Hardcore Rithmatic Theory

[KalynaAnne]

EDIT: I've been informed that the signing report belongs in a different part of the forum and that what I should put here is all of my theorizing with the comments I got today added in, so... In the interest of this not being super ridiculous long, I'm going to link you to the series I've been posting on Tumblr with a note about what each post is about:

 

Part 1  Here I outline how to get all of the binding patterns in the book from special cases of 9-point triangles

 

Part 2 How you could get a 5 point defense as a result of looking at right triangles.  Brandon confirmed the existence of 5 point defenses at the signing today, but said they haven't been well explored in-world.

 

Part 3 How you could get an 8 point defense as a result of looking at right triangles.  Brandon confirmed the existence of 8 point defenses at the signing today, but said they also haven't been well explored in-world.

 

Note: In these tumblr posts (which were written before the signing) I postulate that some bind points might be stronger than others.  Brandon told me that this is not the case. You can always bind more than one thing to a bind point, but binding multiple things weakens the point. It is a much better idea to add a small circle that gets 3 additional bind points. It doesn’t change anything if the point comes from multiple points in the 9-point construction.

 

Part 4 Where I analyze the 9-point circle construction on obtuse triangles and come to the conclusion that they don't give us anything new.

 

Part 5 Lots of thoughts on the changing curvature of ellipses and my guess at the Blad defense. I didn't get a chance to confirm any of this.

 

Part 6 So, there is a generalization of the 9-point circle construction in Geometry that allows for the construction of ellipses.  Here I talk about different triangle centers and how they could be used to produce 9-point ellipses. When I showed Brandon my page of 9-point ellipses constructed from different triangle centers   he stared at them for a moment before answering.  He hesitantly said that, yes, those constructions should be valid in theory, but that they shouldn’t be used in practice.  The sides of ellipses are weak enough that if you expect to need to defend your sides you really should be using a circle.

 

Part 7 More on 9 point ellipses

 

Part 8 Various thoughts on lines of vigor many of which were confirmed today! Things that Brandon confirmed for me at the signing today:

• Yes, Lines of Vigor behave like light waves.
• Yes, this means that higher frequency waves are better for doing damage, lower frequency waves are better for transferring energy (and thus moving things)
• Yes, Lines of Vigor follow the rule that the angle of incidence = angle of reflection when bouncing off Lines of Forbiddance.
• LINES OF VIGOR ALSO REFRACT.  I asked it in terms of whether they slightly change speed and direction when they move between materials like, say, concrete and asphalt.  He said yes and that you also get the wavelength adjusting.

There, ah, may still be more theorizing coming as I get it written up

Edited January 25, 2015 by KalynaAnne

[Kipper]

Hey guys. I asked a bunch of rithmatics questions at the signing today. I'm afraid I don't have a recording (which wouldn't be completely clear because it involved me pointing at drawings in my notebook) or his exact words, but I can pass on what I learned. _{Note: The tumblr posts linked below are meant to help you understand what I mean by constructions, but they were written before the signing and any theorizing not confirmed here is still theorizing.}

• Yes, 5 and 8 point defenses (constructed from applying the 9-point circle construction to right triangles and isosceles triangles respectively) could exist. They haven’t really been explored in world though.
• You can always bind more than one thing to a bind point, but binding multiple things weakens the point. It is a much better idea to add a small circle that gets 3 additional bind points. It doesn’t change anything if the point comes from multiple points in the 9-point construction.
• When I showed him the 9-point ellipses constructed from different triangle centers (see the constructions here ) he stared at them for a moment before answering. He hesitantly said that, yes, those constructions should be valid in theory, but that they shouldn’t be used in practice. The sides of ellipses are weak enough that if you expect to need to defend your sides you really should be using a circle.
• Yes, Lines of Vigor behave like light waves.
• Yes, this means that higher frequency waves are better for doing damage, lower frequency waves are better for transferring energy (and thus moving things)
• Yes, Lines of Vigor follow the rule that the angle of incidence = angle of reflection when bouncing off Lines of Forbiddance.
• LINES OF VIGOR ALSO REFRACT. I asked it in terms of whether they slightly change speed and direction when they move between materials like, say, concrete and asphalt. He said yes and that you also get the wavelength adjusting.

Okay, I totally need to read Rithmatist now. Edited January 25, 2015 by inexorablePanda

[A Joe in the Bush]

This is astounding. I've studied how to make 9 pointers, but I never thought to go further.

