The Rithmatist - Blog Posts

MOAR CHALKLINGS

1/24 Newnan signing report

*dances around* Not only did I get to meet Brandon Sanderson and have him sign my books, I ALSO got to meet Ben McSweeney (see previous post)!

While hanging out in the bookstore waiting for the signing to start, I met a couple of other fans.  One of them had gotten there early enough to get one of the special "get a seat at the front of the Q+A" wristbands but didn't have a question.  I got him to ask what Horneater stew would be like on earth.  It is apparently based on a spicy Korean seafood soup that traditionally is made by just throwing anything acquired from the sea (shrimp, clams, mussels, etc) into the pot whole, shells and all. He gave the name of the soup. It's Korean. I would probably butcher the spelling completely, so I'm not going to try.

When I got through the line, I asked if he could draw the Blad defense in The Rithmatist.  His response was a look that very clearly said "You expect me to remember which one that is by name?" I clarified and he just kind of laughed and drew me one of the 4 point circles and suggested that maybe I could get one of the more complex ones from Ben.

I asked for something about Kaladin when he was signing WoR.  Before I tell you what he wrote, I should mention that he was halfway through writing it when I opened my rithimatics notebook and kind of distracted him. "Kaladin has known multiple lightweilders." 

So. Rithmatics.  I pretty much just started flipping through my notebook asking questions about each page.  Most of the questions got a simple affirmative. For the other things, I'm paraphrasing: 

Yes, 5 and 8 point defenses 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 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.

At this point he started to say that we shouldn't hold up the line too long as I flipped to a page titled Lines of Vigor.  I was going to let it go, but he glanced at the page and told me to go ahead and ask :D

Yes, Lines of Vigor behave like light waves. (I'm so glad I was right on this)

To clarify I double checked that 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.

GUYS.  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.  Ben then commented that he hadn't known that. *flails*

I got "Oh, wow"'s from both of them while I was flipping through the notebook :-).  At that point, Brandon passed me off to Ben and we chatted about inconsequential things while he took one of my spare sheets of paper from my notebook and drew me a picture *flails more*

I'm still on such a high guys, it's crazy. *dances around*

(very minor edit to fix a typo that was bugging me)

GUYS.  Ben McSweeney was also at the Atlanta Brandon Sanderson signing!  And was impressed enough with my notebook of rithmatics that he DREW ME A PICTURE!!!!

Rithmatics: Part 5 - Curvature, Ellipses and a Guess at the Blad Defense

Let's start by considering a snippit from one of the diagrams in The Rithmatist:

(Note: full diagram can be found at http://brandonsanderson.com/books/the-rithmatist/the-rithmatist/rithmatist-maps-and-illustrations/ )

Now we have a problem, namely, circles don't all have the same curvature. In fact (a slight simplification of) the idea of calculating curvature is to determine what radius circle would best approximate the curvature of the line at that point. A circle of radius r has constant curvature 1/r.

The basic idea here is reasonable though – apparently, lines of warding are stronger when they have a higher curvature. You can think of an ellipse as a circle that has been stretched along one axis. This means that if you start with a circle and then stretch it, we can talk about the resulting ellipse being stronger than the original circle where it curves more and weaker where it curves less. Here is what that diagram might look like if we add in the relevant reference circle:

Assuming this interpretation is correct, there are some important implications. The biggest is probably that the size of the circle used to form a defense matters. If you have two otherwise completely equivalent defenses and one of them is a scaled up version of the other, every point in the wall of the smaller defense will be stronger than the equivalent point in the wall of the larger.

Note: There are at least two potential underlying explanations for what is going on here. One option is that there is a certain strength inherent in a portion of a curve of a given curvature. This is the assumption that I am going to work from here. There is also the possibility that there is a fixed total strength for any closed curve of warding and that this strength distributes itself based on curvature. If we stick to circles and assume that strength and curvature are proportional, the two notions are equivalent. The second option is intriguing, but leads to rather messy calculations when we start looking at more interesting constructions. If I stick with this long enough we may eventually get there. I have no idea which option is correct or whether there is a third one I haven't considered.

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.

From an offensive standpoint, this means that the small circles and Mark's crosses added to your opponent's main circle are going to be much harder to affect than their main line of warding. They aren't just in the way – they are actually stronger.

We will talk about ellipses in more depth in future installments, but for now let's close with a guess at what the Blad Defense might look like. All we know about it is that it combines four ellipsoid segments in a non-traditional manner and that it is strong enough that some people think it should be banned from competitions.   

Rithmatics: Part 4, obtuse triangles

In the previous posts we have addressed all acute and right triangles.  In this post, we look at what happens if the triangle is obtuse.

