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Super curious---I actually never took trig (complicated story), so I don't understand the trisecting an angle thing at all. What is it that makes measuring the

Super curious---I actually never took trig (complicated story), so I don't understand the trisecting an angle thing at all. What is it that makes measuring the angle and then dividing that measurement by 3 impossible?

pkd 7y ago
It's not the actual measurement, it's the physical act of dividing an angle graphically into thirds using nothing but a compass and a ruler.
kryptn 7y ago
While dividing by three is easy, dividing an angle by three with just a compass and a straight edge is impossible. Numberphile[1] did a video on this a few years ago [1] https://www.youtube.com/watch?v=6Lm9EHhbJAY https://www.youtube.com/watch?v=6Lm9EHhbJAY (Euclid's Big Problem - Numberphile)
recursive 7y ago
"Measuring" is not permitted. You are granted access only to an idealized compass and straight edge. There's a mobile game called Euclidea based on this which I'd recommend.
- paultopia 7y ago
  Oooh. I was imagining a compass like you're given in school, with labels on it, and ditto a ruler. :-)
  - SamReidHughes 7y ago
    That would be a protractor, which would be allowed if there were 340 degrees in a circle.
throwaway2048 7y ago
An important aspect of this isn't that you cant trisect any particular angle, its that you cant trisect any arbitrary angle, plenty of angles, like 90 degrees, are trivial to trisect.
- xyzzyz 7y ago
  Yes. The better way to phrase it would be that some angles are impossible to construct, and importantly that includes a trisection of some angles that can in fact be constructed.
  - secraetomani 7y ago
    Interesting. So like you have rational numbers and irrational numbers, you have trisectable angles and non-trisectable ones.
    - schoen 7y ago
      This analogy is rather precise—see this comment elsewhere in this thread. https://news.ycombinator.com/item?id=21209466 https://news.ycombinator.com/item?id=21209466
bikenaga 7y ago
The "compass" and "straightedge" are (nowadays) just a colorful/concrete way to describe the problem. When mathematicians consider this, they phrase the conditions in such a way that no physical "compasses" or "straightedges" are involved. For example, you might say: (1) If you have 4 different points P, Q, R, S so that the lines PQ and RS are not parallel, you may "acquire" their point of intersection; (2) If you have 4 different points P, Q, R, S and the circle with center P and radius PQ intersects the line RS then you may "acquire" the point(s) of intersection of the circle and the line; (3) If you have 4 different point P, Q, R, S and the circle with center P and radius PQ intersects the circle with center R and radius RS, you can "acquire" the intersection point(s) of the circles. ("Acquire" a point means roughly that your number system now contains the coordinates of the point.) Notice that this removes the problem of people doing unintended tricks with or drawing horribly complicated diagrams with physical compasses or straightedges, and makes mathematically precise what is allowed. The following is probably more than you want to know, even though I'll omit lots of the details. The idea is that circles are defined by quadratic equations, and lines by linear equations. At every step, you're solving (acquring the roots of) simultaneous linear or quadratic equations. Starting with the rational numbers, as you "acquire" new points, you "extend" the rationals to bigger fields "by extensions of degree 1 or 2". Since degrees multiply, at any point, you've "extended" the rationals by a total degree 2^n. So the result is: Theorem. Any point you construct must (a) lie in an extension of the rationals of degree 2^n; (b) be algebraic, in the sense that its coordinates are roots of rational polynomials. Now take something like trisecting a 60 degree angle. A third of 60 is 20, and to construct a 20 degree angle you need to construct t = cos 20 degrees. But from trig, t is a root of x^3 - 3 x - 1 (which is irreducible over the rationals), so t has degree 3 --- and 3 is not a power of 2. Hence, a 60 degree angle can't be trisected. Squaring the circle means given a circle, construct (the side of) a square with the same area. Take the circle to have radius 1, so its area is pi. The square you need would have side pi^(1/2), but pi (and pi^(1/2)) isn't algebraic in the sense noted above. (Lindemann showed pi is transcendental.) So you can't square an arbitrary circle. Duplicating the cube means given a cube (say with sides of length 1), construct (the side of) a cube whose volume is twice the volume of the original cube (so in this case, the new cube should have volume 2, and its sides should have length 2^(1/3). But 2^(1/3) is a root of the irreducible rational polynomial x^3 -2, so it has degree 3, and again, 3 is not a power of 2. What's remarkable about this is that, once you prove the theorem (which isn't that hard, just a little fussy), you can dispose of those three old contruction problems so easily. (Anyone who wants more details - I've left out a lot - can consult any book on Galois theory. Also, Nathan Jacobson's "Basic Algebra I" covers this [https://store.doverpublications.com/0486471896.html https://store.doverpublications.com/0486471896.html] --- it's a classic of abstract algebra which I like a lot, though a little old-fashioned.)
klyrs 7y ago
Tangential, but interesting: while it's impossible with a compass and ruler, one CAN trisect an angle with origami. https://plus.maths.org/content/trisecting-angle-origami https://plus.maths.org/content/trisecting-angle-origami

pkd 7y ago

It's not the actual measurement, it's the physical act of dividing an angle graphically into thirds using nothing but a compass and a ruler.

