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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. 

Look. One of Wit’s goals in life is to poke fun at people and make them uncomfortable. I’m choosing to believe that that is Wit’s goal here. Jasnah gets it and knows that reacting will just encourage him, so she does her best to avoid reacting outwardly and just inwardly rolls her eyes.

They get along well. They are friends. They are co-conspirators. They challenge each other. Jasnah appreciates that enough to put up with the nonsense. That’s all. 

active footage of me trying to rationalize chapter 99 in my brain

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 ;-)

I hear it’s face day, so here’s my face XD. But also, look at the shiny new hat that I test-knit for @thechronicferuchemist ! It probably should be blocked, but even without blocking I’m super happy with how it came out :D (Her post about the hat is here)

My sister @neranishin and I have projects. She got me a piece of faux fur so we can figure out to make a Soonie pup and I found her fleece that I figure would make good skyeel wings. So there are going to be Cosmere plushies :D

I used fallenwithstyle's chart to knit myself a Pattern hat! The brown is KnitPicks City Tweed DK and the purple is KnitPicks Wool of the Andes sport. They were both sitting in my stash of yarn waiting to be right for something. I need to work on my consistency with keeping the floats loose enough in stranded knitting, but it isn't bad enough to be a problem.The hat also came out slightly longer than I usually prefer, but I'm super happy with it anyway :-). While I was knitting it, it occurred to me that there is potential for an amazing Pattern hat done with cables rather than colorwork. I'm not sure when I will get to it, but designing that pattern has gone in my "projects I will get to at some point" file. 

Sprenhead!

Here’s my finished Pattern hat! I’m really happy with how it came out not just as fanart but as a hat in general (and I said I didn’t need any more hats…).

I changed the chart sightly from the original version I posted, so here is the final version (it doesn’t include instructions for the flared brim or picot hem though, just the colorwork). 

On Shardblades and Horcruxes

I'm working on a longer SA/HP story, but this notion got into my head.  It won't fit in the larger story, but it also won't leave me alone:

In the Harry Potter universe Kaladin finds out about horcruxes when Syl passes through an object without appearing to harm it (maybe he is going on a rampage in the room of requirement which gave him things that he could destroy to relieve stress and he comes across the diadem?).  When Shardblades pierce living things, they kill or cut off the soul rather than harming the physical shell, so it stands to reason that a shardblade passing through a horcrux would destroy the soul piece instead of the object.  He eventually learns about the horcrux in Harry and has Syl manifest as a scalpel so he can remove the soul piece from Harry's scar.

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.

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.

We know kandra can imitate animals - TenSoon is particularly great at using his dog bones to his advantage. But you know what I want? Kandra birds and in particular Kandra vultures 

(BoM spoilers under the cut)

In Bands we find that we are entering the age of flight. This clearly means that we need a kandra who realizes they should learn to imitate a bird so they can keep up. VenDell tells us that small animals (like bunnies - thanks Wayne - also I really want to know more about “that book”) are extremely difficult because they “need a certain mass to hold [their] cognitive functions.” This probably rules out small song birds, but there are large birds of prey that might be big enough. One of the largest birds (discounting ostriches and emu that can’t fly) is the Eurasian black vulture that can get up to 30 lbs. I really like the idea of a kandra vulture. Can this please happen?

Then we can get the kandra vulture who has an intimate understanding of flight because they have eaten birds and absorbed their anatomy and learned to fly to hang out with Ranette and Sophi Tarcsel and create all kinds of wonderful things?