I Don't Particularly Identify With Melody, But I Love Her Character And Think It Is Super Important That

For the Blad design, wouldn't another good idea be to put stabilizing lines of forbidding between the tips of the ellipses, so they are outside the defense? They would stabilize and anchor it, not restrict movement, and also protect from incoming attacks.

That would be great, but unfortunately it won't quite work.  If you connect the tips of the ellipses, this is what will happen:

You could decrease the size of the portion that is cut off by playing with the dimensions of the ellipses, but you are never going to get the tip of the ellipse to stay inside that sharp angle at the top.  The only Line of Forbiddance that touches the bind point at the tip and doesn't intersect the rest of the ellipse is the tangent line, which would be perpendicular to the major axis and thus never reach the second ellipse.  You could contain the entire diagram in a square, but then you also can't attack.

If we could get bind points in places other than the tips, then a variation on your idea might work well.  I have ideas for how we might be able to construct elliptic defenses with more bind points, but writing that up will require more coherence than I have this time of night ;-)

Spy Girls

Shallan:  Daughter of rural minor noble family who goes on a journey and ends up in the court of the current political center of her world where she is slightly overwhelmed by everything and is trying to figure out how to save her family.  In the process, she discovers a plot of some sort in the underworld of the society and create and alter-ego to infiltrate it.

Vin: Daughter of the underworld who finds herself in a new crew where the rules aren't quite what they used to be and she has to adapt as they plot to completely overthrow society. As part of this, she creates an alter-ego in which she infiltrates the court as a daughter of a rural minor noble family who is here to further her family's interests.  

These girls are like two sides of the same fantastic coin and there are two things I really want to read:

1) Shallan and Vin curled up with coffee or tea or something trading infiltration stories and tips with Pattern providing commentary.

2) A scenario where these two are moving in the same court and are initially on opposite sides and they start getting suspicious of each other and eventually one confronts the other (I can't decide which way).  After talking they discover that there is more going on than either realized and essentially end up forming their own faction.

Rithmatics: Part 6, 9 Point Conics and Triangle Centers

Note: From this point on we are drifting farther and farther from what we know from the book. The math is all solid, but its application to Rithmatics is much more speculative.

In rithmatics, the 9-point circle plays an important role in constructing lines of warding and identifying bind points.  We also know that there exist elliptical lines of warding and that they "only have two bind points."  Now, in math we are frequently told things like "You can't take a square root of a negative number", which are true in the given system (real numbers) but not true in general.  The construction for the 9-point circle, as described in the book, doesn't work for ellipses.  However, there is a generalized 9-point conic construction.  To understand it, we need to start with a little bit of terminology.

A complete quadrangle is a collection of 4 points and the 6 lines that can be formed from them.  For our purposes, we will be concerned with complete quadrangles formed from the vertices of the triangle and a point inside the triangle.  The 6 lines are then the sides of the triangles and the three lines connecting the center point to the vertices.

The diagonal points of a complete quadrangle are the three intersection points formed by extending opposite sides of the quadrangle.  If we have a triangle ABC with center P, then the intersection of AB with PC is a  diagonal point.

If you take the midpoints of the 6 sides of a complete quadrangle and the 3 diagonal points of that quadrangle, these 9 points will always lie on a conic. This conic is the 9-point conic associated with the complete quadrangle.

Note that if we choose our point in the center of the triangle to be the point where the altitudes meet (known as the orthocenter), then this construction is exactly what we have been doing to create 9-point circles.

There are four classical and easily constructable triangle centers - the orthocenter, circumcenter, centroid, and the incenter.  There are over 5000 other possible notions of the center of a triangle, but most of them cannot be easily geometrically constructed and they get increasingly complicated. 

Let's look at each of these 4 triangle centers and the conic they produce for a particular triangle. We will use a 40-60-80 triangle in each case for illustration purposes, but the results will be very similar for any acute triangle with 3 distinct angles.

