Merry Christmas! And I Hope That Any Of You Who Don't Celebrate Christmas Have A Wonderful Day As Well

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. 

Inspired by a conversation with Ashiok in the chat :-)

Hey Cosmere fandom, it’s book rec time.

The book series in question is the Imperial Radch Trilogy by Ann Leckie. In order, the books are Ancillary Justice, Ancillary Sword, and Ancillary Mercy.

It’s a space opera with significant focus on character and relationships. The POV character is Breq, an aro-ace space warship AI in a human body who sets out on an impossible journey to kill the Lord of the Radch (the local 3000 year old space emperor) to revenge her beloved Lieutenant. Along the way she (unintentionally and reluctantly) collects people (humans, ships, aliens,…) in a gloriously messy found family.

* It’s super queer 

* Characters actively struggle with depression, anxiety, addiction, etc.

* Almost everyone is a PoC

* For the sake of Propriety everyone wears gloves and bare hands are super scandalous. 

These books are amazing and you should strongly consider reading them. The audio book versions are also fabulous.

If you’ve already read them, I’ve been Radch posting on my other blog (RithmatistKalyna) and I would love to talk with people both about the books/characters/tea sets/etc and all of the Cosmere crossover potential.

Your questions got me thinking and realizing that I have additional questions:

Szeth is likely going to summon his spren as a sword eventually. Is it going to make Nightblood jealous? Will they see it as a way to destroy twice as much evil? Szeth dual wielding blades? Or Nightblood getting frustrated because Szeth really needs a long spear for a sky battle right now and he is using the sprenspear instead of a sword?

Or, would Szeth summon his spren as a shield while using Nightblood as a sword?

What is the conversation going to be like when Nightblood and the spren talk?

Or, y’all, Szeth and Nightblood and the spren in a sort of threeway poly nahel bond. Maybe the bond could dampen the spiritual cracks in Nightblood so that they don’t automatically suck all of the investiture out of anyone holding them? 

I wonder what it’s like to be Szeth’s spren.  Like, the only thing they have to do is stick close enough that he can surgebind.  Nightblood takes care of everything else.  Did some Highspren just go: “This is gonna be the easiest assignment ever heck yeah” and then sleep through the rest of the book?  Are they annoyed at not having anything to do?  Were they tired of working with more conventional Skybreakers?  What do they think of Szeth siding with Dalinar over Nale?

I have questions guys

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

Oathbringer Speculation: Timbre

The descriptions of Timbre would fit with the name “lightspren”. We know that she communicates with Venli by pulsing to different rhythms. During the first shadesmar boat trip, Shallan speculates that the Reachers/lightspresn are using vibrations (ie pulses) to communicate.

This suggests that Timbre is a Reacher, but I suspect that she’s not just any Reacher. We meet Ico in Shadesmar and learn that his father is a deadeye and his daughter “ran off chasing stupid dreams”. Then Timbre tells Venli that her own grandfather was lost to human betrayal. Putting all of this together, I strongly suspect that Timbre is Ico’s daughter. I’m not sure what implications this will have for the future. But it’s a thing.

Citations: (Note: Page numbers come from the Kindle edition.)

What was that small spren that had crept out from beneath Eshonai’s corpse? It looked like a small ball of white fire; it gave off little rings of light and trailed a streak behind it. Like a comet. (pg 340) 

“The copper vibrates,” Shallan said. “And they keep touching it. I think they might be using it to communicate somehow.” (pg 931)

 “Wait!” Adolin said. “Ico, I saw something moving back there.” Ico locked the door and hung the keys on his belt. “My father.” “Your father?” Adolin said. “You keep your father locked up?” “Can’t stand the thought of him wandering around somewhere,” Ico said, eyes forward. (Pg 946)

Ico speaking:  “My daughter used to work there, before she ran off chasing stupid dreams.”  (Pg 948)

Timbre pulsed to Irritation, then the Lost. “That many? I had no idea the human betrayal had cost so many of your people’s lives. And your own grandfather?” (pg 1196)

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.

In the chasms in Way of Kings, Dunny creates an insult poem by making the rhythm fit the song Mari’s Two Lovers. Rock then goads Dunny into singing it before joining in.  Ashiok and I have been working on lyrics for the song.  This is what we have so far, but this is a Rosharan drinking song and we are pretty sure it has many, many more verses, so we are releasing it into the wilds of the cosmere fandom to see if anyone else happens to have any verses to add. Current lyrics under the cut.