Obtuse triangles and the 9-Point Circle Construction

In an obtuse triangle, two of the altitudes fall outside of the triangle. This appears to be a problem, but we can work around it.  The 9 point circle construction we have been using so far is the special case of a more general 9 point conic construction that starts with 4 points.  This more general construction produces a circle whenever the 4 points are the three vertices of a triangle and its orthocenter (the point where the three altitudes intersect).  To find the orthocenter of an obtuse triangle we have to extend the altitudes to find where they intersect outside of the triangle.  We then use the three midpoints of the sides of the triangle, the three points where the altitudes intersect the opposite side (or side extension) and the midpoints of the segments connecting the orthocenter to the three vertices of the triangle.  As you can see in the diagram below, this ends up being the same triangle you would get from considering the acute triangle formed by the orthocenter and acute vertices of the original triangle.  This means that obtuse triangles can give us a different perspective on our circles, but will not produce any new patterns we couldn't get using acute triangles.  The advanced rithmatic theorist should be aware of this but for basic rithmatics it is fine to ignore obtuse triangles.

Rithmatics: Part 3, the potential for 8 point defenses

In the previous installments we have covered acute triangles with 3 distinct angles, equilateral triangles, and all right triangle.  In this post we look at acute isosceles triangles.  

The Potential for 8 Point Defenses and the Hypothetical Owl Defense

Let's look at what happens when we have an acute isosceles triangle.  The altitude from the different angle will bisect the opposite side, which means the midpoint of the base will count as two points.  However,  the 7 remaining points are all distinct this gives us 8 separate points.  

It is interesting to note that, as with the 4 point construction, we have one side of the circle that is tangent to the triangle.  The 4-point Sumsion Defense has a tangent Line of Forbiddance at one of the bind points.  As this is the only defense we know with such a tangent line, it seems reasonable that its existence could be related to the fact that the 4 point circle is the only one we have examples of that has one tangent side with its corresponding triangle. (All three sides of the triangle for the 6 point construction are tangent, but including all of them would trap you - it isn't clear whether it would be a problem to include just one or whether this actually matters at all.  This is something else I hope to ask about at the signing.) Following this logic, I used the Sumsion Defense as starting point for constructing a potential 8 point defense.  It ended up looking rather like an owl :-)

Rithmatics: Part 2, the potential for 5 point defenses

In the last post we looked at what happened with acute triangles with three distinct angles, equilateral triangles, and the isosceles right triangle.  In this post we consider non-isosceles right triangles.

The Potential for 5 Point Defenses

Let's consider what happens when we look at non-isosceles right triangles.  As with the 4 point case, the right angle vertex counts as 3 of the 9 points.  The difference here is that the altitude from the right angle vertex no longer bisects the hypotenuse, which gives us a 5th point.  The three side midpoints and the right angle vertex still form a rectangle.  It is interesting to note that the resulting circle has 3 arcs of the same length - the arcs corresponding to the short sides of the rectangle and the one connecting the short leg of the triangle to the hypotenuse.  This seems like it would be important to keep in mind when constructing defenses based on such circles.  Also note that, as with 9 point circles, there are infinitely many variations on the 5 point circle since different right triangles can lead to different bind point spacing.

In the image below I include a speculative idea I've had.  Since there are vertices which count as multiple points in the 9 point construction, it seems from a mathematical perspective like you ought to be able to bind multiple things to this point.  Whether or not this actually works rithmatically is currently unconfirmed. I will hopefully be able to find out at the upcoming Atlanta signing.

Rithmatics: Part 1

Hello!  I'm rather fascinated with Rithmatics (the magic system in Brandon Sanderson's The Rithmatist) at the moment, which means you are going to be getting a series of mathy posts.  They will all be tagged with #rithmatics .  I've been encouraged to add the cfsbf tag.  If this bothers anyone, please let me know.

In this first post we explore how all of the binding patterns for circular defenses can be derived from 9 point circles. You might be able to get everything from the pictures, but I give explanations as well.

9 Point Defenses

Let's start by talking about 9 point circles.  Start with a triangle.  Absolutely any triangle will do, but for 9 point defenses we want acute triangles (all angles less than 90 degrees) where all of the angles are distinct.  Mark the midpoints of each side and draw in the three altitudes (start at each vertex and draw the line perpendicular to the opposite side) of the triangles.  Mark the points where the altitudes intersect the sides of the triangles (there are 3 such points, one for each altitude).  Note that all of the altitudes meet a single point. Mark the midpoint of each segment connecting P to one of the vertices of the triangle.  This gives you three more points for a total of 9.  These 9 points will be distinct and lay on a circle.

This explains how to get the bind points for any 9-point defense.  However, not all defenses have 9 points.  These turn out to be very special cases of 9 point triangles where some of the points coincide.  

6 Point Defenses

To get 6 points, start with an equilateral triangle.  Any time two angles of a triangle have the same measure, the altitude from the third angle will bisect its opposite side.  Since all of the angles are the same here, all of the altitudes bisect their opposing sides.  This gives us a "9-point" circle with 6 evenly spaced points.

