Latest Posts by kalynaanne - Page 3

So... I was experimenting with the watercolor brushes in FreshPaint and playing around with ways to get different effects for skies...and then this happened... so here, have a happy playful skyeel flying through the sunbeams with its little fishy spren friends.

MOAR CHALKLINGS

P is for Prickletac.  

V is for Vinebud

Badger and I were contemplating what color axehounds should be and then they said there should be sparkle axehounds and that reminded me of My Little Pony and then Badger drew this and it is EXCELLENT. 

my little axehound, my little axehound ~

Ohhh... this looks like fun...

Cinderella's fairy godmother is a lightweaver who sent her spren to the ball with Cinderella to maintain the illusion.  The midnight curfew is there because she could only infuse her spren with so much stormlight and the illusion will fail when it runs out.  Maybe the spren stays with the glass slipper and uses the little bit of stormlight it still has to continue maintaining just that part of the illusion...

Princess and the Pea is an easy one, relatively speaking. The princess in question is a tin-compounding Twinborn who got stuck in a touch loop at an early age. The easiest way I can think of for that to work is that she doesn’t know she’s doing it - perhaps she fundamentally doesn’t understand...

My immediate reaction to seeing this:  Grab my ever present notebook and copy it down so I don't have to keep flipping tabs.  Open the Coppermind.  Translate.  Grin.

an urgent PSA to the cosmere fandom

Hey guys, let's think about bendalloy ferrings, aka Subsumers, for a minute.

We know they can store nutrition and calories in a metalmind which means they can eat as much as they want when there is lots of food and then tap it later.  

Think about a Subsumer at a Scadrial eating competition.  "Sure, I can scarf away as many hot-dogs as you want. No problem."

Think about a Subsumer at an all-you-can-eat buffet.  They get to try everything and then go back for as much more as they want of everything they like. Such restaurants would have to have special rules for subsumers...maybe a pay by the hour thing?

Imagine if Lift gets a hemalurgic spike that lets her become a bendalloy ferring. She would have the best stormlight storage system.   She could eat loads, store most of it in her metalmind(s), and then tap into it whenever she needs stormlight.  Unlike spheres, it wouldn't leak away and if she is tapping it from the metalmind she isn't sucking the energy out of her body.

On the darker side of things, you probably also get the occasional Subsumer who is desperately afraid of gaining weight and always stores everything they eat in their metal minds and never taps them.  Most people don't know that they are a ferring and everyone sees them eating, so people don't realize there is a problem until it has gotten really bad.

Rithmatics digression

We've talked about 9-point circles.  Here is a further exploration of 9-point conics in case you are curious:

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!!!!

Hello tumblr Cosmere fandom!

This last weekend was the MIT mystery hunt and this got me thinking.  Branderson's world building has given us all sorts of lovely systems that could be used as the basis for puzzle-hunt style puzzles.  Are there other puzzle hunt type people around the tumblr cosmere fandom?  If I were to actually turn one/some of the puzzle ideas floating around my head into reality, would there be people interested in poking at it/them?

Rithmatics: Part 8, Intro to Lines of Vigor

Let's start with what we know for sure from the books:

Lines of Vigor can only affect chalk creations

You have to draw at least two waveforms to shoot a Line of Vigor

Lines of Vigor can damage or move Lines of Warding (and other chalk creations)

"smaller curves" are stronger [1]

A "large arc" lets them move other lines [2]

Lines of Vigor can bounce off of Lines of Forbiddance

We have previously established that the source of the illustrations in the book is playing fast and loose with the definition of curvature, so the notions of "smaller" and "larger" in the above are not entirely clear.  I know what I want them to mean though...

Theory Time

You know what this sounds like to me?  This sounds like electromagnetic waves.  EM waves are often modeled with transverse waves (the sine curves all of our examples of LoV look like are transverse waves). Higher frequency radiation, such as UV rays is better at penetration than lower frequency radiation, like infrared.  Infrared, on the other hand, is much better at transferring energy, which could very reasonably lead to a transfer of momentum here and thus lead to things moving.  As long as velocity is constant, frequency and wavelength are inversely proportional.  This means that the tighter the curve is the higher the frequency. Theory: Lines of Vigor are based on EM waves.

Possible Implications:

A EM wave with a larger amplitude has more energy.  This would suggest that a Line of Vigor with a short wave length and high amplitude would be better at punching through things and one with the same wavelength and small amplitude.  The effect could be even more pronounced with high amplitude long wave length waves that are trying to push things around

EM waves follow the law of superposition - if you stack two on top of each other you can add them at every point to get a new wave.  This would suggest that there might be more interesting LoV. Instead of just sin(x), we could get things like sin(x)+sin(x/2).  The more complicated ones could be tricky to get right, but might be able to do interesting things.

Related to the previous point, you could get interesting effects if LoV could have interference patterns like EM waves do...

EM waves travel with different speeds through different materials and you get refraction when they move between materials.  Since LoV only affect chalk, it is possible they are only affected by chalk, but if the material they move across affects their velocity, you could get refracted LoV when they move from the sidewalk to the asphalt.

Lines of Forbiddance would be playing the role of mirrors. I have lots of ideas here, but those are for a future post or this one will end up ridiculous.

Rithmatics: Part 7, The Incenter and 9 Point Ellipses

In the last post, we determined that the incenter (the intersection of the angle bisectors of the triangle) is probably the most viable center option for constructing 9-point ellipses. Today we will look at what we can get by using different triangles.

The first thing to note is that for an equilateral triangle the incenter is the same as the orthocenter.  This is because the angle bisectors are also the altitudes of the triangle.  This means that the resulting conic is going to be the 6 point circle we are already familiar with.

