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.
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.
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 thoughts: Eshonai
What if, just maybe, part of the point is that we are supposed to be angry and bitter about Eshonai’s death? Maybe, given what we know right now, it is supposed to come across as completely pointless.
It seems like no one (except Timbre and sometimes Venli) cares. Eshonai’s death happens off screen and the only reason we get confirmation that she is dead is that Ulim guides Venli and Demid to her to retrieve the plate. We learned all the way back in WoK that scavenging from corpses goes against Listener culture, and yet here we are with Venli being forced to do so anyway. The only reason we even find Eshonai’s corpse is that Venli is being forced to do things that go against her culture. Eshonai is dead. Their culture is being erased.
There are various parallels between the Listeners and the native peoples of the Americas. Far too often their plights are ignored. Indigenous women are murdered or go missing at disturbing rates and yet it almost never makes the news. Their deaths are horrible and pointless and it seems like no one cares. When they do get talked about it is almost always (at least from what I have seen) as an after thought. Much like Eshonai. Maybe that’s part of the point.
But you know what? Even with Eshonai dead, this isn’t the end of her story. Just because her death happened off screen doesn’t mean that will continue to be true. One of the next two books is going to give us Eshonai flashbacks. We are going to get to know her and her world better. And, I suspect, we are going to get to see that awful battle from her perspective and be with her as she falls into the chasm.
Ulim says it looks like she drowned in the flood waters, but my guess is there is more going on. She had plate, a blade, and storm form. We know she was at least somewhat trained with the plate and blade. Kaladin and Shallan survived spending a high storm in the chasms when they were both a mess and Kaladin didn’t have his powers and Shallan was wary of hers. Eshonai should have had the tools to survive. I can imagine a heartwrenching scene coming where Eshonai knows that she could save herself using the power that she has in storm form, but fighting that power and rejecting it. Or Eshonai being given the chance during that first Everstorm to become a fused and choosing to die instead. Some kind of intense internal struggle where she makes the choice to die rather than give in to something she can’t accept.
Her death will still be tragic. We can (maybe even should) still be angry that she died, but until we get her flashback chapters I’m going to cling to the hope that there is more going on than we currently know.
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So, I just had a probably crazy idea.
What if Nightblood isn't actually completely native to Nalthis? What if Nale is bringing it back home when he gives it to Szeth? Nightblood could have been a Rosharian Shardblade (Szeth doesn't seem to question it being a shardblade) that ended up on Nalthis. Then when Vashar and Shashara awakened it, the process roused the dead/trapped spren in the shardblade. From what we have seen, spren require an active bond with a human to remain sentient in the physical realm. Awakening would interact strangely with this resulting in the spren needing a steady supply of breath to remain active. This could provide an explanation for why it sucks the breath out of people (something that is never really explained in Warbreaker).
Nightblood's inability to fully understand what evil is and place it in a human context runs reasonably parallel to Pattern and Syl's attempts to understand the more abstract concepts of the human world. In Nightblood's case, the spren is still trapped in the sword and since it doesn't have a proper bond with a person, it is even harder for it to learn, so it can't progress as well or as quickly as Pattern and Syl.
The more I think about this the better it is working...
I just finished part 4 of The Final Empire and I need to rage for a bit before I can keep reading. Don't mind me. Ranting and wild theorizing under the cut.
"I represent the thing you've never been able to kill, no matter how hard you try. I am hope."
speech and presented yourself as a STORMING IDIOT MARTYR and LET THE LORD RULER KILL YOU WITHOUT FIGHTING BACK. DAMMIT KELL.
This is why you spent all of that time building your reputation among the skaa and getting them to practically worship you. This is why you were asking Sazed about the Valla. You were reassuring yourself that the movement would continue even stronger after you sacrificed yourself. You hoped it wouldn't come to this, but you knew it would. You've trained Vin as your replacement. She proved her competence in her fight against Shan and her power by seeing through your copper cloud. In the process she helped you understand what happened with Mare so that you could let it rest. SHE JUST ASKED YOU A COUPLE NIGHTS WHEN YOU WERE GOING TO LEAVE HER and you hedged. You didn't know exactly when, but you knew this was coming. You knew it would probably be soon. Dammit Kell.
