Has anyone heard of the lost theorem and its relevance to trisection of an angle using classical method?
I have a 10 min video on it as well as a trisection method, although not using the lost theorem.....But I am looking for some feedback from anyone knowledgeable in this area.....
Thank you.... Jeremy
PS I know about Wantzel....this is briefly addressed.
Well, this guy was more honest than most! He constructs two arcs and a straight line, for which he can trisect (the arcs correspond to 180 and 90 degrees) and uses those three points to construct a circular arc. He then says that, since the circular arc trisects those three arcs it seems "reasonable" that it would trisect any arc between them, "would it not be a steady progression" but then ends up saying "I don't know".
No, that circular arc does NOT trisect any arc.
Hi, ....I take it you do agree they trisect the 180 and the 90 degree arc. Can you give an explanation why the arc of a circle center thusly would not trisect other arcs as postulated? I don't doubt it so much as I am just wondering. Also, can you at least give a guess as to what kind of arc between these points would perform this task. Or is there no arc? But this seems unlikely, I would think there would be some kind of progressive arc that would trisect the ever decreasing arc of the ever decreasing angle. Could it be the arc of a circle that is centered somewhere else than where I chose? And how might that center be determined?
This is the crux of the problem....there should be a steady progression from G to I to P and from H, J,and Q that trisects any arc from 180 to 0. Can we find the arc that defines that progression?
thanks for the apology, but I didn't take it personally, I just thought you made a mistake and would realize it upon reflection. No problem. Kind of you to write. (besides, I've gotten much worse from some who wouldn't even look at the construction)
And I realize there is not supposed to be a classical solution to trisecting an angle, but I think this construction offers an interesting mental exercise, as I describe in my last post reply to HallsofIvy. And I have updated the video with a caption as well, that examines this question. It is thus.....
Shouldn't some kind of curve (2 actually) define all the trisecting points as the angle transfers from 180 to 0 degrees on this template? What kind of curve?
Any thoughts?
Hi linelites, really nice to see your video. I was unaware of the problem so I fiddled some with it.
I took your idea and use the fact that a square can be divided in three equal parts, and Im sticking the square to the base of the bisected angle. Could you have a look at this diagram? what am I doing wrong or not understanding? since I have no problem trisecting any angle with this?
I see you are using the Brunnes Starcut....are you familiar with Malcolm Stewart's new book Patterns of Eternity? Really good.
I think you making the mistake of thinking that trisecting the chord of an angle will trisect the arc. Is this what you are thinking?
J
Hello Jeremy,
Thank you for pointing towards this lost theorem. I played a little with it and tried to place the triangle in a sine/cosine pattern. This gives some interesting insight on the relation of the theorem with the circle, like a right triangle inscribed in a circle gives insight in the Pythagorean theorem. I posted some drawings on Lost theorem about angular proportions.
Arjen Dijksman
Hi Arjen,
I am glad to see you develop this theorem. I am not anywhere near your level of expertise, but I'll try to follow your post. I have tried to reach Mr. Romain, but emails I send always come back.
I suppose you already thought about the special case of this triangle, involving the pentagram, with degrees 72, 36 and 72. I wonder what you might find there.
I must ask: did you look at my construction? I don't suspect it succeeds but I think it poses an interesting question....."there must be some kind of arc that includes points GIP (and HJQ) that defines all the trisecting points of angle arcs as they progress from 180 to 0 degrees on this template....what kind of arc would that be?"
Have you any idea, Arjen, if that arc could be defined, let alone classically constructed?
Jeremy
Hi Jeremy,
Yes, I thought about the pentagram (and any other regular polygon). The sine chord pattern then shows supplementary symmetries.
I also looked at your trisecting construction. I need some time to figure out what kind of curve defines all those trisecting points: is it a circular curve or is it something else? I'm not really an expert. Just an amateur with practice in circle drawings... :-) I'll post an answer about it, if I find something interesting.
Arjen
I guess the progressive arc that trisects the circle arc is a hyperbolic arc, because when you increase the angle above 180°, the progressive arc takes the direction of an asymptote near to 60°. But I would have to investigate it more in order to characterize that hyperbola.
Arjen
Thanks Arjen, I get the gist of that. I have a book on strait edge and compass construction of hyperbolic curves called Patterns in Space by Robert Stanley Beard, so maybe there's hope, (of course, such words are heresy, so I retract them
BTW, I don't know how valid the approach, but Romain also hints at a solution to the heptagon using the lost theorem, in a triangle with angles of x, 2x and 4x, totaling 7x, which thus gives a 1/7 of 180 degree angle.