[Cheese United]

Well wouuld you look at all the smart people! Actually, I hope some of this stuff makes it into the azlatianainain. It would be nice for us "Not as smart as Kalyna Anne/The only Joe" type people to have that type of thing explained by Brandy Carpenter Kid

[KalynaAnne]

Well wouuld you look at all the smart people! Actually, I hope some of this stuff makes it into the azlatianainain. It would be nice for us "Not as smart as Kalyna Anne/The only Joe" type people to have that type of thing explained by Brandy Carpenter Kid

If you have questions about any of it I would be happy to break things down further!  Just let me know what you are interested in knowing more about :-)

[A Joe in the Bush]

I'm not that Smart, I just mean I studied and figured out how to make the 9 pointers. I wish I had her smarts.

[Tarontos]

Totally called the 8 and 5 point circles. And dude 9 point ellipses are awesome never actually thought of that, So my question to you make hyperbolas and parabola that follow those rules it could be fun thought experiment wise.

Also hooray for another person as obsessed as i was with this.

 

 

9 Scalene

8 iscosoles

6 equalteral

5 right 

4 right iscolese

2 180/0/0

i think that those are every type of circle you can make but i could be wrong

the nine point ellipse sound fun

 

Did you ask how a line of silencing would interact with a line of vigor by chance.

i while back i made a theory that they should dampen the waves as well, similar to sound.

 

what about a smooth curve on Blad defense sbasicly at the points you have sharp turns you smooth it with a partial circle and attach the extra circle via bind point

Edited January 27, 2015 by Tarontos

[KalynaAnne]

Hello fellow Rithmatics nerd :-)

 

I showed up to the signing with a binder full of rithmatics notes and asked more questions than was really reasonable, but I had decided to start with the more basic things (solid foundations are important), so I didn't get to lines of silencing.  I also haven't really thought about how Lines of Vigor would interact with Lines of Silencing - that's an interesting theory.  I did get it confirmed that LoV behave like light. Different things affect light and sound, so I'm not sure if I would expect it to work or not. I'll have to think more on that.

 

Unless we can make circles using something other than the 9-point circle construction, those are the only options. (There is another theorem known colloquially as the Mystic Hexagon Theorem - see http://en.wikipedia.org/wiki/Pascal%27s_theorem - that feels like it is in the same spirit that might be able to lead to other things. I haven't really explored it much in this context. The statement of the theorem is straightforward enough geometry, but the proof requires algebraic geometry, so it isn't particularly widely known even though it is really nifty)

 

You can definitely use the 9 point construction to produce hyperbolas (though this doesn't guarantee that they can be LoW).  Start with a triangle. If you use the orthocenter to form the quadrangle (see the first post I made about ellipses) you get a circle.  If you pick any other point inside the triangle you get an ellipse.  Extend the legs of the triangle.  If you pick a point in one of the sections that is not adjacent to a side of the triangle, that new point can be used with two of the original points to form a triangle containing the third, so you get an ellipse (or possibly circle if you are tricky about it).  If you pick a point in one of the sections that is adjacent to a side of the original triangle, then the three vertices and your new point form a convex quadrilateral and the 9-point construction produces a hyperbola. If you pick the fourth point actually on one of the sides (or side extensions) of the original triangle, then the 9-point construction leads to a pair of parallel lines - which is a degenerate conic. I'm fairly certain this covers all of the cases and that you can't get a parabola from the 9-point construction, but it's late and possible that my brain isn't functioning at full capacity.  (Also, my scanner is being dumb right now, but I'll add pictures to clarify things when I get it working again.)  

 

Note:

Non-degenerate conics can be a : circle, ellipse, hyperbola or parabola

Degenerate conics can be: two lines (which can be intersecting or parallel), a single line (when the two lines coincide), a single point, or the empty set.

 

Crazy theory time:

What if LoW are closed conics and LoF are open conics?  

 

This would mean that Circles and Ellipses can both be LoW, but that other conics on their own can't. You might be able to do something funky piecing parts of curves together to get other LoW.  I tried to ask about the Blad defense at the signing and got a look that fairly clearly seemed to say "you expect me to know that off the top of my head?" I'll have to try again armed with a sheet of possibilities for piecing together LoW rather than asking about Blad specifically next time he comes to Atlanta. [on that note, I like your idea for smoothing it, but it doesn't fit with the description in the book  where we are told that the Blad defense is formed of 4 elliptic curves.  Getting bind points to line up properly is going to take some serious work though.] There's also something funky going on with curvature and the strength of LoW, but I haven't quite thought that part though enough to explain my thoughts yet.

