Tarontos he/him Posted May 30, 2013 Report Share Posted May 30, 2013 (edited) Spoilers Basic theory is that every bind point set is represented by a Triangles. These are the reference images (note i made these images) http://www.weebly.com/uploads/1/4/9/0/14907026/1369915093.png 9 pointer http://rithmantics.weebly.com/uploads/1/4/9/0/14907026/6701685_orig.png 6 pointer http://rithmantics.weebly.com/uploads/1/4/9/0/14907026/4716856_orig.png 4 pointer Edit: An important assumption is that two point circles are just incomplete six or four pointed circles, Edit:Two point circles can be made by (0,180,0) triangles. credited to happyman First lets look at a four pointer, a square can be made by drawing a circle through any triangle, my example is a 45/45/90 because it is easy to find the circle. A six pointer can be made by using the altitudes of an equilateral triangle and the mid points. The nine pointer is were this theory comes from. the nine point definotion, which is , the nine point circle is drawn on an acute triangle, the points on the nine point circle are determined by three things, the mid point of each side, the intersection of the altitude of each side and the side, and the intersection of the resulting circle and the altitude. All of these circles an be made by placing them on a triangle and marking the bind points. This also explains why there are no five point, seven point or eight point circles. because the bind points can not be covered by any triangle. i assume that if you want a three point you can just make a six point using every other bind point. Edited September 6, 2013 by Tarontos 5 Quote Link to comment Share on other sites More sharing options...

happyman he/him Posted May 31, 2013 Report Share Posted May 31, 2013 (edited) I agree with this. There is actually an illustration in the book (sorry, don't have it with me) where one of Prof. Fitch's drawings is a triangle that resuts in a six-point circle. (Equilateral is easiest, I believe.)As for two-point circles, I would add that there is an element of math known as degeneracy. For instance, a straight line with two ends is technically a triangle with internal angles of (0, 0, 180). The length of one of the sides is the sum of the length of the other two sides, and the two short sides lie on top of the long side. Of course, if this were true, you could place two bind points anywhere on any circle, and it would be technically valid. Of course, it's possible that the strength of the circle is maintained by the area of the circumscribing triangle...Edited to replace "circle" with "triangle" at a key point, and thus have this post make any sense. Edited June 3, 2013 by happyman 1 Quote Link to comment Share on other sites More sharing options...

Tarontos he/him Posted June 1, 2013 Author Report Share Posted June 1, 2013 For instance, a straight line with two ends is technically a circle with internal angles of (0, 0, 180). Triangle, I didn't think of this but it makes the theory fit better. Yeah the two point circle is hard to explain, I would assume that it is just the nature of the circle that makes the bind-points be on the Diameter. I agree with this. There is actually an illustration in the book (sorry, don't have it with me) where one of Prof. Fitch's drawings is a triangle that resuts in a six-point circle. (Equilateral is easiest, I believe.) http://brandonsanderson.com/images/rithmatist/RITHMATIST_SPOTS_PartC_webres.jpg,this is a link to the page with the images this is for future reference, and is the complete images from the Rithmatist, http://brandonsanderson.com/book/The-Rithmatist/page/69/The-Rithmatist-Interior-Illustrations What i really wonder is how the ellipse fits in, why you cant have a four point ellipse, or so on. 0 Quote Link to comment Share on other sites More sharing options...

Tarontos he/him Posted June 4, 2013 Author Report Share Posted June 4, 2013 (edited) First I know i'm double posting but edits do not actually show up as a recent update, and i had a new thought to do with this. Five and eight point circles, so noting the initial post, that all of the circles are really nine-pointers. isosceles triangles, following the nine point format will have eight unique bind points. a right triangle, again using the nine point format has five unique bind points. Edit added reference images five point: http://rithmantics.weebly.com/uploads/1/4/9/0/14907026/3823222_orig.png eight point: http://rithmantics.weebly.com/uploads/1/4/9/0/14907026/9999704_orig.png Edited June 5, 2013 by Tarontos 0 Quote Link to comment Share on other sites More sharing options...