4 Point Defenses

This time we want an isosceles right triangle (you might know it better as a 45-45-90 triangle).  In right triangles, the legs are also altitudes, which means that the vertex at the right angle is also the point where the altitudes intersect each other. It is also the point where each leg "intersects" the other and the "half way point" between the intersection of the altitudes and itself, so it counts as 3 of the 9 points.  The resulting 4 points form a square and so are evenly spaced around the circle.

2 Point Defenses

This is the strangest case.  Here our triangle is degenerate - one of the sides has length 0, which means that the "triangle" is just a line.  To see how to follow the 9 point construction in this case, we can look at a limit.  Start with a really skinny isosceles triangle.  If you follow the construction, you get three points grouped near each approximately half way up the triangle.  The other 6 points are clustered down near the narrow base.  Now pretend the narrow point is a hinge and slowly close it. As you do, the three points in the middle get closer and closer together, the base gets narrower and narrower and the 6 points near it get closer and closer together.  In the limit this gives us a line segment and a circle which uses half of the line segment as a diameter

I don't particularly identify with Melody, but I love her character and think it is super important that she exists (and a young adult novel is exactly the right place for her). Across his books, Sanderson has given us a whole collection of fantastic women who represent a wide range of personalities and strengths. Melody is our stereotypical high school girl. She loves unicorns and pegasus and flowers and drawing and is completely unapologetic about it. She dislikes math and is convinced that she is hopeless at it. She desperately wants to live up to expectations but hates the form they take and thinks they are impossible. She feels lost and alone. She is also amazing.

Spoilers for The Rithmatist under the break.  

She really does struggle with math. It doesn't come easily to her. At the same time though, we get to see that with the right teacher and the right motivation and working at the right pace, she can learn the math. It isn't something she needs to or should just give up at. Even by the end of the book, she still isn't great at math. She has improved, but it is a level of improvement that is reasonable given the amount of time she has been working. We see promise for her to improve more as she continues to work at it. It feels real.

Despite this (in some sense because of it) ends up playing a very important role. Everybody knows that rithmatics is all about the math and the precision of getting your lines and curves and binding points in exactly the right place. It is essentially a science. There are Lines of Making and you can sort of affect what they are good at by their shape, but controlling them is essentially an exercise in programing. You have to know ahead of time what you want them to do and you have to give the instructions carefully. Chalklings are notoriously difficult to work with.

Melody sees things differently. She struggles with the science of rithmatics, but excels at the art. Her chalklings are not the rough sketches thrown in almost as an after thought. Every one is a work of art. They are elegant and detailed and at least approximately anatomically correct. She believes in them. She whispers instructions and they do her bidding. For Melody, working with chalklings, the thing everyone knows is a lost cause, comes naturally. Her role is just as important as Joel's in their final battle, and the fact that she can do magic and he can't is only a very small part of why. Her wonderful unicorns were just as important as Joel's fancy defense circle and carefully placed shots.

Melody is the woman on the programming project who makes sure the user interface is intuitive and functional even if she doesn't do much of the actual programming. She is the mathematician or the physicist who struggles with the more involved computations, but can easily see the symmetries that turn a nasty problem into a much more straight forward one. She is the inventor who sees beautiful, functional things in the natural world and asks why we don't just do it that way. She is Important even as she is very much a stereotypical girl.

Melody is there for all of the high school girls who are convinced they can't do math (or other traditional subjects) and that their passions don't matter. She is there to show them that if they work at it, they can succeed at the areas they feel hopeless in and that their passions do matter. At the same time, she reminds the rest of us that the more unusual perspectives and talents are important. They can provide solutions that simply do not occur to more conventional people.

I just finished Brandon Sanderson's The Rithmatist. Now, any time I read a fantasy novel with an interesting magic system, I imagine what it would be like to have said magic (don't we all?).  This magic system seems particularly well suited to me - it calls out to both my math and artistic sides.  I think I could make a rather good rithmatist.  The freaking 9 point circle is an important part of the magic system.  I can't even.  

I bet that the 6 point defenses don't actually require that the 6 points be equally spaced.  I bet they could actually be anywhere as long as you build the rest of the defense properly.  The Mystic Hexagon theorem has got to be relevant to this magic system.  I bet you can even use the Mystic Hexagon theorem to build 8 point defenses where the extra two points come from the line dictated by the Mystic Hexagon.  And when that line doesn't intersect the circle, it would be a natural (possibly required) place to put a Line of Forbiddance. There will probably be rithmatic drawings showing up here...

Also, my math notes often have non-math doodles in them.  <3 chalklings

Melody and Joel are fantastic. I like them a lot. There are things I need to process before you get more about them though.