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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

Because Syl would be fascinated by lava lamps and try to emulate them.  Clearly.

From Words of Radiance, Ch. 41.

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.

Shallan's Journey: Chapter 5, In Which We Get Out of the House

“Ok, what sort of disguise did you have in mind?” Hermione asked when they returned to the library.

Shallan gave her an enigmatic smile as she turned to her sketchbook and quickly drew Veil. She then took her partially used sphere and wove the image around herself before turning back to Hermione. “Something like this.”

Hermione's eyes widened and she had Shallan walk around the room while she watched for anything that might tip someone off that the girl was wearing a disguise. When she couldn't find anything, she nodded thoughtfully, “It is certainly convincing. Now we need to make sure it won't be easily dispelled. We can cast glamours, which have a similar effect, but if someone suspects them they are easy to counter.”

Sirius had assured them that the ministry would not be able to detect any spell work done in the house, so Hermione raised her wand and started with “Finite Incantatem.” Nothing happened. She smiled, “Well, that's a good start.” She then pursed her lips as she doggedly ran through a sequence of increasingly complex spells used to counter a variety of different glamours and illusions. None of them had any effect. Hermione tapped her chin with her wand as she considered other possible ways to disrupt Shallan's illusion before nodding to herself, “Nox”. This time the illusion seemed to flicker and Hermione raised her eyebrows in question.

“That one worked, but I was able to bring it back quickly,” Shallan held up the now dun sphere, “if you did that again I probably wouldn't be able to though.”

“Hmmmm....well, Nox is not a common way to counter illusions,” Hermione tilted her head to the side and muttered mostly to herself, “It's usually just used to extinguish your wand after you use Lumos. I only thought to try it because I saw you cast an illusion with Lumos. As long as you stay out of situations where someone might be extinguishing their wand that shouldn't be a problem...”

While Hermione was analyzing how much of a weakness the vulnerability to Nox would be, Shallan was contemplating her dun sphere. She quickly cut in when Hermione paused, “Hermione, can I borrow your wand to try something while you think?” Hermione nodded and handed it to her while still mostly lost in thought. After a couple of tries, Shallan managed to correctly cast Lumos and then turned her attention to the sphere, willing the light into it. She smiled in satisfaction as the sphere began to glow again. Fantastic. Now I don't have to worry about not being able to recharge my spheres. She grinned mischievously and turned back to her sketchpad. After finishing the sketch, she beckoned Pattern over. “Pattern, we know you can maintain an illusion. Is there any reason you wouldn't be able to help maintain an illusion I cast on someone else?”

Pattern buzzed thoughtfully, “Mmmmm... I should be able to.”

“Ok, let's try it,” Shallan infused Pattern with light from Hermione's wand and then sent him over to ride on Hermione's shirt as she used her sketch to build an illusion.Shallan called out to pull her friend out of her thoughts, “Hermione, go look in the mirror.”

Shallan followed Hermione as she headed down the hall to the full sized mirror in her room, amused that her friend was still half lost in mentally running through potential challenges they needed to prepare for. A brief glance in the mirror shocked Hermione back to reality as she turned back and forth between Shallan and the mirror. “You... you made me look exactly like you!” She reach up to touch the soft red curls she saw in the mirror and frowned as she felt her normal hair.

Shallan saw her confusion and explained, “It's just an illusion, so you can reach through it, but as long as I'm careful about creating it, that shouldn't be a problem. It does mean that it is safer not to create illusions of different clothing though.”

Hermione briefly considered this and nodded. “We should be able to find nondescript robes to use as part of the disguise.”

Shallan grimaced and pulled her sleeved safehand to her chest. She knew that she was going to have to wear the local clothing when she went out, but she had been trying not to think about it. Just as Veil couldn't wear a proper havah and still blend in as a darkeyes, wearing a dress with a proper safehand sleeve or even wearing a glove here in the middle of summer was going to draw attention she didn't want. She had almost gotten used to seeing the other girls with their safehands uncovered, but it still made her somewhat uncomfortable.

Hermione saw the gesture and smiled sympathetically. Shallan had tried to explain Vorinism to the others and, while they didn't fully understand, they tried to be respectful. “You know, there are everyday robes with long, wide sleeves. If we find you robes with sleeves that are a little too long you can probably keep your hand hidden.”

