johni92

Update for White Sand as a series of graphic novels, starting in 2015.

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.

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.

It seems like five-point and 8-point circles should be viable. The two-point is the most difficult to explain in terms of a nine-point circle. If you look at the limit as the vertex angle of an isosceles triangle approaches 180 degrees, you end up with a circle of infinite radius, with 5 points converging to lie on a straight line. This can't exist on a circle, but if you also take the limit as the side lengths approach 0, they converge to a single point. The other three points lie an infinite distance away, along three parallel lines, which, again, can converge to a single point for zero side lengths. I suppose this can be seen as the two-point circle, but it is somewhat more abstract than the others. Note: taking the limit as the base angles of an isosceles triangle approach 90 degrees and as the length of the base approaches 0 and the lengths of the other two sides approach infinity gives an identical result.