at all. My idea which I failed to clarify in the OP is that the endpoints (what should have been the bind points) have been "pulled" into the 3rd dimension by its degenerate nature, and no longer act as regular bind points but are instead the generators of the Forbiddance force field in 3D space.

I've compared them to black holes, obviously an imperfect comparison because chalkings attacking the endpoints don't get sucked into 3D space, but it does make me think that a sort of "Hawking radiation" might be giving the body of the line its inexplicable strength.


On 6/29/2015 at 8:22 PM, ccstat said:

I think it is also important to note that the line of Revocation is not simply a triangle wave, but also has a straight line superimposed on it along the axis, so something else is happening there besides an infinite series.

I do in fact have an explanation or two for the straight line part of Revocations, but I didn't include it in my post because it seemed out of scope at the time. I was going to put it in a separate thread, but since you asked I might as well discuss it here now.

In the book, if you look at the illustration of Joel/Melody's Line of Revocation as it smashed through an enemy chalkling, you'll notice that it moved teeth-end first (i.e. the side where the jagged line transitions into the straight line), with the straight line trailing behind it. This makes sense aesthetically because you'd expect the "teeth" to be the one hitting the enemy, but it also serves as a clue as to how the Line was drawn: It was obviously drawn with the teeth end outwards (facing away from the user's own circle).

Recall that ordinary Lines of Vigor are drawn starting outward moving in. Lines of Revocation could not be drawn the same way, because the side facing outwards is the transition point between the jagged line and the straight line. By necessity, it would be drawn from inwards out.

This gives us two options. The first option is that the straight line is drawn first, from inwards moving out, then returning with the jagged part moving in. This option is problematic because of this scene in the book:

Quote

His arm moved, the rest of him completely still. He lowered the tip of his rifle, and Joel could see a bit of chalk taped to it. Harding drew a Line of Vigor on the ground.

Only it wasn’t a Line of Vigor. It was too sharp—instead of curves, it had jagged tips. Like the second new Rithmatic line they had found at Lilly Whiting’s house. Joel had almost forgotten about that one.

This new line shot forward like a Line of Vigor, punching through several of Harding’s own chalklings before hitting the defenses.

As you can see, Joel thought Harding was drawing a Line of Vigor. If Harding had drawn a straight line first, then surely that would've been mentioned, and that would look more like someone drawing a Line of Forbiddance, not a Line of Vigor.

So we're left with the second option: the jagged waves were drawn first, from inwards moving out, in the opposite direction that Vigor waves would be drawn. This gives me a couple of ideas about the straight line:

One is that it's a "trigger" for the Line of Revocation to commence. Vigors on the other hand are triggered just by lifting the chalk, so this could be another difference brought about by degeneracy.

Another possibility, which I think is more amusing than likely, is that Revocations are triggered automatically as soon as a certain number of triangular waveforms are drawn, even if the user hasn't lifted his chalk yet; this might leave a line of chalk in its wake such as it shoots off, completing the Revocation "look" that we're familiar with, but it would be purely accidental.

Edited by skaa
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Recall that ordinary Lines of Vigor are drawn starting outward moving in.

Really? I was under the impression that the reverse was true. Where does it talk about the drawing direction of LoV? I'd be interested to see the quote or picture.

Edited by ccstat
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Really? I was under the impression that the reverse was true. Where does it talk about the drawing direction of LoV? I'd be interested to see the quote or picture.

It's in the illustration at the end of Chapter 13 entitled "LINES OF VIGOR PART ONE: BASIC USAGE".

Step one: draw a Line of Vigor, starting outward and moving in.

When the chalk is lifted, if the line completes at least two waveforms, it will shoot straight out and continue on until it strikes something.

Edited by skaa
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On 6/30/2015 at 0:03 AM, skaa said:

The city-sized circle at Nebrask is apparently not large enough to be rendered useless by this structural weakening... Perhaps it has something to do with the Earth's own curvature affecting very large circles?

I'd just like to sort of quickly elaborate on this point about Earth's own curvature being a possible solution to the problem of Warding strength. (To be honest I just figured it out yesterday, hence this three-year necro. Yes, I'm a slow thinker. :lol:)

In the original post I listed down three possibilities for the Warding strength "reference circle": the incircle (the circle with radius equal to the ellipse's semi-minor axis), the circumcircle (with radius equal to the semi-major axis), and what I call the pericircle (a circle with the same circumference as the ellipse). Based on my Warding strength formula I then chose the circumcircle as my favorite, as it leads to the strongest Lines of Warding.

But as @ccstat pointed out, using the circumcircle as the reference doesn't explain why Brandon said that Lines of Warding weaken as they grow larger. Brandon also said that this weakening "isn't as fast as the curvature would indicate".

So what's really going on here? Recently, I've realized that there is actually another possible reference circle that I forgot to put in my list: What if the basis of Warding strength is the circle whose area is the same as the area of the ellipse?

If that's the case, something very interesting happens.

 

###

 

Notice that if the world was flat, a circular Line of Warding will always have an area-based reference circle that is identical to it no matter its size, and we're back to the same problem as before...

Fortunately, the world isn't flat (despite what some people say ;)). It isn't a perfect sphere either, but it's close enough that we can use a sphere to approximate the math for our purposes.

The formula for the area of a circle on a sphere differs from the one used for a circle on a flat plane. This is because on an ideal sphere, drawing a circle wouldn't lead to a flat disc but rather to a spherical cap. The surface area of this spherical cap is given by this formula A = τRh, where R is the radius of the sphere, h is the height of the cap, and τ is our beloved circle constant tau. (Side note: It would've been cool to post this on Tau Day, but I just couldn't wait.)

This spherical cap's surface area is obviously bigger than that of a disc on a flat plane, so a Line of Warding on a spherical surface would in fact have a bigger actual area within it than the circle area suggested by its radius (which presumably is the area of its reference circle). This difference actually increases the larger the Line of Warding is. As we know, smaller curvature ratios lead to weaker Lines of Warding. So now we can finally see why larger circular Lines of Warding are weaker: it's because their reference circles are smaller than they are, leading to a smaller curvature ratio with its reference circle.

Note that for normal-sized circles on a planet-sized sphere, the surface area is virtually flat. There's also the fact that Rithmatists normally draw on man-made flat floors. In both cases, the reference circle is still equal to the a circular Line of Warding. The weakening only starts to be noticeable when the circle gets large enough for the spherical cap to be noticeably tall.

Edited by skaa
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Wow. I can't believe someone would put that much thought into a question I've been asking since I read the book. Thank you very much :)

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