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alias_sqbrDisclaimer: this is based on half remembered mathematics from fifteen years ago and some refresher reading of wikipedia. Kentsarrow asked me about this over a year ago, but I ran out of puff, then fred_mouse asked for a post on "maths" and I decided to finally finish the damn thing. I would double check everything but doing so would require forcing my brain to relearn things it would rather block out (I love maths but have some very bad associations with brain hurting) but here is a short less handwavey proof for anyone who knows what a field is.
Should be understandable with high school calculus, I think? But if you don't get it, blame me not you :) One thing that may go over the head of those who haven't studied maths at uni is the DEEP RIVALRY between algebraists and geometers that made me super gleeful when I learned about this stuff (I have always sucked at geometry)
So! The ancient Greeks elevated geometric construction to an art form, figuring out how to create various shapes and perfectly divide angles into various ratios using only very basic tools. They abstracted these constructions by imagining a perfectly flat plane and two perfectly accurate tools: a straightedge for drawing the line passing between two points (but not measuring the distance) and a compass for drawing perfect circles of a given radius. They also came up with some rules for how such a plane would work, such as the fact that there is a line between any two points.
In around 300BC the Greek mathematician Euclid compiled all this information (as well as some number theory and other stuff) into one book, "Elements". Euclidean geometry is beautifully elegant, boiling all of geometry down into five simple rules (or axioms), and then showing how all the other known geometric facts and techniques can be extrapolated from these rules:
1: It is possible to draw a straight line between any two points
2: Any straight line can be extended off to infinity in either direction
3: It is possible to draw a circle of any given radius
4: Any two right angles are identical
5: Any two lines which are not parallel will eventually meet
(Einstein later proved that the last rule isn't actually always true in the real world, since gravity causes space and lines to bend, but we pure mathematicians tend not to care about such petty details)
Here's the list of basic things you can do with a straight edge and compass. Combining these with Euclid's rules, you can construct all sorts of proofs and shapes.
But there were some questions Euclid couldn't answer. For example: is it possible to cut an angle in thirds using just a straight edge and compass? And, given a circle of a particular radius, how would you construct a square with the same area?
These questions puzzled mathematicians, amateur and professional, for thousands of years, and there were many crackpot claims to have solved them. Noone came up with an answer until the late 18th and 19th century, when various mathematicians (Gauss, Galois, and Hilbert) turned these geometric problems into albebraic ones and finally solved them using a form of mathematics called Galois theory.
The solution: instead of thinking about points and lines, think of everything in terms of points. So, for example, here is the traditional way of thinking of the angle trisection problem:
1) Given lines PA and PB create a line a third line PC using ruler and compass construction such that the angle APC is a third of the angle APB.
But disproving (1) is equivalent to disproving:
1a) Given points P, A, and B find any point C using ruler and compass construction such that the angle APC is a third of the angle APB.
And then instead of thinking of the points as floating in some abstract geometric plane, think of them as specific complex numbers in the complex plane:
1b) Given P=0+0i, A=1+0i, and any B=cos(theta)+i*sin(theta) construct a point C=cos(theta/3)+i*sin(theta/3).
If we can find any complex number B=cos(theta)+i*sin(theta) where it is impossible to construct a corresponding C=cos(theta/3)+i*sin(theta/3) then we have proven that it is impossible to trisect arbitrary angles.
So how do we prove the impossibility of constructing a point?
To answer these types of questions we need to be able to express ruler and compass constructions as algebra. It all gets pretty complex, but here's a simple example:
Geometric construction of a line (using a ruler):
Given points P and A we can find all the points on the straight line PA between P and A.
Algebraic equivalent on the complex plane:
If we start with the set {p+qi,a+bi} we can find all the points in the set {p+qi + k(p+qi-a-bi): 0< k <1}
It's pretty easy to prove whether any given point on the complex plane is contained in this set. For example, if we set P=0+0i and A=1+i then the set of points on the line between them is {0+0i + k(0+0i-1-i): 0< k <1}={k+ki: 0< k <1}. So 0.5+0.5i is on the line with k=0.5 but 1+5i isn't, since otherwise we'd have 1=5=k.
So! The problem now is this:
Let S1 be our initial set of points (eg in equation (1b) S1={0, 1,cos(theta)+i*sin(theta)})
Let S2 be the set of points we need to construct (eg in equation (1b) S2={cos(theta/3)+i*sin(theta/3)})
Let R(S1) be the set of points obtained by applying any combination of ruler and compass constructions to S1.
Is S2 contained in R(S1)? If it is, then the construction is possible. If it isn't, then the construction is impossible.
Unfortunately, Galois theory doesn't tell us R(S1). But it does tell us a nice, easy to work with set G(S1) which contains R(S1). It's pretty easy to test if S2 is a subset of G(S1). If S2 isn't a subset of G(S1) it definitely isn't a subset of R(S1), so we've still proven that the construction is impossible.
AND...if we set B=cos(60)+sin(60) and C=cos(20)+i*sin(20) (in degrees) we can prove that {cos(20)+i*sin(20)} is NOT contained in G({0, 1,cos(theta)+i*sin(theta)}). So it is impossible to trisect 60 degrees, and thus impossible to trisect an arbitrary angle using just a ruler and compass.