Shallan shrugged. That would be better than having to wear short sleeves like the other girls, but it still wasn't the same... She wasn't good enough with English to try to explain the distinction though, so she let it go with a sigh. They weren't ready for the trip yet anyway, so she still had time to talk herself into it.

They spent the rest of the day working on designs for faces for Shallan and Sirius to wear that were close enough to their real faces that the illusion would be hard to detect, common enough that they wouldn't stand out, and, in Sirius's case, different enough that he wouldn't be recognized. They also experimented with how much stormlight/wandlight Pattern needed to maintain an illusion on another person and how long they could expect it to hold. Through all of it, Shallan detached herself from the idea of actually making the trip and focused on it as an abstraction and scholarly question.

That night, as she sat on her bed brushing out her hair with only Patten for company, Shallan finally allowed herself to grapple with her feelings about going out to acquire a wand.

“I can't just stay in this house. I can not. I won't let myself be locked away and hidden by anyone or anything, even my own sensibilities. And, well, it also bothers me that they are keeping Sirius hidden away here. I know his situation is different, but I also at least somewhat understand what it's like and it's not ok. But at the same time, even with Hermione's explanations,I don't know the local culture or how to blend in andto go out with my safehand uncovered... Pattern, I don't know what to do...”

“Mmmmm,” Pattern buzzed, “Hermione said that the robes should be able to hide your hand, yes? And the people here wouldn't pay any attention to it anyway...”

Shallan sighed, looked down at the bed, and wrapped her arms around herself,“Yes, but... that's not the point. Iwould know that it wasn't properly covered and, in addition to it feeling wrong, it would be a distraction.”

Pattern bobbed around confused “Mmmm, if the sleeve will cover your hand anyway, why can't you wear your glove?”

Shallan raised her eyebrows as she looked up slowly. “That, Pattern, is a very good question.”

A couple of days later, Shallan, Hermione, and Sirius were all reasonably happy with the disguises and the plan. At one point they considered getting approval from the Order for their trip, but decided it would be better to ask for forgiveness than permission. Everyone seemed strangely determined to keep Sirius locked up in the house. They had the twins, who were happy for any excuse to cause mischief, create a distraction that allowed Shallan and Sirius to sneak out of the house. Hermione summoned the Knight Bus for them and they set off for the Leaky Cauldron and Diagon Alley. It was a busy enough day at the Leaky that they had no trouble casually waiting for someone to open the passage to the Alley and then following them through.

Their first stop was Gringotts and, as they made their way there, Shallan looked around the street in wonder. She itched to pull out her sketchbook and to begin recording everything she saw, but constrained herself to frequently blinking and taking memories to draw later. Hermione had warned Shallan about the goblins, but she still did a double take when she saw the first one as they arrived at the bank. Sirius presented the key to the Black family vault to one of the tellers and they were able to retrieve the money they would need (and quite a bit more) with no difficulty. Shallan was none too sure about the cart ride to and from the vault and the whole idea of the bank was strange to her - spheres were far too useful to lock away in a vault somewhere. She coped, as usual, by reminding herself to view everything as a scholar and took copious mental notes about the system and stored a few images to draw later.

After leaving the bank, Sirius guided them farther down the street to the Ollivander's. As they walked into the wandshop, Shallan looked around and turned to Sirius in confusion, “Is there no one here?” Sirius shook his head, but before he could answer an old man with a deeply lined face and fluffy white hair suddenly appeared in front of them with wide peering eyes, “Sirius Black. I can't say I expected to see you here.”

Shallan stumbled back a step as her heart started racing. He can see through my illusion. We didn't plan for this...

At this point Ollivander's attention turned to Shallan and his eyebrows flew up. He glanced back curious to Sirius, “Not only did you manage to move unnoticed through the Alley, you also have a companion who is, shall we say, not from around here...”

Sirius rolled his eyes,“Yeah, yeah. My friend is why we are here. She needs a wand. Though, I could use a new one too if you are willing to sell me one.”

Ollivander looked over Sirius pensively, “Yes, I suppose you could. Such a shame. Whatever happened to it?”

Sirius gave a half shrug, “Not really sure.”

“Mmm, I see.” Ollivander nodded to his tape measure which leapt up and slithered toward Sirius. “Your right hand is still your wand hand?”

“So far as I know.”

After several failed fits, Ollivander turned to Sirus, “Here, try this one. Dogwood and dragon heartstring, 9 ¼ inches, rigid.”