 

Now, LoF.  I got it confirmed that a LoV behaves like light and bounces off of a LoF like it is a mirror.  Since a line is a degenerate conic and *much* easier to draw accurately than other conic segments, it would be easy to believe that someone discovered that straight line segments work as LoF and that since then everyone has just assumed that is the only option.  However, in real world physics, elliptic, hyperbolic, spherical and parabolic mirrors are all legitimate and useful things.  It would be amazing if we could have conic segment LoF.  This would open so many new strategies for working with LoV...  

 

Also, I really want there to be Lines of some sort that behave like lenses when they interact with LoV.  They likely wouldn't be practical for dueling, but they could lead to other interesting applications of rithmatics (we know they use it for at least some non-dueling things).

Edited January 28, 2015 by KalynaAnne

[ccstat]

Great work, KalynaAnne! I'm sorry I didn't run into you at the signing.

 

Crazy theory time:

What if LoW are closed conics and LoF are open conics?  

Oooo, I like where this is going.

Also, the effect on LoS on LoV is a fascinating question. I can't decide what I think would happen.

 

And here's a thought for a great defense against LoV: Rather than one long Line of Forbiddance like this _________, construct a series of short LoF at right angles, to give you a zig-zag pattern like this /\/\/\/\/\. This generates a retroreflector, so any LoV sent at you go approximately back where they started, striking your opponent or his/her constructions.

(That thought came from wondering how to utilize a parabololic LoF to aim them all at one place, but the scale you would need seems too large to be practicable, and we don't have confirmation that open conics are LoF.)

 

A question about your 9-point constructions: You tried the incenter, orthocenter, circumcenter, and centroid, which all gave 9-point ellipses (or circles). Aside from the fact that the constructions in the book use the orthocenter, is there any reason to use a "center point" (by whichever definition) at all? I haven't done any geometry to check this, but my impression is that any point internal to the triangle would similarly generate a 9-point ellipse. Do the "centers" have a special property?

And if all internal points work, and obtuse triangles have external orthocenters, we could generalize to say that any complete quadrangle is valid.... (Okay, I thought about what I just said, and obviously it doesn't matter which 3 points in the quadrangle you choose as your starting triangle, the result will be the same. So I think it is self evident that this must be the case.) My formal geometry is a little fuzzy though, so please correct me if my assumptions are false here.

 

EDIT:

One more question (though this probably doesn't have an answer yet). In the final chapter Joel says, "That was one of the great things about an Easton Defense--a large circle with nine bind points, each with a smaller circle bound to it. Each of those smaller circles could theoretically hold up to five bound chalklings." If our suppositions so far are correct, any circle can be a 9-point circle, but he seems to think that the smaller ones were limited to 6 bind points. It's possible that their theory is wrong here, since it is clearly incomplete in other areas, but I have a hard time believing nobody has tried making the subsidiary circle a 9-pointer.

Assuming that there is a limit, does this suggest that smaller circles have less binding strength to lend? Or that some of the lines/angles used to construct the smaller circle must be shared between it and the one it is bound to? That makes the most sense to me, but I can't see an easy rule that would force the outlying circles into the special cases (with reduced bind points).

Thoughts?

Edited January 30, 2015 by ccstat

[KalynaAnne]

Hi ccstat!  Thanks for recording the signing! We'll have to do a better job of coordinating meeting up next time there is a gathering of Atlanta area Sanderson fans :-)

 

Ooooh I like the retroreflector idea! You have to be careful any time you use LoF since you can't get your offense past them either, but that could be a great defense.  If you have multiple opponents, drawing in a retroreflector line to fend off one/some of them while you focus on others could be effective.  There would be a balance then in how many zigzags you want to use since each segment presumably has to be dismissed individually if you want to get through later and also because you don't want their LoV's to be likely to hit the intersection points...   hmmm... there is more thinking to be done on this topic.