Dunny’s poem:

Hey you big buffoon, 

you smell like a wet hog,

so go out by the moon, 

And jump yourself in the bog.

Mari’s Two Lover’s (lyrics by Kalyna and Ashiok)

Gather round and hear

this tale I have to tell

of our young Mari dear

and the men she loved so well

Mari was a lass

Though Not the brightest sphere

She often cut her class

to a wander far and near

There she met a lad

who took her out for wine

But she arrived with Khad

and left with his brother Stine

Khad and Stein were twins

who looked  so much the same

it was beyond her ken

that the two had different names

The twins had a tell:

two eyes of different hues

they were distinguished well

cause their eyes were switched, it’s true

Their town had a dance

And Mari begged to go

While there she whirled and pranced

Always watching for her beau

As she spun around

She fell/came into Khad’s arms

She was extremely proud

Khad had fallen for her charms

Then she came to Stine

And danced with him a turn

She stared into his eyes

Oh but still she didn’t learn

Khad had brown and blue

While Stine had blue and brown

But our dear Mari knew

Her beau was the best in town

Over in a lait

An axehound pup bit Stein

To keep up the charade

Stein cut khad with some fork tines

Now this song must end

I sadly have to go

But Mari and her men

Oh their tale goes on you know

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.

Oathbringer Speculation: Soulcasters

We know that people who consistently use soulcasters over a long period of time are...changed. My theory is that they are being slowly turned into spren and sucked into Shadesmar and further that this is the source of Syl’s comment about how the power coalesces slowly  and parts become sentient and a spren is born.

First off, let’s visit our friend Kaza in Interlude 4. We learn that she is slowly turning to smoke, but it seems to hold together well enough that she can wear a glove over her smoke fingers and still use her hand. She is forgetting the “ordinary passions of human life” and increasingly not needing to eat or drink. The she says that “I have begun to see the dark sky and the second sun, the creatures that lurk, hidden, around the cities of men.” It is clear that she is starting to see into Shadesmar and it isn’t too much of a stretch that in Shadesmar there might start also being evidence of her. The Aimian cook notes that she is “barely human anymore.” At the end of the interlude she chooses to “go with the smoke,” but the other times that she almost goes it sounds like she almost goes all the way into shadesmar and doesn’t come back, not that she almost turns into smoke in the physical realm and drifts off. Later in Celebrant (chapter 102) Kaladin finds himself in a tent with a “single bewildered spren made of smoke,” confirming that smoke spren are a thing.

In chapter 81 Kaladin meets a grain soulcaster and notes that “The woman had an inhuman look to her; she seemed to be growing vines under her skin, and they peeked out around her eyes, growing from the corners and spreading down her face like runners of ivy.” Then in Celebrant we meet spren that “were made entirely of vines, though they had crystal hands and wore human clothing.”

In chapter 105, we meet a soulcaster that makes stone and learn that his “skin beneath [his cloak] was colored like granite, cracked and chipped, and seemed to glow from within.”   In Celebrant there “were other spren with skin like cracked stone, molten light shining from within.”

We don’t have a description of the Azish soulcaster that makes bronze, but it seems like a good bet that their description would match that of the Reachers, who “looked like humans with strange bronze skin—metallic, as if they were living statues.”

In Chapter 35 of Words of Radiance we meet a soulcaster that doesn’t quite fit any of the descriptions of spren that we meet in Celebrant, though she could potentially be in an earlier stage of the granite type: “Prolonged use of the Soulcaster had transformed the eyes so that they sparkled like gemstones themselves. The woman’s skin had hardened to something like stone, smooth, with fine cracks. It was as if the person were a living statue.”

We also hear about Honor and then the Stormfather making Honorspren, so soulcasters wouldn’t be the only way that sentient spren are formed, but I’m fairly convinced that it is at least one way that spren are born. 

For reference and as a side note, in Celebrant they meet Cryptics, Honorspren, Reachers (bronze), Cultivationspren (vines), Inkspren, the ones whose skin turns to ash, the glowing granite ones, the ones made of smoke and possibly also ones made of fog/mist, though I’m not completely convinced those aren’t the same as smoke. If the fog/mist ones are different from the smoke ones, then this gives us 9 different types of sentient spren to correspond to the 9 non-bondsmith orders of the Knights Radiant.