1:23:00 conic sections can definitely get you to cubic numbers, as there was work done on that during medieval times by middle eastern and european mathematicians. As an example, the cube root of 2 can be made from the x-coordinate of the intersection of the parabola y = x^2 with the hyperbola xy = 2. Both curves are conic sections. It can also be shown that any algebraic number can be constructed using the intersection of a sufficient number of conic sections in a sufficient number of dimensions.
The puzzle occurs at 42:37. Can you solve it? I offer a proof that every triangle is isosceles (due originally to W. W. Rouse Ball). Can you spot the error? If so, commet here. If you've seen the puzzle before, kindly make only elliptical comments, so that others might enjoy figuring it out for themselves.
There is no guarantee, for a general triangle, that the first angle bisector in the picture and the perpendicular from the opposite edge intersect inside the triangle. In the given drawing, imagine fixing the top left angle and the top right side, then extend the left edge until the midpoint of the bottom right side moves past the angle bisector. Then the intersection is outside the triangle. Now you can continue with all the other steps, but the conclusion of being isosceles no longer applies to the original triangle. If you draw this skinny triangle, you will see that one side ends up being "II+V", and the other "II-V" using the naming in the lecture. Thank you for the wonderful series, Prof. Hamkins.
The first perpendicular bisector doesn't intersect the angle bisector inside the triangle. If it's an actual isosceles triangle, the intersection point is where the angle bisector meets the opposite side. Otherwise, the intersection point lies outside of the triangle.
The intersection of the angle bisector with the opposite side is to the left of the middle point (one can prove this for a certain relationship between the other angles using the law of sines), so the construction can't look as you draw it.
Just posting to nod to my man Poincare independently developing the modern transformation invariance viewpoint along with Lie and Klein (and shamelessly plugging my paper): "There is a clear affinity between Poincar ́e’s view and that of Klein and Lie, ex-pressed in the Erlangen program. Indeed, there was interaction between Poincar ́eand Lie in 1882 when Lie was in Paris. Lie wrote to Klein that Poincar ́e held theconcept of a group to be the fundamental concept for all of mathematics (Hawkins,2000, p. 182). Jeremy Gray notes that it is very likely that Poincar ́e’s use of groupsin his analysis of Fuchsian functions was independent of Klein’s Erlangen program(Gray, 2005, p. 551). It is not clear whether Klein or Lie ever indicated anything likePoincar ́e’s philosophical view that the group is prior in conception to the terms oc-curring in the axioms of geometry. Poincar ́e’s position is that group theory definesthose terms, that the objects to which they refer are constituted by invariant sub-groups, but an alternative view would be that the objects of geometry are presentedor constructed independently while groups afford a complete means of classificationwithout necessarily constituting the objects that comprise the spaces thus classified.This view is consistent with a mathematical interest in the use of groups to classifygeometries, but inconsistent with Poincar ́e’s fully developed philosophical position." philpapers.org/rec/SHIPOT-6
Actual straightedges and compass have a bounded finite size, and one might wonder whether such less-than-ideal tools are as powerful in construction as the idealized counterparts. I asked a question about this on MathOverflow at mathoverflow.net/q/365411/1946, and you can find an answer there.
At 18:25 you mentioned you have posted the question to MathOverflow. But you actually posted it on MathStackExchange. Here is the link of that question: math.stackexchange.com/questions/3377988/what-is-the-next-number-on-the-constructibility-sequence-and-what-is-the-asympt
1:23:00 conic sections can definitely get you to cubic numbers, as there was work done on that during medieval times by middle eastern and european mathematicians.
As an example, the cube root of 2 can be made from the x-coordinate of the intersection of the parabola y = x^2 with the hyperbola xy = 2. Both curves are conic sections. It can also be shown that any algebraic number can be constructed using the intersection of a sufficient number of conic sections in a sufficient number of dimensions.
Bit of an α comment there, nice.
I'm surprised I'd never heard of the Tarski axioms before, what an amazing result!
The puzzle occurs at 42:37. Can you solve it? I offer a proof that every triangle is isosceles (due originally to W. W. Rouse Ball). Can you spot the error? If so, commet here. If you've seen the puzzle before, kindly make only elliptical comments, so that others might enjoy figuring it out for themselves.
There is no guarantee, for a general triangle, that the first angle bisector in the picture and the perpendicular from the opposite edge intersect inside the triangle. In the given drawing, imagine fixing the top left angle and the top right side, then extend the left edge until the midpoint of the bottom right side moves past the angle bisector. Then the intersection is outside the triangle. Now you can continue with all the other steps, but the conclusion of being isosceles no longer applies to the original triangle. If you draw this skinny triangle, you will see that one side ends up being "II+V", and the other "II-V" using the naming in the lecture. Thank you for the wonderful series, Prof. Hamkins.
The first perpendicular bisector doesn't intersect the angle bisector inside the triangle. If it's an actual isosceles triangle, the intersection point is where the angle bisector meets the opposite side. Otherwise, the intersection point lies outside of the triangle.
The intersection of the angle bisector with the opposite side is to the left of the middle point (one can prove this for a certain relationship between the other angles using the law of sines), so the construction can't look as you draw it.
The angle bisector and perpendicular bisector are agoraphobic, so they stayed inside for Rouse Ball's lunch.
Just posting to nod to my man Poincare independently developing the modern transformation invariance viewpoint along with Lie and Klein (and shamelessly plugging my paper):
"There is a clear affinity between Poincar ́e’s view and that of Klein and Lie, ex-pressed in the Erlangen program. Indeed, there was interaction between Poincar ́eand Lie in 1882 when Lie was in Paris. Lie wrote to Klein that Poincar ́e held theconcept of a group to be the fundamental concept for all of mathematics (Hawkins,2000, p. 182). Jeremy Gray notes that it is very likely that Poincar ́e’s use of groupsin his analysis of Fuchsian functions was independent of Klein’s Erlangen program(Gray, 2005, p. 551). It is not clear whether Klein or Lie ever indicated anything likePoincar ́e’s philosophical view that the group is prior in conception to the terms oc-curring in the axioms of geometry. Poincar ́e’s position is that group theory definesthose terms, that the objects to which they refer are constituted by invariant sub-groups, but an alternative view would be that the objects of geometry are presentedor constructed independently while groups afford a complete means of classificationwithout necessarily constituting the objects that comprise the spaces thus classified.This view is consistent with a mathematical interest in the use of groups to classifygeometries, but inconsistent with Poincar ́e’s fully developed philosophical position."
philpapers.org/rec/SHIPOT-6
Thanks very much for this.
Actual straightedges and compass have a bounded finite size, and one might wonder whether such less-than-ideal tools are as powerful in construction as the idealized counterparts. I asked a question about this on MathOverflow at mathoverflow.net/q/365411/1946, and you can find an answer there.
56:30 so you're saying the earth is flat.
Oh dear, I was hoping to keep my views secret on this matter. :-)
At 18:25 you mentioned you have posted the question to MathOverflow. But you actually posted it on MathStackExchange. Here is the link of that question: math.stackexchange.com/questions/3377988/what-is-the-next-number-on-the-constructibility-sequence-and-what-is-the-asympt
Yes, thank you for this link. Here is the link to the corresponding entry at the Online Encyclopedia of Integer Sequences: oeis.org/A333944.