 

Ah. General 9 point conics. I have two pages of case by case analysis showing everything you can get that I'll attach to this post.  Prior to talking to Sanderson at the signing, I was half expecting that there would only be one special kind of point that would give Rithmatically valid ellipses.  That does not appear to be the case.  Those particular centers have the advantage of being easily constructable with just a compass and straightedge, so they seemed like good candidates.  There is also a particularly nice symmetry to the 6 point ellipse that comes from using the centroid.  I've got some more stuff here that I'm playing with, but it isn't organized well enough yet to try to explain over the internet :-)

 

It's possible that the issue isn't so much that you can't have small 9-point circles but rather that it isn't practical.  The smaller they are, the closer together the points are going to be and I could see there just not really being space to add that many chalklings around it.  None of the defenses we are shown ever actually use more than 4 bind points on the small circles.  The most interesting diagram on this topic though is the Eskridge Defense.  My understanding would suggest that the point where the circles touch has to be a bind point for both circles (whether there is anything else they need to match up with or not).  For the small circle in the upper left, it is reasonable that we are treating it as a 6 point circle - that would at least come pretty close to allowing bindpoints at both ends of the LoF and at the point it touches the main circle.  For the lower left circle though...The only way to get those three points is if it is actually a 9 point circle.  One of the things I'm currently working on is trying to determine what kinds of point patterns (right now starting with three points) can be extended to 9-point circles.  Determining where all 9 points would need to be if you know three of them is likely to take far more time than would be practical.

 

There is also something interesting going on with curvature. I have a couple different ideas here that I'm poking at and teasing out the implications of. I'm trying to come up with a theory that would explain why Sanderson was so confident in stating that 9-point ellipses aren't practical for defenses even if they are Rithmatically valid.

 

 

9-point conics analysis in the spoiler tag.

 

 

[A Joe in the Bush]

KalynaAnne In part 5 of your original post, you theorize that a Larger circle would be easier top break than a smaller circle, But doesn't the great circle at Nebrask depend on being Strong? If circles were weaker the bigger they got, there would be no point even having a Circle at Nebrask, since the wild chalklings could destroy it so easily.

 

What I was refrencing:

 

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.

[KalynaAnne]

KalynaAnne In part 5 of your original post, you theorize that a Larger circle would be easier top break than a smaller circle, But doesn't the great circle at Nebrask depend on being Strong? If circles were weaker the bigger they got, there would be no point even having a Circle at Nebrask, since the wild chalklings could destroy it so easily.

Yeah... there are a lot of things that don't make sense to me about what is going on in Nebrask (we know so little...). In the duel between Fitch and Nalizar at the beginning of the book, a straight hit by one of Nalizar's LoV's nearly breaks Fitch's circle. We see other places where chalklings don't take very long to do significant damage. If the great circle at Nebrask is a basic LoW (even at the strength of our familiar smaller circles) and there are any significant number of wild chalklings (plus Forgotten and whatever else is out there), I have a hard time seeing how they hold it, even with large numbers of Rithmatists monitoring it. It would be like... you know those old arcade games where you have to shoot down the invading aliens or asteroids with your laser gun? Where you can run along the ground (around the edge of the circle) but can't fly after them? You can keep those up for a while, but eventually something is going to get by you. I know they have lots of Rithmatists out there working together, but even so... Also, regardless of how large the circle is, you still (presumably) only get at most 9 bind points and you are still, presumably going to need to anchor it with LoF. You can get 9-point circles where some of the bind points are arbitrarily close together, but there are still going to be stretches where your path is blocked by a LoF. There is also the fact that we are used to seeing LoW attacked from the "outside" (where the curvature is positive) and at Nebrask it seems like it would be attacked from the "inside" (where the curvature is negative) and there's no telling how that would change things. I would LOVE to see a diagram of the current defense scheme in Nebrask.

Basically, I don't have a good answer to this, but I have various tentative half-baked theories:

• We know that there are extra things Rithmatists learn when they are ready to head to Nebrask that they don't get to learn while they are still in school (and thus that we don't know about). It is possible that there is some sort of strengthening ward or something that gets used on the great circle.
• 2) Maybe the great circle is the last line of defense and there is a ring of smaller not-quite adjacent circles inside it that the Rithmatists defend from. It could be that it is mostly there to buy an extra few crucial moments to deal with chalklings that get past the main line of defense.
• Maybe there isn't actually a giant chalk circle and there is actually a much more complex set up. Maybe we talk about the "Great Circle at Nebrask" in a metaphorical sense because in this world defenses are synonymous with circles.

