phillip1882 Posted February 15, 2012 Report Share Posted February 15, 2012 okay i have to admit i am impressed by joe's second diagram.it looks ligitamate, i can't find a ready flaw.the contruction steps are this.start with angle A.extend the base, and contruct a line perpendicular to it at point A.constuct arc C. where arc C meets the angle line, label B.divide line AC into five parts. mark the 1/5 point from C, D. also mark the 3/5 point, E.contruct a perpendicular line at point E.where it meets arc C, construct a perpendicular line of length AD. call this line FG. From point G, construct a line that goes through points B and D, and hits the extended base at H.Angle AHB = 1/3 *angle A. (note: line DH = 2*AC, which is provable.) Quote Link to comment Share on other sites More sharing options...
phillip1882 Posted February 15, 2012 Report Share Posted February 15, 2012 okay i'm havign holusinations, this obviously has SOME flaw, for example, if the angle A is less than 75 degrees.ingore my above post. Quote Link to comment Share on other sites More sharing options...
Turtle Posted February 15, 2012 Report Share Posted February 15, 2012 ...the contruction steps are this.start with angle A.extend the base, and contruct a line perpendicular to it at point A.constuct arc C. where arc C meets the angle line, label B.divide line AC into five parts.... how do you accomplish the five-part division? as to the trisecting device, even though it has a straightedge component, the fixed vertices effectively make it a ruler so it fails to meet the criteria of trisection by straightedge and compass alone. Sounds as trivial as bisecting an angle, which you probably learned in grammar school, but apparently it is impossible. Greek mathematicians started trying thousands of years ago and Pierre Wantzel finally proved it can not be done in 1837. Yet the battle continues!... some battle... apparently. Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted February 16, 2012 Author Report Share Posted February 16, 2012 Regarding the comments on Diagram-3 at < http://users.tpg.com.au/musodata/trisection/trisecting_any_angle.htm >:Please note it is not claiming to be a trisection of a 'given' angle but trisects any 'chance' constructed angle.Joe. Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted February 16, 2012 Author Report Share Posted February 16, 2012 ...... as to the trisecting device, even though it has a straightedge component, the fixed vertices effectively make it a ruler so it fails to meet the criteria of trisection by straightedge and compass alone...... You are correct in that the physical device at ZOL but 'represents' the draftsmanship of Diagram-1, just as Diagram-1 draftsmanship but 'represents' the elementary geometry of any 'chance' trisection. Quote Link to comment Share on other sites More sharing options...
Turtle Posted February 16, 2012 Report Share Posted February 16, 2012 You are correct in that the physical device at ZOL but 'represents' the draftsmanship of Diagram-1, just as Diagram-1 draftsmanship but 'represents' the elementary geometry of any 'chance' trisection. :lol: that's one tough noggin ya got jay. 'represents' does not trump 'is'. your further mincing of terms when you say 'chance' rather than 'given' does not get you over the hurdle either. 'given' is short for 'given arbitrary angle' which means exactly 'by chance' or choose any angle you want. once you choose it, you give it as in drawing it. Quote Link to comment Share on other sites More sharing options...
phision Posted February 16, 2012 Report Share Posted February 16, 2012 okay i have to admit i am impressed by joe's second diagram.it looks ligitamate, i can't find a ready flaw.the contruction steps are this.start with angle A.extend the base, and contruct a line perpendicular to it at point A.constuct arc C. where arc C meets the angle line, label B.divide line AC into five parts. mark the 1/5 point from C, D. also mark the 3/5 point, E.contruct a perpendicular line at point E.where it meets arc C, construct a perpendicular line of length AD. call this line FG. From point G, construct a line that goes through points B and D, and hits the extended base at H.Angle AHB = 1/3 *angle A. (note: line DH = 2*AC, which is provable.) Why bother with points E, F and G or the associated lines? :unsure: If you draw the line through points B and D, you will still find point H on the baseline! It can seen more easily from this simplified drawing that, for a slightly more acute angle where, point B was the same distance from the baseline as point D, then the line through these points would never cross the baseline ... and for an even more acute angle point H would be at the other side of A. This is not a general solution! :Bump2: Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted February 16, 2012 Author Report Share Posted February 16, 2012 :lol: that's one tough noggin ya got jay. 'represents' does not trump 'is'. your further mincing of terms when you say 'chance' rather than 'given' does not get you over the hurdle either. 'given' is short for 'given arbitrary angle' which means exactly 'by chance' or choose any angle you want. once you choose it, you give it as in drawing it. You are correct: In Diagram-3 the point H will always determine the size of the angle, but one will not know its size until point P is located, which is determined by knowing the size of the line segment M-N. The link does not claim that the diagrams are trisecting a 'given arbitrary angle' as all the diagram angles are truly 'chance' constructions, but happen to be within automatically trisecting models. Somewhere on the line AB in Diagram-3 there is a point H which mathematically and geometrically exactly corresponds to the point to construct 60 degrees and exactly trisect it within the automatically trisecting model. My terminology is merely directed at making the true distinction between 'draftsmanship' and 'geometry' which are not the same thing. The mathematicians on this site already know this. Joe. Quote Link to comment Share on other sites More sharing options...
