Recently I have been studying Lie algebra and group theory. I revisited the possible straight edge and compass solution to the ancient problem of trisecting an angle in light of what I learned.
Possible Trisecting solution on this blog
As a result I realized that I had made a mistake in the characterization of this method as being a solution, when it is a near approximation. So I was wrong and this is why:
I thought of a method which perfectly divides the chord, but does not divide the arc exactly, of course. I say of course now, but that is the way with learning new things, they seem so obvious once they are understood and are unknown before that.
It seems that it ever so slightly "might" be possible to trisect the angle , but it would also imply a solution to another problem in higher order equations and wouldn't be a quick "cool" solution so it gets put on the back burner until some new associated information comes along.
I can't help but think about things and though it is not a mathematical trisection such that it is "perfect", I do not see that it could ever be "perfect" with a pen or pencil and straight edge. It is possible to get the actual third within +- the distance of a line width with the defined tools and how can it be any better than that. It is a real world problem and not a matter of the cos(theta). So it becomes a pedantic objection when a mechanical procedure of limited duration does produce the result within the limits of measurement. I have been working with my atomic force microscope and considered using this technique with the probe and my 3D positioning mechanism and it would still not be exact as it would be +- .1 Angstrom. It would seem that trisection with a compass and straight edge is an obscure mathematical oddity that has long since become pointless or obsolete in the face of quantum mechanics.
It does however make me wonder about a root method for high powers that was bouncing around my head and if it comes to something I will post the results here.
Also I spent some more time with thirds and it can be done at least 3 ways, but one isn't a mathematical solution, just physical, like Archimedes' principal.
Galois and group theory at UNSW Wildberger
These showed up last year and I hadn't watched them yet, but UNSW has some great teachers that are informative and entertaining. That is in Australia. There are more that deal with more complex aspects, but this seems to be a good introduction "if" you have already completed and understood linear algebra. Strang at MIT is not as entertaining and sometimes labors points, but knows his stuff. Reminds me of Leonhard Euler in appearance.
It is interesting and has a beauty in it that I suppose arises from symmetry and transform such that it might be the intellectual equivalent of music.
(Very interesting stuff)
MORE and MORE:
So I was thinking about this and based on the theory from Euler and others, it is intuitive that this is so. I don't want to be a Galois , but this is how I think. It was said that he just leaped and it was hard to follow and I can't tell you literally why this is so, it just seemed it would be so based on how I perceived this to work. People who know what this is will know that it has everything to do with the internet and much more.
The proof is too long to fit in the margin (chuckle). :)
Don't tell me it is sloppy code as I know that, but this is just to see if it is true what I think.
from fractions import gcd import math def is_prime(n): if n % 2 == 0 and n > 2: return False for i in range(3, int(math.sqrt(n)) + 1, 2): if n % i == 0: return False return True for x in range(2,20): for y in range(2,20): if gcd(x,y)==1: if is_prime(y): z=(x**(y-1))%y if z!=1: print 'Failed 1x',x,y,x**(y-1),z z=(x**(2*y-2))%y if z!=1: print 'Failed 2x',x,y,x**(y-1),z #else: #print 'Success at 2x',x,y,x**(2*y-2),zNo matter what I do, blogger screws up the python indent and I don't have time to figure out why.
So here is a PIC
KmPlot is really handy for dealing with functions and in this case it is v=sqrt(2ad) with first and second derivative.
There are new graduate level thermodynamics lectures from MIT that are up now from 2014.