Orthocenter: We already know about the orthocenter (that is what most of this series has been focused on so far).  For reference, here is what the 9-point circle for this triangle looks like:

Circumcenter: The circumcenter of a triangle is found by finding the midpoint of each side of the triangle and drawing in the perpendicular bisectors.  The points where the perpendicular bisectors meet is the circumcenter.  Note: This point is also the center of the circle that can be circumscribed around the triangle.

Unlike with the orthocenter, the lines we use to construct the circumcenter (the dashed lines in the diagram) are not part of the complete quadrangle, so we have to finish the quadrangle after we have identified the circumcenter.  The resulting conic is an ellipse.

Centroid: The centroid of a triangle is formed by finding the midpoint of each side of the triangle and connecting it to the opposite vertex.  The intersection of these median lines is the centroid.

The lines used to construct the centroid are part of the complete quadrangle, but we have the interesting situation where the centers of each side are also the diagonal points of the complete quadrangle.  This means that, regardless of the triangle used, we will only ever have 6 distinct points.  The resulting conic is an ellipse that is tangent to all three sides of the triangle.

Incenter: The incenter of a triangle is the intersection of the  angle bisectors of the triangle.  

Note that the lines used to construct the incenter of the triangle are also the additional lines of the complete quadrangle.  In addition, as long as the angles of the original triangle are distinct, the 9 points in the construction will all be distinct.  The resulting conic is an ellipse.

In Summary:  There are lots of ways that we could potentially construct a 9-point ellipse from a triangle.  Of these options, I would guess that the construction using the  incenter of the triangle is the most likely to produce valid rithmatic structures.  I lean this way because, as with the orthocenter, constructing the incenter also constructs the complete quadrangle and its diagonal points.  Furthermore, the 9 points of the construction will all be distinct (except in special cases). As such, we will explore 9-point ellipses constructed with the incenter more thoroughly in the next post. 

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

From Words of Radiance, Ch. 41.

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.

Shallan's Journey: Chapter 2, In Which Shallan Meets a Fire Chicken

A sudden whooshing sound made Shallan spin around and her jaw fall open in shock at the resulting vision. A burst of flame quickly resolved itself into a person and the most brilliant chicken she had ever seen! After a few moments, during which she and the new man studied each other, Pattern drifted into her view and buzzed “Shallan. Your mouth. You are staring.”

Her jaw snapped shut and she quickly looked down as she began thinking frantically. This new man was certainly pale enough to be Shin, but he had far, far too much hair and she had never heard of Shin being able to travel in fire. For that matter she had never heard of anyone being able to travel in fire. Oh storms! She had no idea what was going on or what she should do next... but... but she had gotten through unfamiliar situations before... she just had to decide how to play this. Surely it wouldn't be worse than her first few meetings with the Ghostbloods. She closed her eyes and took a deep breath to center herself. Confidence. She could do this. Pattern's gentle buzzing as he floated near her head was reassuring. He was often good at helping her sort through things and if everything went completely storms up she would at least be able to defend herself. She held her sketchbook close to her chest with her sleeved safehand and her traveling bag to her side with her freehand as she looked back up at the man and the fire chicken with what she hoped was an expression of wary confidence.

She realized that he seemed to be just as curious about her as she was about him. Well, she supposed that if he was typical of the people here, then she probably appeared quite unusual. His sparkling purple dress thing was nothing like men would wear anywhere on Roshar.

Dumbledore wasn't accustomed to being speechless, but he truly wasn't sure how to react to the young woman he found when he arrived. She seemed startled, so he kept his face serene as he stood quietly to give her time to compose herself – and him time to try to make sense of the situation. She was wearing a style of dress he had never encountered despite his many years and extensive travels and seemed to be accompanied by a small whirling mass of black lines. Most curious indeed. When she looked back up and didn't seem inclined to either attack or run away, he took a step toward her and gestured in greeting.

“My dear girl. I noticed unexpected activity here at Stonehenge and came to investigate. Would you mind telling me who you are?”