Regarding curvature, the part you reference is one of the parts of my theorizing that I am least confident in...more tentative half baked ideas:

In one of the diagrams in the book (this one) we are told that an ellipse "will be stronger where it curves more than a circle and weaker where it curves less than one." As I mentioned in part 5, this statement is naive and doesn't actually say much. At the signing, Brandon acknowledged that 9 point ellipses would be Rithmatically valid but also made it clear that it would not make sense to use them. The implication seemed to be that if you were in a situation where you expected to need to defend the sides of your LoW, you should have started with a circle. (Note that the usual method for determining the curvature at a point in real world math is effectively determining what radius circle would best approximate the curve at that point. We say that the curvature of a circle with radius a is 1/a.) Some possibilities for (at least part of) what is going on :

• The simplest reasonable explanation I've come up with - the one from part 5- would be that every curvature is assigned a strength and that the strength proportional to curvature. Assuming my calculations are correct, this would mean that if you have an ellipse with semi-major axis a and semi-minor axis a/2 (so it is twice as long as it is wide), then the curvature at the strongest points would be 4/a and at the weakest points would be 1/2a. That is, it would be 4 times as strong as a circle with radius a at the strongest points and half as strong as a circle of radius a at the weakest points. Another way to think about it is that the strength of the ellipse at the weakest points would be the same as the strength of a circle whose radius is twice the semi major axis of the ellipse. The closer together the two axies are in length, the closer both factors will be to 1. If you were in a situation where you were expecting the brunt of the attack from one direction but there was still the possibility of an occasional attack from the side, it seems like a 9 (or 6) point ellipse where you could bind a couple of defensive chalklings and could add more stabilizing lines could still be a good idea.
• It could be that the basic premise of 1, that every curvature is assigned a strength, is correct, but that the relationship is more complicated than a simple proportionality. This could lead to weaker sides for the ellipse, but it would also lead to larger circles getting weak faster as well.
• There could be a certain total amount of strength (which I tend to think of in terms of charge) in a Line of Warding. That charge then could be attracted to areas of higher curvature. In a circle, where the curvature is constant, it would spread out evenly. In an ellipse, it would concentrate in the areas of higher curvature and leave the longer "sides" weaker. Depending on how this "attraction" worked, it would mean that larger circles are still weaker than smaller circles (since the charge is more spread out), but that the long side of an ellipse could be weaker than a circle of equivalent curvature.
• We could have something similar to 3, but where the total amount of charge in the LoW depends on the length of the curve (every inch of chalk contributes a unit of charge, or something) rather than being fixed for lines of warding. This possibility would mean that all circles could have the same strength. It would also lead to the possibility of long narrow ellipses with extraordinarily strong ends and equally weak sides. This option is the one that currently feels to me like it fits best. The idea of a LoW being somehow charged also fits thematically with the LoF that produce something reminiscent of a magnetic field and LoV that behave like EM waves. It is also rather more complicated to work with and raises more questions (How exactly is the total charge related to length? How does the "attraction" component work? How does the charge distribution affect the way LoF work in stabilizing the LoW? etc...)
• There could be something else going on entirely.

I wish I were a Rithmatist so I could draw a bunch of circles and ellipses of different sizes and shoot equal LoV at them from equal distances but different angles and record all of the results and analyze them. I actually have a whole list of Rithmatics experiments I wish I could run. Ah well.

Edited January 31, 2015 by KalynaAnne

[ccstat]

Thanks for the generalized 9-point conics explanation! Clear, concise, and helpful. I wasn't sure I was doing it right, especially when it came to the hyperbolas, and I totally forgot about the degenerate cases.

 

Okay, this post ended up being much longer than I intended, so I'm spoilering sections for length. Thanks for getting me thinking about this!

 

About curvature and Nebrask's Great Circle

Possibility #4 seems most likely to me as a general Rithmatic principle. I really don't know what is going on in Nebrask, though. I have envisioned the circle there as being large enough that locally it appears nearly indistinguishable from a LoF. Therefore, it may begin to take on hybrid qualities, including enhanced strength.

 

Postulate on Bind Point Requirements (referencing the Eskridge Defense)

The most interesting diagram on this topic though is the Eskridge Defense.  My understanding would suggest that the point where the circles touch has to be a bind point for both circles (whether there is anything else they need to match up with or not). [...] For the lower left circle though...The only way to get those three points is if it is actually a 9 point circle.  One of the things I'm currently working on is trying to determine what kinds of point patterns (right now starting with three points) can be extended to 9-point circles.  Determining where all 9 points would need to be if you know three of them is likely to take far more time than would be practical.

