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Discovering the Line of *Spoiler*


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For the same reasons as before, I am not putting the words Line of Silencing in the actual title.


How did Trent discover it?


Thanks to KalynaAnne, we know a lot about the fundamentals of Lines of Warding, and other lines.


Lines of Warding are circles derived from significant points within a triangle.


Lines of Vigor appear to be simple sine curves.


Lines of Forbiddance appear to be simple line segments.


Lines of Making appear to be a fundamentally different kind of line, and I'm going forward on the assumption that as they are a different category, anything we know about them does not pertain to the Line of Silencing.


So. How did Trent discover it? It could have been by simply seeing it somewhere; on a cliff in Zona Arida, or at Nebrask, or in some obscure tome. I highly doubt it was by just drawing random things until he became convinced that one of them was rithmatic.


Or. He could have discovered it by looking at something geometric, and extending it somehow. I propose this is what he did. I think he noticed the arithmatic spiral, and decided to find a significant form which could be made from it.




That's the first turn or so of one arm of a basic arithmetic spiral. I can use it to construct a Line of Silencing... I just can't think of why doing it is obvious. Why someone would do it, and then insist, this must be rithmatic. A trait of the arithmetic spiral is, if you draw a ray from the origin outward, every line segment between times it crosses the spiral will be similar. I also note that the spiral is called arithmetic, which just sounds like the word rithmatic, but I'm unwilling to assume someone would assume this (and be right) based on happenstance of nomenclature; both names derive from the same root, but that means little. Also, technically this is only the positive values of the spiral; for all values you'd need to add the reflection in the y-axis, which would mean it would cross itself every time it crossed the y-axis.


Anyway. Take this (or, I believe, any) fragment of a spiral; double it, and rotate the double 180 degrees (or, if you prefer, reflect it through its origin) and you will get a doubled spiral.




Starting to look familiar? Take four of these segments (two rotated 90 degrees) and stick them together to get:




This might imply that more complicated LoS are possible, if you have more turns in the curve of the basic shape.




Maybe that one would dampen sound even further? Lower threshold for the decibel at which it triggers?


However, this is all pretty hollow. With a starting point, and the fact that I know where it goes, I can connect the dots... what's missing is, why is this OBVIOUS? How could Trent have started from the spiral and made his way to the LoS? What's so obvious about taking eight copies of a spiral, reflecting half of them, and looping them all together?


In addition, I would like to study more about the involution of a circle. Apparently, it's based on a circle, looks like a spiral, and has become fundamental in making gears work more efficiently. This has all the hallmarks of rithmatics; one unseen shape affecting the resulting shape, and it even ties in with gears, like the six, four and nine-toothed gears inside a dollar coin. This seems far more promising. I'm going to do more research and see what I can come up with.

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I did some work with the involute of a circle.


I don't understand the involute very well, but from what I understand, it's this. You build a spiral as per normal, but instead of rotating out from one single point, it is built from a rotation point along a central circle. This has the interesting effect of making every curve not just equidistant like an arithmetic spiral, but also parallel to each other, which the other spiral is not. It also means that (0,0) is not present.




That one is slightly less easy to see than my pics of arithmetic spirals... if someone better at drawing than me wants to improve them, mazel tov.


Now. What happens when we take the involute, and join it together the way we did spirals?




That... doesn't seem like it at all. Something's wrong. Remember, an involute never touches the origin. So what if we instead join them via the circles they are involutes of?




So, what might this mean? We know a very poorly drawn LoW or LoF will still function. And the ones found at the crime scenes seemed lopsided and imperfect, anyway. Maybe an ideal LoS has gaps to represent the missing circles? The way a nine-point circle has nine salient points due to the invisible triangle, maybe four invisible circles mark the gaps between the four branches? Closing that gap means it still works, just imperfectly; that's why you can still whisper or talk a bit before it triggers, and that's why they ran out of power by morning.


There's still so much I don't understand. A four-point LoS works if you join the branches at right angles... but why? Why not use 45 degrees and make an 8-point LoS? A spiral continues to infinity. There has to be a reason, something about stopping the angle where you do, that convinced Trent this was the rithmatic expression of a spiral.What am I missing?


Is this the kind of question I could ask Ben McSweeney during his AMA? I've only been part of one AMA ever and it was Mr. Sanderson; what I like best about him is, he's atypical. Which means I have no idea the etiquette of a normal AMA. Should I just ask him what his favorite kind of puppy is or something generic like that? Or can I ask him questions about rithmatics?

Edited by Oudeis
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Two points:

The fibonacci sequence is closely related to music. Not only does it form the foundation of how we separate musical notes (13 notes in a span of an octative, scales are composes of 8 notes, the 5th and 3rd of which create the foundation of chords, etc), but also in how we arrange musical (and actual) time. There have been composers who have specifically included these concepts in their art, and even those who have not have often produced works in which this sequence (and the golden ratio) are important. Supposedly, many insruments are also constructed with the Golden Ratio in mind.


So, yes, it makes sense that a shape related to the fibonacci spiral would affect sound.


Second, there may also be a connection to the Quadrivium. These are part of the old classification of the Liberal Arts. It comprised arithmatics, geometry, astronomy, and music. The first three have already been displayed as being important to rithmatics: it would make sense for the 4th to have a greater role than had previously been known.

Of course, none of that means that the Line of Silence is actually related to a fibonacci spiral, but a closer look at the later makes a good argument for it. Based on what we know about the other Rithmatic Lines (save those of Making), they are closely tied to geometry, so we should expect the Line of Silence (and Revocation) to do likewise. The Golden Ratio in turn produces the Golden Triangle, which looks to be important for Lines of Warding and bind point. The same Golden Ratio produces a Golden Spiral, which may be the basis of the Line of Silence. The conundrum comes, though, from the two spirals per segment of the Line of Silence. For that, I have no idea.

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