Hey Cosmere Fandom, It’s Book Rec Time.

F is for Frillbloom.  For more on frillbloom, visit the coppermind!

Ok, so after the skyeel came out yesterday, I decided to try a chull today.  I'm not entirely happy with it, but here, have it anyway.

Hey y’all, it’s Kaly reads a poem time again.  This one is called Entropy’s Embrace.  It isn’t quite a cosmere specific poem, but it is a great poem, was written by our friend worldsingers and it does have some lovely cosmere evoking imagery.  You can find the text and a review of the poem here.

OMH THAT IS ABSOLUTELY STORMING ADORABLE!!!!! You’re amazing :D

“anything about axehounds or chulls or skyeels or other stormlight critters”

For kalynaanne! Here’s an axehound pup taking a nap in the afternoon sun. :D

https://www.kickstarter.com/projects/1268017280/potions-a-curious-tale/widget/video.html

Potions: A Curious Tale - Adventure through an enchanted world where your wits are your greatest weapon and combat is not always the answer. - http://kck.st/1oIP0qQ

The leading creative director of this project is Renee Gittins. She was a frosh at Harvey Mudd my senior year so while I didn’t know her well, she and I have several mutual friends who are excited about this. From what I’ve seen on the kickstarter, this game looks like it has a lot of potential so I wanted to spread the word.  

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

xkcd’s data recovery program gave me the prompt “Somewhere, this exists.”  hehehe

Shallan's Journey: Chapter 3, In Which Pattern Declares English Full of Lies

Shallan heard her name and glanced up warily. The group was clearly asking Albus about her, but she had no idea what they were saying and was already trying to process too many things to want to add anything else. The stove woman noticed her looking up and came over with a plate of...something, a fork and a glass filled with a drink that appeared to be some sort of orange juice or wine. Shallan realized that she was hungry and took a deep breath to calm herself before offering the woman a small smile and accepting the plate with a “Thank you” even though she knew it wouldn't be understood. She inspected the food and realized quickly that she had no idea what it was, though at least the utensil was mostly familiar. She took a bite and contemplated it. It was very different, but at least it was edible. The woman gave her an encouraging smile and said something Shallan couldn't understand before bustling back over to the discussion at the table.

As she ate, Shallan worked on talking herself down. You had already figured out you weren't on Roshar. The plant and animal life here is very different – the plants don't retreat into the ground and there are chickens that travel in fire. Of course the food is going to be different. The whole world is different. That also means they are going to have different sensibilities and different ideas about who should do what jobs. Shallan groaned quietly. On an intellectual level that seemed straightforward enough, though it wasn't going to make it any easier to navigate this world. Maybe it would be easier to handle if she viewed the situation as a scholar. She reached this conclusion as she finished her meal, so she carefully set the plate on the ground next to her, opened her sketchbook and began to document what she had seen so far. In addition to helping Shallan organize her thoughts, the acts of writing and drawing always helped to calm her.

Hermione rolled her eyes at the argument raging around her. She supposed it could be possible for the Death Eaters to try something similar to whatever was going on as a way to get a spy into the order, but she really didn't think they were that clever. Dumbledore's story also had too many details that made it clear that this new girl was not from anywhere they had ever heard of for the spy theory to be plausible. She watched the new girl as she pulled out a notebook and appeared to begin writing in it. Hermione smiled. That was something she would do. If you don't know what is going on, read, and if there is nothing to read, write down anything and everything.

Hermione noticed that Dumbledore was now standing quietly off to the side and shaking his head as he watched the argument unfold. She knew he would get his way eventually, he always did, but she supposed that if they didn't get the argument out of their system now it would just start again later. It probably would anyway - from what she could tell the Order seemed to spend an awful lot of time arguing. She pushed her chair back from the table and quietly approached Dumbledore. “May I take the new girl, Shallan, I think you said, up to the library?” He nodded his agreement to her, but kept his attention on the argument.

She smiled as she walked over to Shallan and collected her plate to take it to the sink. Shallan was sufficiently absorbed in her writing that she barely seemed to notice. When Hermione returned she greeted the girl, “Hi Shallan,” and when Shallan looked up pointed to herself “Hermione... Her-my-oh-nee.”

Shallan smiled tentatively back and repeated carefully “Her-my-oh-nee...Her-my-oh-nee...Hermione.” Storms! she thought, at least their language seems to use the same sounds, but they go together in such strange ways.” She watched curiously as the other girl, Hermione, mimed sitting at a table and writing and then gestured towards the stairs. Shallan nodded, tucked her sketchbook, which she was now treating as a notebook, and pen into her bag and stood. She didn't mind sitting on the floor, but she also wouldn't say no to having a proper table or desk to work at.

Shallan followed Hermione up two flights of stairs and a little ways down a hall to a room she decided must be Hermione's bedroom. Here Hermione dug a new notebook (it was always a good idea to have notebooks on hand, and they were so much more practical than doing everything on scrolls) and a few pens and pencils from her trunk before gesturing further down the hall. At the end of the hall Hermione opened a door and gestured for Shallan to enter. Her jaw dropped open as she looked in the room. “Pattern! Look at all of the books!” She stepped into the room and looked around in amazement as Pattern detached himself from her dress to explore the library as well.

Hermione's eyebrows shot up as she heard Shallan exclaim excitedly and saw something seem to float away from her. As Shallan looked around the library, Hermione walked cautiously but curiously toward the... thing? creature? that was now floating around the room. As she approached it she asked, mostly rhetorically, “Hello, and what might you be?”. She was somewhat surprised when the... whatever it was... buzzed back at her with something that almost sounded like it could be words. “I don't know why you are talking to me. Surely you know I won't know your language, you storming girl.”