 

That illustration stood out to me, too, and at first I was going to agree with you about that lower left auxiliary circle being a 9-pointer, with only 3 utilized. However, I now think that all 9 points (or fewer, for special cases) must be explicitly utilized to generate a valid defense. Basically my reasoning boils down to the fact that if you could use a subset of points, any 2 points on a circle would be valid bind points, and there would be no need to specify that a 2-point circle uses a diameter. Instead, Rithmatists would be devising defenses with 2-pointers positioned however they wanted. Most sets of 3 points (I suspect all, but I can't prove it for sure) are also valid subsets, and they would have discovered arbitrary 3-point defenses as well. I concluded then that the special cases are the only lower-order possibilities, because those are the only ones that legitimately use all 9 points (due to redundant positioning).

 

In applying this to the outlying circle on the Eskridge illustration, I rely on this page about bind points. Basically, if you contact a LoW at a non-bind point, it weakens the circle at that spot, making it easier to breach. The smaller circle contacts the larger circle at a bind point of the larger circle, so the main defense is not weakened, but the smaller circle is vulnerable at that junction because the Rithmatist has defined its bind points at the ends of the marked diameter. This isn't a serious defect, because she wouldn't care if her opponent's chalklings or LoV broke into the smaller circle--she would not be in danger or lose a duel. Presumably, she could elaborate it into a 9-point circle, but until she defines the other points correctly the contact with the main circle will remain a weakness, not a bind point.

 

This brings up a tangential question for the predictive problem mentioned above: does it simplify anything if you start with two points of a diameter? Does that meaningfully limit the set of possible configurations? I feel like from a practical standpoint that would be the most likely starting point to want to elaborate from.

 

About the Blad defense, and a possible problem with the above postulate

The most serious problem with my postulate here is your (excellent!) proposal for the Blad defense. From the description in the text, the principles you apply there appear valid even if you are wrong in some particulars. The most notable difference is that the Blad defense does not use a continuous conic! Instead, it combines "four ellipsoid segments." A fundamental assumption in nearly all defenses is that the central circle/ellipse must be entire, because any gaps can be exploited by chalklings to endanger the Rithmatist inside. However, that does not mean that constructions using only segments of a circle or ellipse are not Rithmatically valid. 

 

The best support for this (though I don't have specific quotes) is my impression that once a circle is breached, its functionality is not destroyed. Bound chalklings still draw strength from the circle, etc. The only problem is that bad things can now get at you--the magic hasn't stopped working.

 

If this is the case, it suggests the possibility (though does not require) that subsets of bind points may be used. Of note, your proposed Blad defense does utilize the 2-point special case for the crossed ellipses, so even though segments of the original ellipses are missing, all bind points are accounted for. Whether a similar defense could be constructed using 3 or 5 half-ellipses would depend upon my postulate above, since the opposite side would be missing for each marked bind point.

 

For the time being, I remain of the opinion that all 9 bind points must be explicitly present to make a valid Rithmatic construction.

 

Open and Degenerate Conics

Working from the idea that conic segments are sufficient for a useful LoW, provided that a sufficient number of bind points (possibly all 9) are present, this opens up the possibility of using hyperbolas and parallel or single lines. The whole thing doesn't have to be there, even for finite conics, so a portion of an infinite conic should be valid. The question arises then how a linear LoW would be distinguished from a linear LoF. Does the fact that it derives from an open conic make it a LoF? Does the Rithmatist's intent imbue the line with the correct quality? Does explicitly marking bind points make the difference? (Is that why they are instructed never to let LoF touch except at corners? Because introducing a single bind point destroys the LoF quality, but without adding in the additional points it never becomes a valid LoW?)

I don't know any of the answers, and I'm not sure whether these will be addressed in the story. It seems esoteric enough that Brandon will probably focus on other aspects of the magic first instead.

 

Possible constructions related to the Blad Defense:

I have been thinking about a number of different ways to use segments of circles and/or ellipses. Some satisfy my explicit bind point postulate, others violate it extravagantly. The most compelling, though, for both its simplicity and its adherence to the postulate, strongly resembles the Blad defense, only with circles. Certainly banned from the dueling ring, this promises to be a versatile defense with many bind points (8 available) but easier to construct on the fly than a 9-point defense. Two variations are shown below, based on 4-point and 6-point circles.

 

4-pointer:

This is extremely similar to KalynaAnne's version of the Blad defense. You sacrifice the ellipse's strength of curvature, but you gain additional bind points and increased integrity at the junctions, since the intersection of the segments occurs at a valid bind point. You also gain the ability to anchor it more extensively with internal LoF. (I may have gone a bit overboard here.)

 

The dashed lines are not necessary to include, but I don't see any reason why adding them would be harmful. In fact, adding the central circle as a "home base" in case an outer circle is breached would be a simple and effective extra measure of protection.

