I'm Slightly Late, But Here You Go!  Have A Little Story :-)

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.

I finally got around to reading Sanderson's Shadows for Silence in the Forests of Hell (I've had the anthology checked out from the library for a couple of weeks). It has a lullaby. My brain provided a tune for the lullaby (I do not know where the tune came from, I don't recognize it and if it sounds like anything it was unintentional).  Getting this file into a form tumblr would take was a stupidly convoluted process and I'm not entirely happy with the recording quality (I really don't know what I'm doing with recording), but, here, have a cosmere lullaby.

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. 

Here, have another of Iredomi's poems!  The text can be found in his original post here on sebrukiscrossbow. 

Hey y’all, it’s Kaly reads a poem time again.  This one is called Entropy’s Embrace.  It isn’t quite a cosmere specific poem, but it is a great poem, was written by our friend worldsingers and it does have some lovely cosmere evoking imagery.  You can find the text and a review of the poem here.

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.

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.

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.

If you have questions about editing or the Coppermind in general, I would also be happy to help/encourage you :-) 

In the vein of wikipedia’s wikiprojects, I have recently made a few pages which will help people who are looking for a way to contribute to the coppermind but don’t know where to look to find things that need doing. They are a set of pages which list the status of various articles, so you can see which articles really need help. As a new contributor, adding content to the articles in the good or stub categories of your favorite series would be a reasonable place to start.

Alcatraz

Cosmere wide

Elantris & The Emperor’s Soul

Legion

Mistborn: Trilogy, Wax & Wayne, Crafty RPG

The Reckoners

Rithmatist

The Stormlight Archive (warning: this page is huge, there’s just so many articles relating to it it’s still not very usable)

Warbreaker

For more details on what the various categories mean, jump under the cut.

Read More