Shallan turned at Hermione's voice and Pattern's buzzing and saw her examining Pattern with a bewildered look on her face. She laughed slightlyand walked over to join them. “Pattern,”she said pointing at her spren. Then she looked at Pattern pointedly, “This is Hermione. She seems to appreciate writing and books. We should be nice to her. She might be able to help us learn their patterns.”

Pattern swirled around more quickly and responded, “She can't understand what we say anyway.” Shallan raised her eyebrows at him and Pattern began to swirl more slowly again before drifting slightly closer to Hermione and buzzing, “Hermione.”

By this point Hermione was staring wide eyed at the two of them. Her mind was racing as she tried to make sense of what she was seeing and hearing. Dumbledore had said something about Shallan's companion, but seeing it was something else entirely. Apparently Shallan had a constantly changing sentient, talking fractal as a friend, or at least companion. Hermione knew that figuring out how to communicate well with Shallan was going to be a challenge when she decided to rescue her from the chaos downstairs, but now she began to think that she was in even farther over her head than she had realized. She wasn't sure if this Pattern was going to make things easier or harder and bit her lip in concentration wondering where to start before finally nodding and gesturing to the room around her, “library.”

Shallan nodded and made a similar gesture around the room and repeated “library.” They repeated this process with a table, chair, book, pen and pencil before Shallan realized there was no way she was going to remember everything and pulled out her sketchbook. She sat down at the table and began making a list. Hermione sat next to her and watched as Shallan drew small pictures representing each of the words they had covered and then drew out an elongated symbol next to each. After finishing with the library words, Shallan added a small portrait of Albus with his name and Hermione with hers. She then sketched the woman who brought her food and looked questioningly over at Hermione as she pointed to the picture. Hermione broke out into a grin as she realized the elongated symbols must be Shallan's form of writing.

“Oh that's brilliant!” She then pointed to the picture and said “Molly.” Shallan repeated, “Molly,” back and wrote it in next to the picture. Hermione grabbed her own notebook, flipped it open to the first page drew a quick sketch of a book and wrote “book” next to it. She then gestured between the two notebooks. Shallan grinned and nodded as she pushed her notebook over so that Hermione could write each of the words using her script. Pattern hovered over them buzzing with excitement. When she had finished, he commented, “Mmmmm...patterns..not perfect, but close...”

Shallan looked up at him curiously. “What patterns do you see?”

“Mm, her small symbols mostly pair with yours. But not all... chair has too many...”

Shallan thought about this for a second before nodding to herself and flipping to the next page of her notebook where she wrote out each letter of the Alethi alphabet while Hermione watched curiously. Shallan then pointed to the first symbol and spoke its sound while looking at Hermione expectantly. Hermione repeated the sound and drew the appropriate letter or blend of letters next to it. When they finished, Hermione pursed her lips. The chart they had just made was mostly accurate, but not completely. She pointed to the letter “g”, which she had written next to the symbol Shallan had indicated was for the hard g sound and gave both the hard and soft g sounds. Shallan raised her eyebrows. Hermione thought for a moment and then pulled over her notebook and said “gem” as she wrote the word, underlined the “g” and repeated the soft g sound. She then repeated the process with the word “gap.”

Shallan sighed. Of course. Hermione's script wasn't completely phonetic. That would have been too easy. Well, at least it did actually use letters and she wasn't going to have to learn a huge collection of glyphs. They made another pass through the chart with Hermione giving some of the most common alternate ways to form the sounds. Pattern watched this process somewhere between amused and annoyed. “Mmmm, this is a terrible pattern. Full of all kinds of lies.” Shallan glared at him.“That may be, but it is going to let us understand the people here.”

They spent the rest of the afternoon naming objects to slowly increase Shallan's vocabulary. Shallan found the process to be a strange combination of tedious and exhilarating. There were so many words to try to remember, but learning something new was always exciting.

Dinner that night thankfully involved many fewer people than lunch had. Hermione introduced Shallan and Pattern to Molly, Ginny, Ron, Fred and George and made the twins promise to at least give Shallan time to adjust before including her in any pranking. She told them about how they had spent the afternoon and asked Mrs. Weasley if they happened to have any pictures books that might be useful for helping Shallan learn English. It turned out that the Weasley's had a large selection of picture books that could read themselves to children, so there was an expedition to the burrow to collect them and bring them back to headquarters.

Ron had no patience for spending the day helping Shallan and Pattern with English, and Hermione angrily kicked out the twins after Pattern caught them teaching Shallan lies, but Hermione, Ginny, Shallan, and Pattern spent the next several days focused on getting to a point where they could communicate things that required more than gestures and pointing. Whenever Shallan felt like her capacity for new words and grammar was saturated, they took breaks during which she would teach the other girls a little Alethi.

Pattern naturally caught on more quickly and so occasionally wandered off to discover what kinds of lies the twins were creating. The fact that he was able to warn Ginny and Hermione of the pranks the twins were setting up helped the girls to accept him and quickly become fond of him despite the fact that Shallan was unable to satisfactorily explain the idea of spren. Ginny giggled that having Pattern as a friend was way better for spying than those extendable ears her brothers had created. Shallan laughed along and thanked the Almighty that there weren't other Cryptics here to bond with the boys. She could only imagine where that would lead.

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.   

Frillbloom!  As with shalebark, frillbloom is a family of plants rather than a specific type.  Here is how I imagine a few different varieties, in both open and closed form.