 

6-pointer:

In this case, the dashed lines may or may not be necessary, depending on whether the "all bind points accounted for" postulate holds or not. If not, you may omit them, but it is possible that center point is required.  If you do include the internal portions of the three outer circles, you could conceivably use some very internal LoF for anchoring. Those probably get in your way too much, but it's an option.

I've varied things a bit on configuration of each circle to demonstrate the flexibility of placement for LoF, LoM, and subsidiary LoW. I have no expectation that the ones chosen here are an optimal combination.

 

I feel it is particularly worth noting that while a single Rithmatist could command such a defense, related constructions with two or more interlocking circles connected at valid bind points provide the possibility of powerful team tactics.

Paired Matson Defense:

Depending on the circumstances, this may or may not be better than two conventional defenses separated by a modest distance. I haven't decided yet when doing something like this would be worth it, but it is important to recognize the possibility. One contributing factor to the decision: presumably, if the circles are helping to anchor each other, fewer total LoF need to be drawn. That should allow greater freedom of movement within the circle for each Rithmatist.

 

One more observation on bind points:

In drawing the above constructions, I referenced the illustrations and realized something that I hadn't paid attention to before. Nearly every defense in the book allows a single bind point to connect with both an anchoring LoF and either a chalkling or an outer circle. For whatever reason I had in my mind that the one-point-one-connection rule extended to LoF, but it clearly does not. Does that stem from the fact that LoF are a different class altogether? Or is it the chalklings that are the special case, since they apparently draw power from the circle at that bind point? Note that multiple LoF and an outer circle connect to a single bind point in the 9-point Easton Defense diagrams. There may be an easy answer, or there may be a lot more going on than we realize.

[KalynaAnne]

I'll reply more fully later (I've only got about 10 min right now and I'm certain I will need more than that), but I wanted to point out that the diagram for the Sumsion Defense indicates bindpoints for optional chalklings.  It may just be that it counts as a 2 bind point circle without them and that if you add one you have to add both to make it a 4 point circle, but it is an interesting situation.

 

I'm glad to have someone to bat ideas around with :-)

[Tarontos]

All this talk of curvature got me thinking of something called a super ellipse

essentialy 

it is an elipse that instead of being

(x/a)^2+(y/b)^2=1

is

(x/a)^n+(y/b)^n=1

interestingly as n approaches infinity you have a rectangle.(so a rectangle is a degenerate super conic)

So the fun part is the strength of the SE(this is going to get confusing it has the same abbreviation as my major) has very flat tips with sharply curving corners.

So what it looks like eventually is 4 lines with some very sharp curves linking them together.

So i wonder if a rectangle would almost be able to have bind points at it's corners.

and the strength on the flat parts would either be negligible or fantastic as they approach a line.

ok so the only thing SE have going for them is increadably strong corners so a defense using an incomplete rectangle (missing the corners) could be utilized to send chalklings out. i'm not sure weather these would actually mesh with the Triangle well do to the difference in curvature but it may be a sound rithmatic construct(i know at least some of the four pointers would work but that's all i can really confirm).

Nebrask:

IT is interesting that it manages to hold- but it could just be related to relative curvature of a body. 

Edited February 6, 2015 by Tarontos

[Divine0Flame]

This thred ruined this section of the forum for me. My brain, trying... to much geometry. PAIN!! Rithmatics is significantly more complicated than I first thought.

Multiple things show that you can let LoF touch at more than thw corners/ends. I am namely refrencing Joles's use at the end of the book where he has Melody construt a "maze" to trap the wild LoM. This disproves that putting a bind point on a LoF makes is it a straight LoW. It is likely however that you must draw the line with the intent to make a LoW. Why is this so complicated!

Edited February 27, 2015 by Divine0flame

[ccstat]

Brandon answered some of our questions on his reddit AMA. KalynaAnn and I both asked follow-ups, so hopefully we'll get some additional info soon. Here are the relevant posts:

Question:

You confirmed at the Atlanta signing that 9-point constructions could be applied to ellipses to generate valid Rithmatic defenses. Could the same be done using open comics such as hyperbolas and parabolas? (Or do open conics become lines of Forbiddance instead of lines of Warding?) Also, the disparity in strength between the sides and points of an ellipse is attributed to their difference in curvature. Does this mean that a large circle is inherently weaker than a small circle since its local curvature is less?

Answer:

Yes to all questions, though 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.