Elend, you lovable naive fool of a nobleman, you need to find Vin and help her hold herself together so she can take over. Or maybe so the pair of you can take the lead if you can rally your friends and make them actually care enough to act. She has just had her brother's training that everyone will abandon her reinforced though, so she's probably going to try to push you away in a form of emotional self defense. I hope you don't let her. She is unbelievably strong, but there is power in having people you truly care about that she doesn't fully understand yet (The very last thing Kelsier said to Vin was "You still have some things to learn about friendship, Vin. I hope someday you realize what they are." Dammit Kell. That sounds like you knew you weren't going to be there. Maybe you had already decided that today was the day to get yourself martyred afterall.)
Other Thoughts:
* IF there really is an 11th metal (which I'm skeptical of), then there needs to be a 12th. They come in pairs.
* There is more to gold that Vin currently understands. Gold is disturbing and creepy and no one wants to burn it which also means no one has fully explored it.
* Feruchemisty is going to become important. Vin's experiment with Sazed's metal is going to come back. It could have just happened to answer the question of whether the two can interact (because I was definitely questioning it before it happened) but I will be surprised if it never surfaces again.
* In fact, some combination of Feruchemistry and Allomancy could very well be what lets the Lord Ruler survive things like having a spear run through his chest. Oh Storms. What if the reason he has been so intent on killing Keepers is because there is a way to store their life force in a metal (Gold? If I had to guess it would be gold) and then he can steal that life force by burning the metal. It would be like Allomancy meets Feruchemistry meets Chromatic Breath and the immortality of the Returned in Warbreaker.
I read a lot and have a lot of books that I enjoy, but a friend recently introduced me to the Stormlight Archive and it has inspired a level of obsessiveness that I haven't had since I met the Harry Potter series back in middle school. My new obsession led me to the Cosmere Challenge tumbler and then the current prompt led to this. Hopefully someone else will also find this entertaining :-)
Rough premise: Something has caused Shallan, Kaladin, Adolin and Renarin to be thrown into the Hogwarts world with their character development through the end of Words of Radiance but Hogwarts aged. I'm imagining that it is Harry's 4th or 5th year and that Renarin is Harry's age and the others are older. I have multiple plot bunnies in my head that require variations on the premise. Regardless, here is what I imagine happening when they are sorted:
The sorting hat barely touches Adolin's head before calling out “Gryffindor!” Adolin doesn't know quite how to handle the new situation he has found himself in and falls back on what he does understand as a defense. Unfortunately, this means he comes across as a superficial person who is extremely cocky and a huge flirt and the general impression that everyone gets is that they have been saddled with a new McLaggan. It takes several months and lots of set backs while Adolin learns how to interact with a world where the fact that he is a lighteyes Highprince is irrelevant, but he eventually finds his way and is accepted into the Gryffindor fold.
The sorting hat is less sure what to do with Renarin. It recognizes his quiet intelligence and wit and considers Ravenclaw, but Renarin wants Gryffindor. He has been told from the time he was very young that he is not cut out to be a warrior to the point that he has internalized the idea that he can't be. Despite this, he has always wanted to be able to follow in his father's and brother's footsteps. He is not at all confident that he truly belongs in Gryffindor, but he does know that that is where he really wants to be. The sorting hat considers and tells him that it would probably be truer to himself for Renarin to go to Ravenclaw, but that Gryffindor will also help him grow and respects his choice. Once he is in Gryffindor, Neville recognizes him as a kindred spirit and takes him under his wing. He is shocked the first time he has one of his minor epileptic fits and the other boys just shrug it off – after all, it is nothing compared to Harry's Voldemort induced visions. It boggles his mind that, for once, he is more popular than his brother and that it is more common here for him to need to defend Adolin's character than the other way around.