Shallan looked back at him in confusion. His language was unlike anything she had ever heard before. She opened her mouth planning to say something formal and productive, but what came out was “Who on Roshar are you and where are we?” Pattern buzzed in a way that sounded almost like a snicker. “Mmmm...There's my Shallan. The strange man probably can't understand us anymore than we can understand him.” Shallan glared at Pattern.

Dumbledore's eyes widened as he watched this exchange. She spoke a rather curious sounding language and her friend seemed to be able to communicate directly with her. This was most unusual. He began contemplating how he could communicate with her. There were spells to ease translation, but they all required the caster to have some knowledge of both of the languages they were working with, a condition he usually easily met. They would not work here though.

Shallan narrowed her eyes in thought. She had no idea how to get back to Roshar (it was clear by now that she was no longer there) and as a Lightweaver she was fairly certain her surges wouldn't be particularly helpful for getting back. As much as she didn't like it, this most likely meant that she was stuck here until Jasnah figured out what had happened and was able to rescue her. Unless Jasnah showed up soon, it wouldn't be practical for her to just sit here waiting for her. This meant that, at the very least, she needed information. Right now her best hope of information was this strange man. Shallan made eye contact with him and then pointed to herself. “Shallan.”

The man smiled and nodded to her as he repeated, “Shallan,” and responded with a similar gesture and “Albus.”

Shallan nodded slightly, “Albus.” Now what... Shallan bit her lip and then sat down as she opened her sketchbook and pulled out a pen. They couldn't communicate in words, but pictures didn't have a language. Her pen flew across the page as she quickly sketched Urithiru and the portal they had been experimenting on. She sketched the palace of Jah Keved and added lines that would hopefully suggest that she was hoping to travel there. She then drew a large slash through Jah Keved and firmer lines from the portal into a solid black blob with further lines leading to a quick sketch of her current surroundings. As she finished, she stood and turned the sketchbook toward him hopefully.

Albus studied the image thoughtfully. Whoever this girl was, she had quite a talent for drawing. That quick sketch of Stonehenge was quite remarkable. He turned his attention to the rest of the picture. There was an interesting tower that she seemed to be suggesting was the starting point of her journey. Next to it seemed to be a traveling device of some sort with the indication that it was supposed to take her to the other large building. Instead it apparently took her into a void and then dumped her here. Dumbledore nodded thoughtfully and slowly pulled out his wand.

Shallan watched with fascination as the man, Albus, she reminded herself, took out an oddly uniform stick and began weaving it through the air. To her astonishment, he began to form the image of a large stone castle in the air between them. Was this man another Lightweaver?! Was that stick some strange fabrial? The image zoomed into a room where there was an image of Albus sitting at a desk... Shallan squinted, was he...writing? The image man then looked up and over at a small image of the fire chicken. The fire chicken image flew to the image of the man, which appeared to grab the fire chicken's tail before they both disappeared in a flash of fire. Albus waved his arm and the castle also vanished.

He then wove an image of what appeared to be a home with several other people around and then added an image of himself, the fire chicken and...and Shallan... appearing in a burst of flame.

Shallan frowned. Aside from questioning the brilliance of going anywhere in a strange land with a strange man she had just met, Shallan was worried about what would happen if Jasnah made it here and she wasn't nearby. She turned back to her sketchbook and flipped to the next page. This time she started with an image of Jasnah next to an image of Urithiru with her hand up to her eyes as though looking out of the page. She followed it with an increasingly hasty series of places around Roshar with a flowing line connecting all of them in sequence. Then a picture of Jasnah at a desk surrounded by books and finally an image of Jasnah at in their current location looking sad and confused.

Albus nodded slowly. Shallan seemed to be indicating that someone would be frantically looking for her and didn't want to miss them when, if, they made it here. While he was contemplating the problem Shallan turned to Pattern. “Do you know of any way we could leave a message here for Jasnah? Maybe in Shadesmar?”

“Mmmmm... Jasnah will be tracking you, not just your path. Moving around shouldn't interfere much.”