Follow-up by me:

Wow, thanks! Am I interpreting your answer correctly to say that a hyperbola could be used to make either a line of forbiddance or a line of warding, depending on what the rithmatist does/intends?

Follow-up by KalynaAnne

Hello! Rithmatics is wonderful :-) As a math nerd, I just wanted to point out that you can't actually use the generalized 9-point conic construction to get a parabola. It can lead to circles, ellipses, hyperbola (and in degenerate cases a pair of lines, which can be intersecting, parallel or coincident). If hyperbola - where you clearly can't draw the entire infinite lines- work though, I'm curious as to whether a partial circle would work. If a Rithmatist were backed up against a wall, could they draw a semicircular defense? Thanks!

Thanks Kalyna for correcting me on the parabola business. It sounds like it would work for a LoF though.

Edited May 27, 2015 by ccstat

[Juanaton]

Have you considered how to draw a Mark's Cross? That is something that really bugs me since it seems to me that it should be impossible to draw two LoF that cross, as is described. For that matter, the same problem seems to apply to the Taylor defense. I considered that it could be drawn using two right angles meeting at the center of the circle, but that seems...odd.

Edited August 10, 2015 by nsweigart

[ccstat]

Good point. I have two versions that I can't decide between. The simple explanation is that you use 3 lines instead of two, and they don't quite touch. That would mean it looks like -|- more than +.

The other possibility is that it takes a second for the drawn line to project its force field vertically, which gives a brief grace period to draw the full cross. Given that Harding's bullet was stopped by a hastily drawn LoF, this can't be a usual feature. However, since we know the Rithmatist's intent matters, it is possible that when they are planning to construct a Mark's Cross they subconsciously (or maybe intentionally) instruct the first LoF to not become complete until they draw the intersecting line.

Edited August 11, 2015 by ccstat

[Ookla the Absent]

My favourite rithmatic question was can you make a chalkling inquisitor to spike other chalklings.

The basic answer is if they came up with inquisitors, yes.

[Young Bard]

Quick question on the 4 point circle.

 

This is probably me being a scatterbrain, but it seems to me like there should be a 5th point, right in the centre of the 4 point circle, if your theory is correct.

 

Basically, you have:

 

3 midpoints

3 altitude points

3 altitude midpoints.

 

For the 4 point triangle, you have only place two altitude midpoints. The third one would go right in the middle of the circle, between the corner and the midpoint.

[WeiryWriter]

That's not quite how it works actually (because bindpoints only exist on the circle itself, not inside or outside of it), it's more like:

 

3 - side midpoints

 

3 - point where the altitude meets the side

 

3 - point where the altitude crosses the circle (not necessarily its midpoint)

 

A while back I actually made some diagrams that will eventually be used in an update of the Rithmatics page on the wiki.

[Young Bard]

OK, got it. Thanks for the clarification.

[Juanaton]

I kind of hope the 8 point defense doesn't show up in universe since it makes it incredibly simple to get a defense with nearly as many bindpoints as a 9 point defense. A regular octogon is a valid configuration for the bindpoints. For that matter, any grouping of 6 points symmetric with respect to the first two (which are endpoints of a diameter) is a valid 8 point configuration. It's an incredibly powerful, adaptable, and simple defense to create.

Edited August 18, 2015 by nsweigart

[Juanaton]

Quick question on the 4 point circle.

 

This is probably me being a scatterbrain, but it seems to me like there should be a 5th point, right in the centre of the 4 point circle, if your theory is correct.

 

Basically, you have:

 

3 midpoints

3 altitude points

3 altitude midpoints.

 

For the 4 point triangle, you have only place two altitude midpoints. The third one would go right in the middle of the circle, between the corner and the midpoint.

 

Actually, the altitude midpoints don't matter at all. The important points are: 1) Side midpoints, 2) Altitude bases, and 3) the midpoints of the line segment from each vertex to the point where the altitudes intersect.

 

In a right isosceles triangle the point where the altitudes intersect is the vertex of the right angle. So the side midpoints for the non-hypotenuse sides are also two of the type 3 points. The third type three point is halfway between the vertex of the right angle and itself...in other words, it is also the vertex of the right angle.

 

So there was never supposed to be a point in the center of the circle.

 

Of course there is a 5 point circle (I think it's mentioned in this thread, but if not...). You get it by plotting those same nine points for a scalene right triangle. The 5th point shows up because now the altitude base from the right angle is not at the midpoint of the hypotenuse like it was in the right isosceles. 

Edited August 18, 2015 by nsweigart