Kaladin doesn't believe it at first when they tell him a storming hat is going to sort him into a “house,” whatever that nonsense is. It sounds like a lighteyes thing – house Kholin, house Davar – and he wants nothing to do with it. Further explanations lead to the conclusion that it is more like being sorted into different barracks, which is at least something he can understand. Then they bring the hat out and it is an ancient thing that talks using a rip in its fabric as a mouth. Kaladin seriously considers throwing his hands up and walking out, but Adolin and Renarin are here, which means he needs to be here as well, so he grits his teeth and plays along. They put the hat on his head and he can suddenly hear it talking in his mind. Apparently these people have trapped a spren in a hat. The hat laughs at this thought and responds that it is not a bad analogy, but assures him that it is fine and there is no real hatspren that needs rescuing. At this point, to the surprise of the hat, Syl joins in the conversation. After much confusion, the sorting hat decides it is going to have to treat Kaladin and Syl as a single entity. It almost sends them to Gryffindor, but then realizes Kaladin's strong sense of honor is deeply rooted in his belief in the powers of hard work and doing what is right even when it isn't easy and sends them to Hufflepuff. Syl is pleased by this, but Kaladin is so overwhelmed by the whole business at this point that his only immediate reaction is to grumble that it is going to be hard to keep an eye on the storming Gryffindor Highprinces when he is in a different house.
Shallan gives the hat it's biggest challenge – at least until it discovers Pattern. Her dedication to her family and the hardwork and determination she showed while convicing Jasnah to taker her as a ward make her a solid candidate for Hufflepuff. Her clever use of lightweaving and willingness to do anything to achieve her goals, despite the questionable legality/morality of it, makes her the epitome of a Slytherin. She has the Gryffindor tendency to dive into things without fully thinking them through and a Ravenclaw's passion for learning. In the end it is Pattern's philosophy that “Truth is individual” and that “Your truth is what you see” that tips the scales and makes the hat decide to send them to Slytherin. Once in Slytherin, Shallan concludes that what she learned while figuring out how to navigate the Ghostbloods will serve her well in her new situation. It is going to be a challenge, but, with Pattern's help, she is absolutely up for it. [Note: There is so much potential here I can't even... We will see if I can get any of it to solidify. If anyone else wants to play with it (or any of the rest of it) I'm totally cool with that.]
In an alternate premise where Renarin is the only one who has jumped universes:
The sorting hat recognizes Renarin's courage and inner strength and wants to put him in Gryffindor. However, for the first time Renarin is in a place where no one knows him, where there are no preset expectations for him. In this new world, no one thinks it odd for a man to want to read and write. No one knows that he is the son of a Highprince. No one thinks he is a failure for not being comfortable around swords in general and shardeblades in particular. Gryffindor feels too much like the expectations he has escaped. He thinks Ravenclaw sounds acceptable. He would prefer not to be thrown into any preconceived classification, but if he has to have one, scholar sounds like a breath of fresh air. He thinks this might be an expectation he could actually live up to. The sorting hat grumbles about it and tells him that Gryffindor would be good for him, but accepts Renarin's choice in the end.
I was at a craft store today (this is always a dangerous thing) and was looking for a thing on the baking supplies aisle and I made a connection:
If you ever want vials of things that look like metal that you can actually for real eat, this is where you should go. There is such a thing a powdered food coloring and it comes in metallics. There are also vials of things that are basically silver and gold food grade confetti.
I'm not certain what kind of liquid I would suspend them in and you would still have to figure out how to pull off your special effects after drinking them, but there is a way to have vials of "metal" that aren't poison.
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
So. I found my way to tumblr when I first discovered Brandon Sanderson's books. As a result, this, my main, was all Sanderson all the time. Tumblr won't let us change which blog is the main blog and my brain won't let me make this blog more general, so you'll find my general tumbling (currently including a great deal of Imperial Radch and Murderbot) on my "side blog" RithmatistKalyna.tumblr.com .
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