Takeshi Posted June 5, 2013 Report Share Posted June 5, 2013 What i really wonder is how the ellipse fits in, why you cant have a four point ellipse, or so on. I think this has to do with an ellipse having two focii. If you think of a circle as an ellipse where both focii are at the same point, then lines that cross one focii cross both focii. I also believe the strength of the bind points depends on how close to the "focii" of the circle the intersecting lines of the triangle are (hope that made sense). So in the case of an ellipse, the only "triangle" that cross over two separate focii is the straight line. 1 Quote Link to comment Share on other sites More sharing options...

Topomouse he/him Posted June 5, 2013 Report Share Posted June 5, 2013 Dang, just yesterday I was re-reading Rithmatist and had the same idea about 8 bind-point and the possible importance of triangles but someone beat me. 0 Quote Link to comment Share on other sites More sharing options...

Tarontos he/him Posted June 5, 2013 Author Report Share Posted June 5, 2013 (edited) I think this has to do with an ellipse having two focii. If you think of a circle as an ellipse where both focii are at the same point, then lines that cross one focii cross both focii. I also believe the strength of the bind points depends on how close to the "focii" of the circle the intersecting lines of the triangle are (hope that made sense). So in the case of an ellipse, the only "triangle" that cross over two separate focii is the straight line. I like this but i believe you are referring to the vertices of the ellipse, the foci are the center of a circle and the lines of nine-pointers do not cross the center, the vertices of a circle are any where. in an ellipse the foci are the points that lines are drawn between, while the vertices are the two farthest away points. Note: i am not at home and hence can not attach reference images they may becoming. as promised the image, http://www.weebly.com/uploads/1/4/9/0/14907026/8703525_orig.png Topomouse Posted 05 June 2013 - 05:54 AM Dang, just yesterday I was re-reading Rithmatist and had the same idea about 8 bind-point and the possible importance of triangles but someone beat me. To remember Hoid's speech about noteriety Edited July 26, 2013 by Rubix 0 Quote Link to comment Share on other sites More sharing options...

Ender he/him Posted July 25, 2013 Report Share Posted July 25, 2013 Uhh... I'm not great at math but are you trying to say that no triangle but the line would work on an ellipse? Or am I missing the point? 0 Quote Link to comment Share on other sites More sharing options...

Tarontos he/him Posted July 26, 2013 Author Report Share Posted July 26, 2013 Uhh... I'm not great at math but are you trying to say that no triangle but the line would work on an ellipse? Or am I missing the point? Yes, no 0 Quote Link to comment Share on other sites More sharing options...

johni92 Posted April 21, 2014 Report Share Posted April 21, 2014 As for two-point circles, I would add that there is an element of math known as degeneracy. For instance, a straight line with two ends is technically a triangle with internal angles of (0, 0, 180). The length of one of the sides is the sum of the length of the other two sides, and the two short sides lie on top of the long side. Of course, if this were true, you could place two bind points anywhere on any circle, and it would be technically valid. Of course, it's possible that the strength of the circle is maintained by the area of the circumscribing triangle... Sorry to resurrect an old topic, but I just discovered this after making this post here: http://www.17thshard.com/forum/topic/5990-trigonometric-connections-to-rithmatics/?p=127619 I don't think it's true about the two bind points being able to be anywhere on the circle. Finding the nine points in terms of this triangle gives two points perpendicular to one another, so it makes sense they'd have to be the end points of a diameter of the circle, rather than being just anywhere. 0 Quote Link to comment Share on other sites More sharing options...