When we were working with the orthocenter, right triangles gave us special cases because two of the legs of the triangle were also altitudes.  Since we are using angle bisectors instead of altitudes now, right triangles do not form a special case.  In fact, we will never have a situation where a bisector coincides with a triangle side.  

Another interesting thing to note is that, unlike the orthocenter, the incenter will always be inside the triangle, even if the triangle is obtuse.  This means that obtuse triangles now produce new 9 bind-point patterns.

The most interesting case occurs when we look at isosceles triangles. Here, the angle bisector for the different angle also bisects the base of the triangle.  This means that those two points coincide and we end up with 8-point  ellipses.  Furthermore, the  symmetry of the triangle means that the angle bisector of the unique angle will also end up being an axis of the 8-point ellipse.  If the unique angle is smaller than the two matching angles, it will be the major axis.  If it is larger, it will be the minor axis.  Both possibilities have potential strengths and could be used to design useful defenses.

Next time we will either go off the deep end with a crazy idea I have for Lines of Warding or we will dive into Lines of Vigor. I haven't decided topic I want to write up next.

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. 

Welcome to the 2015 Cross Cosmere Rithmatics Olympiad!

The other night in FNCC I brought up the question of what would happen if someone drew a chasmfiend chalkling.  Things snowballed from there and dm-mo, ElderDragonMystic and I got carried away.  As a result, our friends from across the Cosmere are gathering to compete in games derived from one of Sanderson’s non-Cosmere magic systems.

Welcome to the 2015 Cross Cosmere Rithmatics Olympiad!  Teams are gathering from across the Cosmere to compete in an exciting assortment of rithmatic events! The games will be kicking off soon and we here at the Cosmere Chronicle will follow the action when they do, but in the mean time let’s introduce the teams.  

Roshar is fielding three teams this year.  

The first is their traditional team from the Alethkar Academy of Rithmatics.  This team’s roster is formed of brothers Adolin and Renarin Kholin, their cousin Elhokar Kholin, and Shallan Davar, a young woman who hails from Jah Keved.  As expected, these four all have solid groundings in classical rithmatics and should be a force to reckon with.  Rumor has it that there is some dissension in the group though and that they don’t always work particularly well as a team. It will be interesting to watch them and see how this plays out

The other two teams are a ragtag bunch known only as Bridge 4 that are not associated with the Alethkar Academy.  This motley crew appears to bring together students from nearly all regions of Roshar and has divided itself into a Division 1 and a Division 2 team. The senior team consists of Kaladin “Stormblessed”, Moash, Teft and Sigzil.  The junior team is comprised of Lopen, Rysn, Tien, Lift and Syl and coached by Numuhukumakiaki'aialunamor “Rock” of the Unkalaki.  We know very little about the rithmatic background of any of these competitors, but if what we have seen in the practice fields is any indication, they could shake things up - they appear to be bringing a style unlike anything we have seen before.

Scadrial is fielding two teams, one each for Division 1 and Division 2.

Scadrial’s senior team should look familiar to anyone who follows Intra-Cosmere Rithmatics.  It is, as expected, comprised of husband and wife team Elend and Vin Venture as well as The Survivors, Kelsier and Spook.  In case you don’t  usually follow ICR, Scadrial rithmatics is known for working in pairs and they should help make the pairs dueling exciting to watch.  The Ventures work seamlessly together while Kelsier and Spook always seem to be able to recover no matter what mess of a situation they find themselves in.

Oddly enough, Scadrial’s division 2 team has an odd number of competitors: the established duo Wax Ladrian and Wayne and newcomer Marasi Colms.  We caught Marasi to ask her how it is working in rithmatics on Scadrial without a partner. She informed us very matter of factly that she simply hasn’t found the right partner yet, but that there was no way she was going to let that cause her to miss the opportunity to come to the games this year.  Based on the rest of our conversation, we expect her to hold her own in the theory competition.

Sel is also fielding a team for each division.

To form its senior team, Sel held a local preliminary competition where it ended up choosing a pair each from two very different parts of the planet.  It is comprised of Sarene, Hrathen, Shai and Gaotona.  It will be interesting to see how the four members blend their styles for the team portions of the games, but they all seem quite focused and determined to win.  One thing is certain - we can always expect precision from Sel rithmatics.

Sel’s division 2 team is possibly the most unusual team - its captain, Raoden, isn’t actually a Rithmatist!  When we found Raoden to ask about this, he calmly reminded us that one doesn’t have to be a Rithmatist to be a scholar of rithmatic theory. His team is rounded out by a particularly laid back young man named Galladon and an enthusiastic but clearly intelligent girl named Kaise who is, we believe, this year’s youngest competitor.  

Political unrest on Nalthis has disrupted its rithmatics program, so it has only sent one team this year.  Veteran Vasher is back and should be a powerhouse in any event he competes in,  though rumors suggest that he may only be here for theory this year.  The team seems to be held together by the lovely princesses of Idris, Vivenna and Sisirinah.  It is rounded out by Siri’s quiet husband Susebron. Keep an eye on Nalthis if you are interested in chalklings.  No one is quite sure how they do it, but in the past we have seen chalklings from the Nalthiean competitors that seem almost sentient.

We here at the Cosmere Chronicle look forward to sharing the Cross Cosmere Rithmatics Olympiad with you in the coming days.  We will bring you the results from the events, analysis of different rithmatic styles, and stories from around Olympic Village.

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 ;-)

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.

Merry Christmas! And I hope that any of you who don't celebrate Christmas have a wonderful day as well :-)

Elendel, during the Wax and Wayne era