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Found 11 results

  1. 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.
  2. This theory is based off of the fact that Rithmatic lines only work if you believe they will work and know what they do at the same time. This theory's main point is that everyone is a Rithmatist, but most don't believe they are (or could be), so they cannot use the powers. This explains why Melody's entire huge family is Rithmatists, as her parents married as Rithmatists, so her siblings, as a part of an all-Rithmatist family, thought that they would undoubtably become one as well. For each child, the amount of faith needed got smaller. Joel is a Rithmatist if this theory is to be believed. When he draws the chalk line during the Inception, it is stated that "he knew what would happen. His hand passed over the line." His hand passed over because he didn't believe. The Shadowblazes serve as a test of faith. They probably flee from every person who becomes Incepted, just to test whether the person will believe even after the Shadowblaze departs them. The Forgotten do not possess Rithmatic powers; they simply allow their possessees to know that they are, indeed, Rithmatists. This is why Harding could make chalk lines. _______________________________________________________________________________________ If you think this theory is plausible, give it a comment and perhaps an upvote, and I will send you a cookie via forum.
  3. The Tower of Nebrask stood tall above the plains, wild chalklings flowing out from its every crevice. It had stood for ages untold, and soon its master plan would be coming to fruition. Elsewhere, the survivors of a traitorous plot were devising a scheme of their own. “Where’s Caccoo Moreau?” asked Aiden. “Dead.” replied Kadal softly. “The Forgotten got to him.” “What are we going to do?” prompted Aiden. “We’ve lost so many already.” “I have a plan.” interrupted Mya. “We’re going to end this war, once and for all.” “I’ve already told you, Mya, it won’t work! We don’t have enough manpower!” yelled Kadal. “That’s where they come in…” Mya explained. Elsewhere, a group of classmates were reminiscing about their time at Armedius Academy. “It’s hard to believe it’s been this long since we’ve gotten together.” remarked Tendin Throk. “Feels like yesterday we were all fighting in the Melee.” added Vao Temer. “I still should have won that.” bitterly replied Joseph. “Matist was nearly down for the count, but then,” “Yeah, yeah, we know.” interjected Pete Gazon. They had all heard the story by this point. “Speaking of Matist, where is he?” asked Slavista. “Schmoozing it up with the bigwigs no doubt.” responded Miffed. “Jeesh, don’t they know that this is important? We’re here to remember those we’ve, we’ve...” Apla trailed off, tears welling up. Elysian put a hand on Apla’s shoulder. “We all miss them.” “Hey, let’s not all be down in the dumps! We’re here to celebrate their lives, not mope around.” complained Drake. Raising his glass, Jeeves proposed a toast. “To our good friends. May we meet again in the Master’s halls.” The remaining alumni responded with a hearty “Here, here!” Elsewhere, in the halls of Central Command, the Forgotten planned their next move. “Status report!” barked Samuel Kessen. “Thing are going according to plan.” replied Wyatt. “Isaac has dropped the hammer on the Fifth brigade, and Cole has subverted supply lines for the Sixth.” “What about our stragglers?” continued Samuel. Tory Farth fielded that question. “Don’t worry about them. Shanice has already gotten several of our operatives to monitor them.” He shook his head. “The fools plan plan to coerce an Armedius Alumni group into joining them, and then mount a desperate attack on the Tower. We’re one step ahead though, and have had Matist convert several of his classmates. With all of our sabouters, there’s no way the offensive succeeds.” “And you’re sure the conversion was successful? We’ve had too many mistakes lately. We can’t afford another.” demanded Samuel. “Of course, they were successful.” Joel replied. “Ronald was a fluke. That man never had his lid screwed on right to begin with. You haven’t had any problems with Jain, have you?” “No, of course not.” Sam broke out in a sinister smile. “How long have we waited for this? Nothing will go wrong. At long last, it is time for our revenge.” The Forgotten’s eyes glowed white. Elsewhere, a giant squid was on a mission. He plodded across the land, one thing on his mind. It was time for it to smash. The Tower was waiting. All the pieces were on the board. And now, after all these years, it was time for the game to end. Welcome to Long Game 57, The Tower of Nebrask. This is the conclusion of the 5 other Rithmatist games in this sub-forum. I'm Sart, your GM. @Ax's Boyfriend is helping me out as a co-GM. This game will be starting on July 8th, at 8:00 PM CDT. Rules: Player List: Quick Links:
  4. I was thinking of Lines of Forbiddance and I realized that they might have the ability to create levitation. When looking at the wiki, all it says is that they can’t be moved and the wall it creates is perpendicular to its surface. I have questions though. Can the thing the line is on be moved? Can it be rotated? How far does a Line of Forbiddance stretch outward? Does swinging it and hitting something put a force on you or does it just pass through? What if it is initially drawn on the bottom of something or on a slope? Can this make something float in mid air? Depending of how Lines of Forbiddance work, they could get very powerful, very quickly. Ps. Does this count as introducing myself?
  5. Okay, this is something that has been bothering me for a while. Why are chalklings viable? At first I'd assumed that they were super simple, but the book shows them being really detailed. I don't care how fast you can draw, a decent chalkling is going to take at least thirty seconds. A line of vigor can be drawn in less than three. And don't give me 'chalklings don't need to be precise'. If you're good (and any decent rithmatist needs to be) you don't have to draw slow in order to be precise (although I'll admit that no human could draw with the end of a rifle). Any thoughts?
  6. The question is simple. How are Mark's crosses: crossing lines of forbiddance inside the small circles (which appear all over the novel) made? As soon as you draw one, you won't be able to cross it while drawing the other! Do you have to do it "ambidextrously" like Prof. Fitch? Or is it really 4 lines of forbiddance that meet in the middle?
  7. A collection of chalklings, including a unicorn done in something approximating Melody's style. Drawn in chalk on black cardstock.