Qfwfq Posted February 16, 2012 Report Share Posted February 16, 2012 Joe, to put it briefly you are merely confusing the issue. You have not given a solution which meets the requisites of the ancient problem that was proven to have no solutions and there is no novelty that constructions of some kind exist wherin there are angles with a 3 to 1 ratio. Do you get what I meant when I talked about successive approximations vs. a finite step SE&C construction? Turtle 1 Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted February 17, 2012 Author Report Share Posted February 17, 2012 (edited) Joe, to put it briefly you are merely confusing the issue. You have not given a solution which meets the requisites of the ancient problem that was proven to have no solutions and there is no novelty that constructions of some kind exist wherin there are angles with a 3 to 1 ratio. Do you get what I meant when I talked about successive approximations vs. a finite step SE&C construction? I am not confused and to me the trisection problem is no more than a curious puzzle. I have no interest whatever in the software based endless approximations, as they are not USE&C constructions. My original intention with < http://users.tpg.com.au/musodata/trisection/trisecting_any_angle.htmt > was feedback on the possibility of locating a 'given acute angle' within an automatic trisecting model such as Diagrams 1 and 3. Edited February 18, 2012 by Jayeskay Quote Link to comment Share on other sites More sharing options...
Qfwfq Posted February 17, 2012 Report Share Posted February 17, 2012 (edited) No Jay, I wasn't talking about software. I was talking about methods. There are also mechanical methods that pretty much boil down to the same thing as successive approximations with SE&C and an image you posted seems to be of this ilk. EDIT: sorry, it wasn't you that posted it but still, all I can see that you have done is constructions where an initial angle is tripled. Edited February 17, 2012 by Qfwfq ooops! Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted February 18, 2012 Author Report Share Posted February 18, 2012 (edited) No Jay, I wasn't talking about software. I was talking about methods. There are also mechanical methods that pretty much boil down to the same thing as successive approximations with SE&C and an image you posted seems to be of this ilk. EDIT: sorry, it wasn't you that posted it but still, all I can see that you have done is constructions where an initial angle is tripled. If you look at Diagram-1: You can see that the trisecting angle DCI is not required to construct angle HIJ. If you look at Diagram-2: There is no initial angle to triple. Edited February 18, 2012 by Jayeskay Quote Link to comment Share on other sites More sharing options...
Jayeskay Posted March 6, 2013 Author Report Share Posted March 6, 2013 (edited) The site : http://users.tpg.com.au/musodata/trisection/trisecting_any_angle.htm has been updated. Jayeskay. ................................... The update was incorrect, and has been removed. Jayeskay. Edited March 8, 2013 by Jayeskay Quote Link to comment Share on other sites More sharing options...
Dane Posted April 7, 2013 Report Share Posted April 7, 2013 Quote Link to comment Share on other sites More sharing options...
CraigD Posted April 8, 2013 Report Share Posted April 8, 2013 Welcome to hypography, galet! :) Please feel free to start a topic in the introductions forum to tell us something about yourself. The trisection you illustrate is usually called “trisection using a carpenter’s square”. I believe it has been known for centuries, but the earliest publication of a description of it I was able to find online was 1928, in Scudder, H. T. "How to Trisect and Angle with a Carpenter's Square." Amer. Math. Monthly. There are many online descriptions of it – this is nice one, with several other methods of trisecting angles other than with a compass and straightedge Such trisection methods don’t contradict the Wantzel’s famous 1837 proof of the impossibility of trisecting an angle with a straightedge and compass, nor the several easier ones that have been done since then. It’s impossible to trisect an arbitrary angle using a straightedge and compass, but not using other drawing tools. Quote Link to comment Share on other sites More sharing options...
Dane Posted April 8, 2013 Report Share Posted April 8, 2013 Thanks for the detailed response. I knew that the trisection of an angle impossible with ruler and compass - it should not even try, but I think that my"T" divider elegant tool for practical use. With a few of these tools is possible n - section. Quote Link to comment Share on other sites More sharing options...
Rade Posted April 9, 2013 Report Share Posted April 9, 2013 See here for paper claiming arbitrary acute angle trisection using ruler and compass: http://larrysimpson.zymichost.com/aaat.pdf Quote Link to comment Share on other sites More sharing options...
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