kryptn 7y ago

While dividing by three is easy, dividing an angle by three with just a compass and a straight edge is impossible. Numberphile[1] did a video on this a few years ago [1] https://www.youtube.com/watch?v=6Lm9EHhbJAY https://www.youtube.com/watch?v=6Lm9EHhbJAY (Euclid's Big Problem - Numberphile)

recursive 7y ago

"Measuring" is not permitted. You are granted access only to an idealized compass and straight edge. There's a mobile game called Euclidea based on this which I'd recommend.

paultopia 7y ago
Oooh. I was imagining a compass like you're given in school, with labels on it, and ditto a ruler. :-)
- SamReidHughes 7y ago
  That would be a protractor, which would be allowed if there were 340 degrees in a circle.

throwaway2048 7y ago

An important aspect of this isn't that you cant trisect any particular angle, its that you cant trisect any arbitrary angle, plenty of angles, like 90 degrees, are trivial to trisect.

xyzzyz 7y ago
Yes. The better way to phrase it would be that some angles are impossible to construct, and importantly that includes a trisection of some angles that can in fact be constructed.
- secraetomani 7y ago
  Interesting. So like you have rational numbers and irrational numbers, you have trisectable angles and non-trisectable ones.
  - schoen 7y ago
    This analogy is rather precise—see this comment elsewhere in this thread. https://news.ycombinator.com/item?id=21209466 https://news.ycombinator.com/item?id=21209466

bikenaga 7y ago

The "compass" and "straightedge" are (nowadays) just a colorful/concrete way to describe the problem. When mathematicians consider this, they phrase the conditions in such a way that no physical "compasses" or "straightedges" are involved. For example, you might say: (1) If you have 4 different points P, Q, R, S so that the lines PQ and RS are not parallel, you may "acquire" their point of intersection; (2) If you have 4 different points P, Q, R, S and the circle with center P and radius PQ intersects the line RS then you may "acquire" the point(s) of intersection of the circle and the line; (3) If you have 4 different point P, Q, R, S and the circle with center P and radius PQ intersects the circle with center R and radius RS, you can "acquire" the intersection point(s) of the circles. ("Acquire" a point means roughly that your number system now contains the coordinates of the point.) Notice that this removes the problem of people doing unintended tricks with or drawing horribly complicated diagrams with physical compasses or straightedges, and makes mathematically precise what is allowed. The following is probably more than you want to know, even though I'll omit lots of the details. The idea is that circles are defined by quadratic equations, and lines by linear equations. At every step, you're solving (acquring the roots of) simultaneous linear or quadratic equations. Starting with the rational numbers, as you "acquire" new points, you "extend" the rationals to bigger fields "by extensions of degree 1 or 2". Since degrees multiply, at any point, you've "extended" the rationals by a total degree 2^n. So the result is: Theorem. Any point you construct must (a) lie in an extension of the rationals of degree 2^n; (b) be algebraic, in the sense that its coordinates are roots of rational polynomials. Now take something like trisecting a 60 degree angle. A third of 60 is 20, and to construct a 20 degree angle you need to construct t = cos 20 degrees. But from trig, t is a root of x^3 - 3 x - 1 (which is irreducible over the rationals), so t has degree 3 --- and 3 is not a power of 2. Hence, a 60 degree angle can't be trisected. Squaring the circle means given a circle, construct (the side of) a square with the same area. Take the circle to have radius 1, so its area is pi. The square you need would have side pi^(1/2), but pi (and pi^(1/2)) isn't algebraic in the sense noted above. (Lindemann showed pi is transcendental.) So you can't square an arbitrary circle. Duplicating the cube means given a cube (say with sides of length 1), construct (the side of) a cube whose volume is twice the volume of the original cube (so in this case, the new cube should have volume 2, and its sides should have length 2^(1/3). But 2^(1/3) is a root of the irreducible rational polynomial x^3 -2, so it has degree 3, and again, 3 is not a power of 2. What's remarkable about this is that, once you prove the theorem (which isn't that hard, just a little fussy), you can dispose of those three old contruction problems so easily. (Anyone who wants more details - I've left out a lot - can consult any book on Galois theory. Also, Nathan Jacobson's "Basic Algebra I" covers this [https://store.doverpublications.com/0486471896.html https://store.doverpublications.com/0486471896.html] --- it's a classic of abstract algebra which I like a lot, though a little old-fashioned.)

klyrs 7y ago

Tangential, but interesting: while it's impossible with a compass and ruler, one CAN trisect an angle with origami. https://plus.maths.org/content/trisecting-angle-origami https://plus.maths.org/content/trisecting-angle-origami