akasketch he/him Posted April 22, 2014 Report Share Posted April 22, 2014 I saw this thread and thought, "MAN! Idea stealer!" Then, this: Sorry to resurrect an old topic, but I just discovered this after making this post here:http://www.17thshard.com/forum/topic/5990-trigonometric-connections-to-rithmatics/?p=127619 I don't think it's true about the two bind points being able to be anywhere on the circle. Finding the nine points in terms of this triangle gives two points perpendicular to one another, so it makes sense they'd have to be the end points of a diameter of the circle, rather than being just anywhere. Sir, I thank you, for as dead as my old thread may be, the rules specifically state that if you are providing more evidence to a theory, you are not resurrecting. If anyone is at fault here, it is the creator of this thread, as he STOLE MY THREAD. Just kidding. Kind of. Not really. Okay, not at all, but can you blame me? Anyhow, a two-point circle would be referencing the base and tip of a triangle, and an argument that one could make a 9-point circle and only use two of the points is not possible, as the bind points are determined not only by the Rithmatist's intent, but also how he or she draws the circle. The placement of the final addition to the line of warding would probably solidify where the bind points are, and prove whether or not the Rithmatist's creation is stable or not. The fact that each type of circle (excepting the 1- and 2-point circles) is based on the nine important points of a different triangle (See aforementioned linked thread for comprehensive, albeit confusing, evidence) proves this thread true, and also shows us that other combinations might be possible, though difficult to work with. Research into the actual discovery of the nine-point circle (http://en.wikipedia.org/wiki/Nine-point_circle) shows that there are other important curves and even entire circles associated with it that could possibly be utilized in potential lines of warding. For Cosmere Conspiracy Theorists: *Gasp!* There are 16 tangent circles in an orthrocentric system!! Coincidence? I think not! 0 Quote Link to comment Share on other sites More sharing options...

johni92 Posted April 23, 2014 Report Share Posted April 23, 2014 I don't really buy the idea that a two-point circle would just be referencing the base and tip of a triangle, because every other circle is able to be expressed in terms of the nine points, just with some merging to be the same point, hence why I tried to show that it is actually possible for a two-point to be expressed as a nine-point with them all converged down to only two. It also better explains the requirement for the points to be end points of a diameter than the "just the tip (and base)" explanation. Also, you mentioned one-point circles? It's been a while since I read it, but are they possible? I don't remember. And I'm not sure what its complement triangle would be if it does. 0 Quote Link to comment Share on other sites More sharing options...

akasketch he/him Posted April 23, 2014 Report Share Posted April 23, 2014 (edited) I don't really buy the idea that a two-point circle would just be referencing the base and tip of a triangle, because every other circle is able to be expressed in terms of the nine points, just with some merging to be the same point, hence why I tried to show that it is actually possible for a two-point to be expressed as a nine-point with them all converged down to only two. It also better explains the requirement for the points to be end points of a diameter than the "just the tip (and base)" explanation. Also, you mentioned one-point circles? It's been a while since I read it, but are they possible? I don't remember. And I'm not sure what its complement triangle would be if it does. Just remembered the (0, 180, 0) angles comment. Yes, that works, and makes more sense than my theory. The one-point circle is the most basic form, and it probably uses a triangle with side lengths of 0, as well as (possibly) the starting and ending point of the circle. To clarify what has been stated here, there is a complement triangle for each type of line of warding. 1- and 2-point circles have been mentioned in the thread, one of them here, and all others have links to areas where one could find them, but I shall re-state for convenience's sake. 1-point: side lengths of 0; in other words, all nine reference points are in a single point in space. I realize that there is a rule that any two sides must be longer together than the third side to make a triangle, but the 2-point circle would not work with this rule applied either, so we shall disregard it for now. 2-point: Undetermined. Angle lengths of 0, 180, and 0, with edges on either side of the circle works best. The angle of 180 would have to pass through the center, I assume, though then you have a base with no part of the circle passing through its midpoint. 3: none possible. 4: An isosceles right triangle. 5: A non-isosceles right triangle. Why it is not mentioned in the book, I do not know, unless it was decided too impractical for use. 6: An equilateral triangle. 7: none possible/found. 8: An acute, isosceles triangle. See 5. 9: A scalene, acute triangle. Also: To remember Hoid's speech about noteriety But, one thing he did not mention was what happened if the original was forgotten. *sniff* Just giving you a hard time, my friend! Edited April 23, 2014 by akasketch 0 Quote Link to comment Share on other sites More sharing options...

Tarontos he/him Posted April 26, 2014 Author Report Share Posted April 26, 2014 For the one point circles, neat i never thought of that. and now to break some brains use a sphere and a tetrahedron. follow all the same rules but in the third dimension so lines becomes planes too. so have i broken your brains yet. any one have a 3D modeling software they could test that on As to why the 5 and 8 point circles are not mentioned in the book likely is do to them be advanced rithmantic theory, and at some point joel says that he has trouble getting rithmantic texts. also no hard feelings from either end 0 Quote Link to comment Share on other sites More sharing options...

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