Search the Community
Showing results for tags 'trig'.
Found 1 result
Sorry for the length in advance. When I read this book, the reasoning for the nine-pointer (described via annotated depiction from what I assume is Joel's notebook on page 243) kinda blew over me, and I just wanted to experience the story for once, rather than get caught up in the world's physics. That said, I still promised myself that I would check it out after I was done. I finished the novel, then recommended it to my little brother's friend, who promptly inhaled the darn thing, then asked, "Any other suggestions?" with a huge smile on his face. After this, I finally looked it up on Wikipedia (mostly due to his own interest in Rithmatics), and the results I found then and afterwards were extremely intriguing. First: The 9-point circle is an actual discovery made by Olry Terquem, and has some significance in the geometric world. Second: (From here on out is a thought process) The 9-point circle doesn't quite work for equilateral and right triangles; what do the look like; what are their Rithmatic equivalents? Third: Equilaterals would lend themselves to the six-point circle due to their nature of fusing together three different pairs of significant triangle points that would be fully represented in the 9-pointer (kind of shown in a picture on the top of pages 94-95; I just discovered this disproof of my originality in thought, as well as another in the history section of the aforementioned Wikipedia article). Fourth (fittingly): If both the 9- and 6-point circles can be represented as a relationship between the circle and a single triangle, what about the 4-point circle? The four points form an inscribed square when connected, and a square is essentially two equilateral right triangles stuck together at the Hypotenuse. On a whim, I drew this on a piece of graph paper: I noticed that all nine points were represented, and several at once in the peak (I had the hypotenuse on the bottom), and was then feeling nearly satisfied with my pursuit of Sanderson's use of Trig relations in his novel. Fifth: What about ellipses? The first thing that came to mind was Isosceles triangles, and thus I drew this on the same piece of graph paper: I'm pretty sure that the points at which the ellipse passes through the sides are their midpoints. Obviously, all nine significant points are NOT represented by said ellipse, but it does pass through at least two, probably four, of them. Sixth: This one is best described through simply showing a picture: I was messing around with isosceles triangles, so naturally, I wanted to see what their complement circle would look like point-wise, so I essentially drew up this diagram on my graph paper. That's it for the thought process, but I'm having trouble with a couple. For one, although Lines of Vigor are made from a sine or cosine graph, where did lines of revocation come from? And, I'm completely at a loss as to where the spiraling line comes from. Comments, further proofs, or disproofs? EDIT: sorry for the small pics.