“You are sure?”

“Nothing is sure. Sure is truth, but truth is based on lies and always changing.”

Shallan rolled her eyes. Someday she would learn not to try to get a straight answer out of Pattern. “Do you think we should go with the man and his fire chicken?”

“Mmmm... he makes good lies. ”

Right. That would be what Pattern focused on. Well, Albus had been reasonable so far and she didn't exactly have any other options. Now, how to tell him that she was willing (or at least had resigned herself) to go with him...

Shallan turned back to her sketchbook and made a sketch of the fire chicken with both her and Albus holding its tail and then got Albus's attention. He gave her a questioning look and pointed back at her Jasnah sketch montage. Shallan shrugged and pointed at Pattern. There was no point in trying to explain her rough understanding of what Pattern had tried to tell her. Albus smiled, nodded and looked around carefully to make sure they were alone before saying slowly and carefully, “The headquarters of the Order of the Phoenix may be found at number twelve, Grimmauld Place, London.”

Shallan raised her eyebrows and cocked her head. Was she supposed to have gotten anything from that? Albus sighed and repeated the first word, “The,” and then gestured to her. Shallan repeated it back to him. They went through the whole phrase that way, first with the individual words and then with subphrases until Shallan was able to repeat the whole thing back to him. The words felt strange in her mouth and she had no idea what they meant or what their importance was, but she could say them.

Fawkes, who had been flying a sort of dance in and about the standing stones, now flew over to them and dangled his tail feathers between them. Shallan held out her arm and, to Dumbledore's astonishment, her friend seemed to meld into the sleeve of her dress. This girl came with no end of surprises. Shallan took a deep breath and confidently but gently took hold of Fawkes's tail with her freehand. Dumbledore joined her and they vanished in a flash of flame.

After the events of the last year, Shallan thought she had about had her share of new, unsettling experiences. The process of traveling by fire chicken wasn't actually that shocking – she felt a little warm and a slight tickling before suddenly arriving in a dark gloomy hallway – except for the simple fact that it had happened. Shallan had accepted the fact that there were fully sentient spren. She had, very reluctantly, accepted that she was on track to become one of the new Knights Radiant – the order whose previous incarnation had betrayed all of mankind on the Day of Recreance. She knew that Jasnah could use Shadesmar to travel quickly around Roshar and that Kaladin, that storming obnoxious but fascinating bridgeman captain, could use stormlight to fly.

If the fire chicken had been a spren, she might have been able to just accept that Albus had some unknown surge that let them use fire to travel. The chicken, however, was clearly not a spren. It was a chicken. A flashy, brightly colored, singing chicken, but it was a chicken. Chickens did not travel in fire. If chickens and fire went together they became food.

While Shallan was still mostly out of it as she tried to come to terms with the idea of traveling by fire chicken, Albus gently led her down the hall to a staircase and then down into what appeared to be a kitchen. The sound of several people talking over each other startled Shallan out of her confusion just in time to notice a woman wearing short sleeves with both hands and arms exposed standing at what seemed to be a stove. She scanned the room and saw that there were boys, girls, men and women sitting together at the table eating the same food. She groaned, took a step back so that she was against the wall next to the stairs they had just come down, slid to the floor and buried her head in her arms. Oh storms. This world was nothing like Jah Keved, Kharbranth, or Alethkar.  

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

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.

AHHHH I just discovered that somewhere in the convoluted process of getting the file from SoundRecorder to tumblr the first line got lost... Why didn't anyone say anything?!  Corrected version coming shortly!

Corrected version can be found at http://kalynaanne.tumblr.com/post/111933243197/ok-completely-new-post-because-the-other-one because tumblr apparently won't let me fix this one *rolls eyes*

Ok.  I figured that if I did the lullaby from Shadows, that meant I should also do Shallan's lullaby from Words of Radiance. I sat down and messed around with it until I came up with a tune I like, so, here, have another cosmere lullaby.