OMH THAT IS ABSOLUTELY STORMING ADORABLE!!!!! You’re Amazing :D

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

I'm having trouble getting the next chapter of my Shallan in HP land crossover right, so I've been rereading Shallan chapters trying to figure out why things aren't working.  It's getting there.  In the mean time, have some Pattern thoughts:

“The pattern seemed impossible to capture. Its complex repetitions twisted down into infinity.” (pg 67)

Our dear friend Pattern is a fractal. This much is clear. It takes a while, but Shallan finally gets around to asking Pattern if he has a name beyond the one she has given him. His response:

“Name is numbers. Many numbers. Hard to say. Pattern...Pattern is fine.” (pg 215)

Fractal patterns that are on some level infinitely complex can often be described by seemingly simple equations or sets of rules.

Theory/ Headcanon: Pattern's true name is a mathematical description of his fractal form. The fractal exists in multiple dimensions and his appearance in the physical realm is just one tiny slice of his full form: the intersection of the fractal with the physical realm. His apparently changing patterns are not actually his pattern changing, but instead him moving so that the intersection changes and we get different views of him. His ability to shrink corresponds to moving so that his intersection with the physical realm is smaller.   

Now I really want to know Pattern's true name.

Ok.  I sat down planning to draw a Sarpenthyn.  You know, the critter in the menagerie that everyone but Shallan though was disgusting.  And then this adorable creature came out and I’m so confused. What do y’all think? Could this be a  sarpenthyn?

Character, book, and author names under the cut

Nangong Jingnu/Qiyan Agula- Clear and Muddy Loss of Love/Jing Wei Qing Shang by Please Don't Laugh/Qing Jun Mo Xiao

Dr. Arada/Overse- The Murderbot Diaries by Martha Wells

Harrowhark Nonagesimus/Gideon Nav- The Locked Tomb by Tamsyn Muir

Renarin Kholin/Rlain- The Stormlight Archives by Brandon Sanderson

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!!!!

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

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.  

My thought process on reading this:

Yeeesssss sibling interatctions!  

Mmm, if Lift were Kaladin's little sis, she would be the little sis of all of Bridge 4.  Since Rock is in charge of food, he would be her favorite big bro (or maybe uncle?...hmmm...either way works) and she would hang out around him and try to snitch food constantly.  Rock can see Wyndle and they conspire to try to keep her somewhat in line, but Rock is also never actually mad about her eating.   Sigzil takes it on himself to be her tutor and both starts teaching her to read glyphs and helping her understand her awesomeness by experimenting with it.  Lift isn't thrilled with his tutoring and sneaks off whenever she can. She learns much more about her awesomeness from the games she gets into with Lopen.  She catches Lopen glowing sometime and insists (with much stomping and pouting that isn't actually necessary because Lopen doesn't really need convincing) that he teacher her how to glow because that is so awesome!  At some point he sticks her to the wall and starts to laugh at her but then she becomes awesome and slides down the wall and past him making the bottoms of his shoes awesome as she goes past so that when he tries to turn around he slips and falls.  Lopen is briefly stunned and then bursts out laughing and congratulates her and from then on they are constantly trying to use their powers to get one over on each other.

http://therustyrobot.tumblr.com/post/102898558995/does-anyone-else-really-want-to-see-the-dynamic

does anyone else really want to see the dynamic between kaladin and lift

because I’ve been thinking about it all day and I feel like since most of the radiants so far are pretty tall and lift is. not. she’d want to see what all the tall people were doing so since kaladin’s the tallest she’d end...

Yahoo Groups Will Be Shutting Down - What Each of You Can Do

This post will be frequently updated - check the Google Doc linked here for the latest version. Follow this Tumblr for more information in the days ahead.

Yahoo Groups will be shutting down key features and restricting access to Groups, with user-uploaded content being deleted on December 14, 2019. With this shutdown, decades of fandom history will vanish. But there is something that every member of the fandom community can do now - whether they’ve ever used Yahoo Groups before or not.

I am a member of a Yahoo group - what can I do?

Immediately contact the admins to find out what their plans are.

Download PGOffline, a Windows tool to save the files and messages. Any member of the Yahoo Group can use this tool - you do not have to be an admin to save the mailing list. 

Since many admins are busy or inactive, install the free Windows software program yourself and start downloading - focus on Files, Photos, Links, and Messages. A step-by-step walk through is available here and a video tutorial here.

Submit your plan to download here.

 Export the messages and backup up the files and photos by copying them to another folder on your computer. Then contact Open Doors, the OTW preservation program. The OTW is open to providing storage of Yahoo Groups backups that are assembled by moderators and non-moderators alike. Details are here.

If you are a member of a Yahoo Group and have downloaded the files and messages, and the admins do not respond, please contact your fellow mailing list members. Remember, anyone can save a mailing list messages, files and photos and submit them to Open Doors.

Click here for instructions on how to use the Chrome plugin

If you need help from fellow fans, try asking on the “Save Yahoo Groups” Discord Channel

Mac Users: try this Chrome based plugin

Instructions on how to use the Chrome plugin

Alternatively, if you are familiar with Python, please experiment with the tools found at https://www.archiveteam.org/index.php?title=Yahoo!_Groups

I am Admin - what can I do?

Communicate with your Group members - let them know your plans

Download your Group messages, files, photos, and links using the tools above. Don’t forget to download your members list.

Decide if you want to simply archive the old posts and disband the mailing list or start up somewhere else.

Please take a look at setting up a Dreamwidth community - it allows explicit material, threaded conversations, privacy locks and is free. Image uploading is limited to 500MB for free accounts and 1.5GB paid accounts. Also, if you are considering Groups.io as an option, please note that it does not allow any material that depicts sexual activity, even implied sexual activity or anything that could be considered a fetish.

I am not certain I have time to help download? Is there something else I can do?

If you are a member of a mailing list, submit the mailing list for consideration. It will help volunteers focus their efforts. Don’t forget check here to see if your group is already being downloaded.

Even if you are not a mailing list member, consider creating a page for the mailing list on Fanlore, so that there will be a place for people to talk about the mailing list, its history. Details here.

Login Problems?

Has your “inactive” email been purged? Is your email email address not linked to a Yahoo Group ID?

An “inactive” account is one where you haven’t sent an email or logged in for the past year. Receiving emails does not count as activity. The email address is then released, and can be used by someone else.

If you used your Yahoo email to subscribe to a mailing list, your access to the Yahoo Group is gone (along with all your emails). If your account was deactivated within the past 90 days, you may be able to reactivate it. (more here). Alternatively, you can try contacting Yahoo Mail support.

Outside the 90 day window: you can try creating a new Yahoo email account using the same screen name as long as no one else has snapped it up. Simply sign up as you would for a ‘new’ email account. Reclaim an inactive mailbox. Then log into your Yahoo Group.