I just came across an article about the impossibility of trisecting an angle using just straight edge and compass, but I guess I am not completely sure about the rules regarding the straight edge and compass. I understand that it is not allowed to lift the compass from the page, it can only be used to draw circles or arcs using one leg as a pivot. My question is, is it permissible to change which leg is the pivot such that if you first draw a straight line, then use the compass to draw a circle, starting with both legs on the line and finishing with both on the line--then without lifting the compass use the leg which transcribed the circle as the new pivot, transcribe another overlapping circle (and another and another, on and on) so that you end up with a series of overlapping circles? Or does this violate the rules?
straight edge and compass rules
- Thread starter nickgc
- Start date
Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,587
Why do you think that you can't move the compass? You can move the straight edge as many times as you like to draw straight lines and you can use the compass to draw as many circles as you want and can even change the diameter of the circles.
For the record you can trisect some angles, just not most. For example, since you can draw a 90 degree angle, you can trisect a 270 degree angle.
D
Deleted member 4993
Guest
You can even trisect a 90° angle.
Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,587
Yep. Just too hard to explain via the board.
Only if is rational
compass and ruler rules
I say you can't lift the compass merely because this is what I have read about the original rules for the ancient Greek constructions. I believe by not lifting the compass from the page, it prevents you from using it as a length measure, since the straightedge also is not allowed to have any markings on it. So I simply wondered whether switching between the two legs as the pivot leg was permissible. This would allow the construction of a series of overlapping circles with the center of one on the perimeter of the next. It seems to me this would allow for the trisection of virtually any angle although the # of degrees of the actual angle to be trisected would not necessarily be known.
I say you can't lift the compass merely because this is what I have read about the original rules for the ancient Greek constructions. I believe by not lifting the compass from the page, it prevents you from using it as a length measure, since the straightedge also is not allowed to have any markings on it. So I simply wondered whether switching between the two legs as the pivot leg was permissible. This would allow the construction of a series of overlapping circles with the center of one on the perimeter of the next. It seems to me this would allow for the trisection of virtually any angle although the # of degrees of the actual angle to be trisected would not necessarily be known.
I guess what I'm trying to describe is just a more complicated way of tiling a plane with equilateral triangles. Once you have done that, basically any angle defined by the intersection of any two lines of the tiling can be trisected or actually divided into any whole number of parts. You just need to measure how many line segments out from the origin to each of the points then going 3 or 4 or 5 times further. Sorry hard for me to explain but as an example: imagine 9 such triangles. The 60° angle at one corner can easily be trisected by drawing a line from the origin to the apex of the 3rd triangle out, along each direction. So you can now have a 20° angle. You could using this, go out 20 X further and be able to generate a series of 1° increments, if this makes any sense