We can similarly show that it is impossible to construct a circle with the same area as a given square (the points version is something like: given points A=0+0i and B=1+0i, construct a third point C=1/sqrt(pi)+0i, so that the circle through C centered at A has the same area, 1, as a square with one side AB)
And yet people STILL claim to be able to square the circle. But they are WRONG because of ALGEBRA. *waves little algebraist flag*
Should be understandable with high school calculus, I think? But if you don't get it, blame me not you :) One thing that may go over the head of those who haven't studied maths at uni is the DEEP RIVALRY between algebraists and geometers that made me super gleeful when I learned about this stuff (I have always sucked at geometry)
Ruler and compass constructions
So! The ancient Greeks elevated geometric construction to an art form, figuring out how to create various shapes and perfectly divide angles into various ratios using only very basic tools. They abstracted these constructions by imagining a perfectly flat plane and two perfectly accurate tools: a straightedge for drawing the line passing between two points (but not measuring the distance) and a compass for drawing perfect circles of a given radius. They also came up with some rules for how such a plane would work, such as the fact that there is a line between any two points.
In around 300BC the Greek mathematician Euclid compiled all this information (as well as some number theory and other stuff) into one book, "Elements". Euclidean geometry is beautifully elegant, boiling all of geometry down into five simple rules (or axioms), and then showing how all the other known geometric facts and techniques can be extrapolated from these rules:
1: It is possible to draw a straight line between any two points
2: Any straight line can be extended off to infinity in either direction
3: It is possible to draw a circle of any given radius
4: Any two right angles are identical
5: Any two lines which are not parallel will eventually meet
(Einstein later proved that the last rule isn't actually always true in the real world, since gravity causes space and lines to bend, but we pure mathematicians tend not to care about such petty details)
Here's the list of basic things you can do with a straight edge and compass. Combining these with Euclid's rules, you can construct all sorts of proofs and shapes.
But there were some questions Euclid couldn't answer. For example: is it possible to cut an angle in thirds using just a straight edge and compass? And, given a circle of a particular radius, how would you construct a square with the same area?
These questions puzzled mathematicians, amateur and professional, for thousands of years, and there were many crackpot claims to have solved them. Noone came up with an answer until the late 18th and 19th century, when various mathematicians (Gauss, Galois, and Hilbert) turned these geometric problems into albebraic ones and finally solved them using a form of mathematics called Galois theory.
Think Points!
The solution: instead of thinking about points and lines, think of everything in terms of points. So, for example, here is the traditional way of thinking of the angle trisection problem:
1) Given lines PA and PB create a line a third line PC using ruler and compass construction such that the angle APC is a third of the angle APB.
But disproving (1) is equivalent to disproving:
1a) Given points P, A, and B find any point C using ruler and compass construction such that the angle APC is a third of the angle APB.
And then instead of thinking of the points as floating in some abstract geometric plane, think of them as specific complex numbers in the complex plane:
1b) Given P=0+0i, A=1+0i, and any B=cos(theta)+i*sin(theta) construct a point C=cos(theta/3)+i*sin(theta/3).
If we can find any complex number B=cos(theta)+i*sin(theta) where it is impossible to construct a corresponding C=cos(theta/3)+i*sin(theta/3) then we have proven that it is impossible to trisect arbitrary angles.
So how do we prove the impossibility of constructing a point?
Turning rulers and compasses into algebra
To answer these types of questions we need to be able to express ruler and compass constructions as algebra. It all gets pretty complex, but here's a simple example:
Geometric construction of a line (using a ruler):
Given points P and A we can find all the points on the straight line PA between P and A.
Algebraic equivalent on the complex plane:
If we start with the set {p+qi,a+bi} we can find all the points in the set {p+qi + k(p+qi-a-bi): 0< k <1}
It's pretty easy to prove whether any given point on the complex plane is contained in this set. For example, if we set P=0+0i and A=1+i then the set of points on the line between them is {0+0i + k(0+0i-1-i): 0< k <1}={k+ki: 0< k <1}. So 0.5+0.5i is on the line with k=0.5 but 1+5i isn't, since otherwise we'd have 1=5=k.
Putting it all together
So! The problem now is this:
Let S1 be our initial set of points (eg in equation (1b) S1={0, 1,cos(theta)+i*sin(theta)})
Let S2 be the set of points we need to construct (eg in equation (1b) S2={cos(theta/3)+i*sin(theta/3)})
Let R(S1) be the set of points obtained by applying any combination of ruler and compass constructions to S1.
Is S2 contained in R(S1)? If it is, then the construction is possible. If it isn't, then the construction is impossible.
Unfortunately, Galois theory doesn't tell us R(S1). But it does tell us a nice, easy to work with set G(S1) which contains R(S1). It's pretty easy to test if S2 is a subset of G(S1). If S2 isn't a subset of G(S1) it definitely isn't a subset of R(S1), so we've still proven that the construction is impossible.
AND...if we set B=cos(60)+sin(60) and C=cos(20)+i*sin(20) (in degrees) we can prove that {cos(20)+i*sin(20)} is NOT contained in G({0, 1,cos(theta)+i*sin(theta)}). So it is impossible to trisect 60 degrees, and thus impossible to trisect an arbitrary angle using just a ruler and compass.
We can similarly show that it is impossible to construct a circle with the same area as a given square (the points version is something like: given points A=0+0i and B=1+0i, construct a third point C=1/sqrt(pi)+0i, so that the circle through C centered at A has the same area, 1, as a square with one side AB)
And yet people STILL claim to be able to square the circle. But they are WRONG because of ALGEBRA. *waves little algebraist flag*
no subject
Date: 2014-01-08 10:12 pm (UTC)no subject
Date: 2014-01-10 02:10 am (UTC)Hee, thanks :)
no subject
Date: 2014-01-09 06:41 am (UTC)no subject
Date: 2014-01-10 02:08 am (UTC)Haha, well, that may just be a sign I did it wrong :)
no subject
Date: 2014-01-13 06:05 pm (UTC)no subject
Date: 2014-01-14 12:34 pm (UTC)Yes! It's like an isomorphism of ideas :)
no subject
Date: 2014-01-14 04:19 pm (UTC)