    © KalynaAnne

  8. Found a Shadowblaze at this restaurant. It's not moving... I think it's dead.
  9. Hello, everyone. I know you're all Brandon Sanderson lovers, so here's a question for you. Would you like to help build a wikia site for 'The Rithmatist'? I noticed there wasn't one, so I made one. Here it is: Anyway, If you'd like to help, just message me on the site. It's brand new, so there are hardly any pages, but I'd appreciate anyone who'd like to join. Or even just tell others that it exists. Thanks
  10. Sorry for the length in advance. When I read this book, the reasoning for the nine-pointer (described via annotated depiction from what I assume is Joel's notebook on page 243) kinda blew over me, and I just wanted to experience the story for once, rather than get caught up in the world's physics. That said, I still promised myself that I would check it out after I was done. I finished the novel, then recommended it to my little brother's friend, who promptly inhaled the darn thing, then asked, "Any other suggestions?" with a huge smile on his face. After this, I finally looked it up on Wikipedia (mostly due to his own interest in Rithmatics), and the results I found then and afterwards were extremely intriguing. First: The 9-point circle is an actual discovery made by Olry Terquem, and has some significance in the geometric world. Second: (From here on out is a thought process) The 9-point circle doesn't quite work for equilateral and right triangles; what do the look like; what are their Rithmatic equivalents? Third: Equilaterals would lend themselves to the six-point circle due to their nature of fusing together three different pairs of significant triangle points that would be fully represented in the 9-pointer (kind of shown in a picture on the top of pages 94-95; I just discovered this disproof of my originality in thought, as well as another in the history section of the aforementioned Wikipedia article). Fourth (fittingly): If both the 9- and 6-point circles can be represented as a relationship between the circle and a single triangle, what about the 4-point circle? The four points form an inscribed square when connected, and a square is essentially two equilateral right triangles stuck together at the Hypotenuse. On a whim, I drew this on a piece of graph paper: I noticed that all nine points were represented, and several at once in the peak (I had the hypotenuse on the bottom), and was then feeling nearly satisfied with my pursuit of Sanderson's use of Trig relations in his novel. Fifth: What about ellipses? The first thing that came to mind was Isosceles triangles, and thus I drew this on the same piece of graph paper: I'm pretty sure that the points at which the ellipse passes through the sides are their midpoints. Obviously, all nine significant points are NOT represented by said ellipse, but it does pass through at least two, probably four, of them. Sixth: This one is best described through simply showing a picture: I was messing around with isosceles triangles, so naturally, I wanted to see what their complement circle would look like point-wise, so I essentially drew up this diagram on my graph paper. That's it for the thought process, but I'm having trouble with a couple. For one, although Lines of Vigor are made from a sine or cosine graph, where did lines of revocation come from? And, I'm completely at a loss as to where the spiraling line comes from. Comments, further proofs, or disproofs? EDIT: sorry for the small pics.
  11. The Eskridge defense with dragon-cat chalklings done in chalk on black cardstock.

    © KalynaAnne