2000 years unsolved: Why is doubling cubes and squaring circles impossible?

Today's video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles?

00:00 Intro
05:19 Level 1: Euclid
08:57 Level 2: Descartes
16:44 Level 3: Wantzel
24:00 Level 4: More Wantzel
31:30 Level 5: Gauss
35:18 Level 6: Lindemann
40:22 Level 7: Galois

Towards the end of a pure maths degree students often have to survive a "boss" course on Galois theory and somewhere in this course they are presented with proofs that it is actually not possible to accomplish any of those four troublesome tasks. These proofs are easy consequences of the very general tools that are developed in Galois theory. However, taken in isolation, it is actually possible to present proofs that don't require much apart from a certain familiarity with simple proofs by contradiction of the type used to show that numbers like root 2 are irrational.

I've been meaning to publish a nice exposition of these "simple" proofs ever since my own Galois theory days (a long, long time ago.
Finally, today is the day :)

For some more background reading I recommend:
1. chapter 3 of the book "What is mathematics?" by Courant and Robbins (in general this is a great book and a must read for anybody interested in beautiful maths).
2. The textbook "Field theory and its classical problems" by Hadlock (everything I talk about and much more, but you need a fairly strong background in maths for this one).

Here is a great two-page summary by the mathematician Drew Armstrong of what is going on in this video

(there is a little typo towards the bottom of the page. It should be 8 cos^3 theta + 4 cos^2 theta - (!) 4 cos theta -1 = 0. Replace cos theta by x and you get the cubic equation I mention in the video. )

Enjoy :)

Burkard

[akap]

When I want to double a cube, I just pull out my complex ruler and e^i(pi) compass and get to work. Easy.

[Mathologer]

Finally, a crazily busy semester here in Australia is almost over. Just some end-of-semester exams left to finalise next week, a short trip to Japan and then I should have a bit more time for making Mathologer videos for the rest of the year. Can’t wait.


Anyway, today's video is about the resolution of four problems that remained open for over 2000 years from when they were first puzzled over in ancient Greece: Is it possible, just using an ideal mathematical ruler and an ideal mathematical compass, to double cubes, trisect angles, construct regular heptagons, or to square circles?


Towards the end of a pure maths degree students often have to survive a "boss" course on Galois theory and somewhere in this course they are presented with proofs that it is actually not possible to accomplish any of those four troublesome tasks. These proofs are easy consequences of the very general tools that are developed in Galois theory. However, taken in isolation, it is actually possible to present very accessible proofs that don't require much apart from a certain familiarity with simple proofs by contradiction of the type used to show that numbers like root 2 are irrational.


I've been meaning to publish a nice exposition of these "simple" proofs ever since my own Galois theory days a long, long time ago. Finally, today is the day :)


If you make it to the end of this 40 minute video please leave a comment below and let me know how well this explanation worked for you.

[bobfenz]

I was a physics student at the University of Arizona in the 1970s and stumbled on a retired gentleman (my age now) at a Swensons coffee shop who was prodigiously working on geometric constructions with a ruler and compass. He explained to me what he was doing, squaring a circle, trisecting angles. I looked at his work and was amazed at the number of books he had compiled and his efforts. We became friends over coffee and discussed his progress and life in general. He had many words of wisdom for a young man in very turbulent times and he was also certain he could solve these problems with a ruler and compass.
Then suddenly, he disappeared one day never to return to the coffee shop. Later I was told by a friend that he returned to Long Island and passed away. Another traveler in time trying to tackle difficult problems. His name was Bruce Stegmann from Port Washington, New York. A good friend.

[mheermance]

The infinite series 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 ..., conditionally converges to 1/3. This means that with an infinite number of angle bisections, you can trisect an angle! Normally it takes a while, but my friend Zeno has a magic tortoise that can bisect an angle in half the time as the previous bisection. So the tortoise completes the trisection in a finite amount of time. Pretty spiffy!

[cidlunius]

Hey. Compared to the amount of minutes in 2000 years, this video is remarkably short.

[simonsallen]

I have been watching this while eating a breakfast bowl of cereal. I was thinking about how much work has gone into preparing the video. Even with a very knowledgeable presenter, it must have taken hours. I was pondering this when the answer popped out. About 200 hours. I am not surprised. The video is an incredible achievement. Thank you Burkard and Marty. This is a video I will be revisiting. A lot.

[Ma_X]

There are only two students attending a math lecture. Suddenly four of them got up and left. The professor thought with regret: "Well, now two more will come in and there will be no one left at all."

[mapifisher]

They said it couldn't be done,
But he went right to it
And took that thing that couldn't be done
And couldn't do it.

[rc]

Wow. Such an amazing and carefully constructed video. Also, the important warning about ‘changing the rules’ shows (to me in my opinion) how seriously and solemnly the ethic of education is handled on this channel. You have a very satisfied and excited new subscriber!

[thedude]

I had a Rubik's cube, and then I went on Amazon and ordered another one and in less than 24 hours later, BAM! I just doubled my cube. Solved in less than a day.

Next!

[johannkapserschmidt]

Nice Easter egg at 13:25 with captions on. Well done Karl.

[DanJan]

I will have to come back to this video when my head is fresher :)

[Supremebubble]

Every time I see a video on this channel I secretly hope it is a long one. I feel happy now :)

[MrCheeze]

Not enough comments here are talking about how amazingly accessible you made these advanced topics. Very well done.

[reznovvazileski]

Note to self: If Mathologer says something was explained in a previous video, actually watch the previous video before continuing to watch the video. I made this a lot harder for myself than it needed to be :')

[Niminos]

I want to share something uncommon (maybe, in fact I don't know)


I am a french student in master's degree (Paris XI) who recently went through a course in Galois Theory (soberly named "Algèbre"). It was tough and too short to really understand the ideas. The exam went pretty poorly for me because of that.
These kind of videos are therefore precious resources for me: I can recognize all the tools we had to use (extension fields etc.) in a way that clearly go into why we need them and how they work. That part was rather cryptic before.
Since I also want to be a teacher, that video (with some others, and on many subjects !) is a great gift in that respect. I really wanted to thank you for that. You are a great professor.

[brunoandrades]

These are really the only long videos I can stand; and I actually enjoy them a lot; you explain very clearly and I understand every thing you do. Thank you for such good content

[bismuth]

I dont even know math but your personality and enthusiasm has drawn me to watch

[andrewmcginley]

There were quite a few places where I just went, “that makes sense, ” without proving it to myself, but I felt like I followed the whole video.

[richardschreier]

This video is full of some lovely gems. The construction of a regular pentagon using 4 intermediate circles and one intermediate line was my personal favourite and I have shared this with friends. (To prove the method works, I solved for the coordinates of the key points and used the fact that cos(pi/5) is (1+root(5))/4 and that cos(2*pi/5) is (-1+root(5))/4. Not elegant, but it got me there.) The description of the rooty-expression subfield and the consequence that certain geometric constructions are not possible provides closure for more decades-old gaps in my math knowledge. (I am ashamed to admit that I forgot the rational root theorem and had to re-watch that video.) Thanks for putting this video together. Your enthusiasm for the subject combined with your depth of knowledge is a wonder to behold. No one can get the the "root" of the problem like you can!