As Sirius took the wand it let out a loud bark and released a puff of black smoke that swirled and resolved itself into a black dog with an energetically wagging tail. Sirius grinned broadly. “Ah, this will do quite well.”

Ollivander nodded and turned to examine Shallan as he muttered about dogwood not usually being literally related to dogs. “And which is your wand hand?”

The tape measure slithered over to her and began take all sorts of measurements as Ollivander moved around the shop collecting wand boxes. As the tape measure finshed and made its way back to its home on the counter, he returned to Shallan and began handing her wands to test, telling her about each wand and its potential strengths as he did. None in the first batch gave any reaction, though there were a few that Shallan felt the same almost vibration from that she noticed the first time she held Fred's wand. While Ollivander returned to shelves to collect more wands for her to try, Shallan heard a soft buzzing over his mutterings.

“Mmmmm....patterns...”

Shallan narrowed her eyes and looked toward Sirius who had Pattern hiding on his robes, I thought Pattern agreed to stay quiet, but she asked quietly anyway,“What patterns?”

“In the wands. Some like you better than others. The fir ones...”

Shallan took a deep breath and then cut in to Ollivander's running commentary, “Um, sir, the fir wands have felt closer to right than the others.”

He froze and Shallan spent several uncomfortable moments with him blinking at her before he was off digging through his store of wands again. “Let's see... Let's see... maybe... yes... Here try this one, 10 and a half inch, surprisingly springy fir and unicorn hair.”

This time when she tentatively took the wand, the tip burst into brilliant light.

“Interesting, Interesting... Fir... You must be stronger than you look.” Ollivander peered intently at her, “A survior's wand, particularly inclined toward transfiguration... That will be seven Galleons each for the two wands.”

While Shallan was still trying to figure out how to respond to the disconcerting shopkeeper, Sirius handed over the Galleons and gently guided her out of the shop.   

Note: I had a hard time getting this chapter right.  I'm reasonably happy with it now, but I would also be happy to have feedback :-) I hope you enjoyed it!

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.

Hello and welcome to Cooking with Kaly.  I decided I wanted to make chouta. It was an adventure. 

We know that the meatballs are made by mixing flangria with ground lavis, forming it into balls, battering it and frying it.  To create these meatballs we therefore need something to represent flangria, something to represent lavis, and a way to put it together without egg (since eggs are unlikely to be used in cheap street food on Roshar.)

I happened to have ground turkey in the freezer that I had gotten on sale at some point. It seemed like a reasonable stand in for flangria. Given the range of ways lavis is used, I feel like corn is a reasonable earth analogue, so ground lavis gets to be corn meal.  I mixed the 1.5 lbs of meat with 1/4 cup of cornmeal, 1 Tbsp chili powder, 2 tsp cumin, and 2 tsp oregano, formed it into small balls (maybe a heaping Tbsp of meat per ball?  I didn’t measure), then covered them and put them in the fridge for a while to let the flavors meld.  

Before cooking them, I rolled them in cornmeal, dipped them in water (some but not all of the cornmeal will fall off), then rolled them in the cornmeal again.  I fried them in a thin layer of canola oil in a cast iron skillet until they were golden brown.  The cornmeal gets all crispy and the flavors are great.  I’m a big fan of these meatballs. (note:the plate in this picture is small, the meatballs are not giant. There are also a lot more meatballs not shown)

For the flatbread, I mixed 1 cup mashed potatoes (since tubers seem to be common on Roshar) with 1/4 cup cornmeal, a heaping 1/4 cup flour and 1 tsp baking soda. I kneaded the mixture until I had a dough that I could press out into a decent sized flatbread.  This made enough for 2. Transfer the pressed out flat bread *carefully* to the frying pan and make sure you don’t try to flip it too soon or it will tear and be a mess. This made a decent flatbread. I think the recipe needs work though.

I used the pan drippings from frying the meatballs to make the gravy, but I didn’t have any good broth to use so it came out a little bland.  I think drippings gravy is the right idea here, I just didn’t do it well.

Overall verdict: This definitely feels like street food. It was tasty and has the potential to be amazing once I get the flatbread and gravy right. Nomnomnom. It’s not going to win any awards for being good for you, but that’s not what this is about. 

Pixely critters from hanging out in the Steel Ministry on iscribble.

Ok.  Completely new post because the other one keeps reverting to the cut off version.  This is my interpretation of Shallan's Lullaby from Words of Radiance. So far as I am aware the tune doesn't belong to anything else.