Mathologer - 2019-06-29

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 http://www.math.miami.edu/~armstrong/461sp11/ImpossibleConstructions.pdf Here is a derivation of the cubic polynomial for the regular heptagon construction by (I think) the mathematician Reinhard Schultz http://math.ucr.edu/~res/math153/s10/history09a.pdf (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. ) Here is an interesting paper that explores why Wantzel's results did not get recognised during his lifetime https://www.sciencedirect.com/science/article/pii/S031508600900010X Thank you to Marty and Karl for your help with creating this video. And thank you to Cleon Teunissen for pointing out that the picture of Pierre Wantzel that I use in this video is actually not showing Pierre Wantzel but rather Gustave Gaspard de Coriolis who was also a mathematician and lived around the same time as Pierre Wantzel. It appears that whenever there does not exist an actual picture of some person Google and other internet gods simply declare some more or less random picture to be the real thing. See also this page by the SciFi writer Greg Egan who made sure that no actual picture of himself is to be found on the internet: https://www.gregegan.net/images/GregEgan.htm Enjoy :) Burkard Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer

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

Indeed

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.

:)

And...?

This is the most healthy, uncorrupted commend section on the whole of YouTube

edit: and positive

Townsends has similar!

@Podemos URSS i was gonna write a punny comment like that myself.

+Zael Slyte

what

+Charles Johnson

There always be that one person who always does this.

Fun fact : Mathematician and other people who likes math are generally open minded being and doesn't like confrontation

I love your T-shirt. I can't take my eyes off it. I need one just like it.

Check Dodecahedron - Kwintessens album cover

Autumn Desolation pretty awesome! and its funny how they (whoever designed it) took the shapes and preferred to have them laid on their sides flat, not pointing upwards like on the shirt. its also aligned to the bottom, not the middle which gives it more space. it also looks fantastic. thanks!

It's symbology....

@Analog yeah I did it..

You have to come India to wear it😀

What is that called? I searched nested polygon disk and couldn’t find it

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!

If you are allowed infinitely many steps you can construct any number whatsoever :)

Mathologer Including several numbers that aren’t in the real numbers, including a number that’s larger than any real number and a number that’s larger than 0 and smaller than any positive real number!

@Cephalos Jr. With infinite step, constructing some numbers that aren't real number like infinity and infinitesimal isn't a problem, but I don't think there's a way to construct complex number

@Noname imagination! Add imagination and complex number (like pink unicorns) isn't a problem anymore :)

@Mathologer And also, within a finite number of steps you can construct any number, to within a satisfactory precision. I'd love to see good, well-convergent constructions of some of these.

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.

:)

Me: I am terrible at math and want nothing to do with it.

Youtube: We recommend this video about impossible to solve mathematical equations.

Me: ok.

@mapperific I mean Frank is a master mathemetician, his ability to count to autism is unsurpassed.

Same here 😂

@Ta ble 6 days ago, "@mapperific I mean Frank is a master mathemetician, his ability to count to autism is unsurpassed."

I suggest " I mean Frank is a master mathemetician, his ability to count to TOTAL autism is WITHOUT WORDS."" A VANISHING ABILITY. SPEECHLESS !!! UNDEFINEABLE !!!

tard

@Xanthopathy Highlighted reply Xanthopathy 38 minutes ago, "tard" AND DEFINITELY ADD "TARD"...

2:13 how I feel during a calculus test

It is posible trisect an angle with origami's rules, but can not analiticaly.

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 :')

Yup - Pro Tip FTW :-)

In Spain, when someone gives an absurd and rondabout explanation for something we say he's "squaring the circle"

There's a great mobile game called "Euclidea" that has many levels you solve by constructing a given shape with ruler and compass. I had a great time playing it :)

Hah, I literally had that open in another tab to play around with throughout this video

@Brian W any angle 360°/N where 3 doesn't divide the +ve integer N is trisectible.

@Grizzly01 I do not believe it is possible to divide a circle into 360 equal segments geometrically, if it is could you please direct me to the relevant youtube video.

@Brian W I never said it was.

I was just putting forward a well-known way of checking if a particular angle may be trisectible or not.

For example:

60° = 360°/6

6/3 = 2, so 60° is not trisectible (using straightedge & compass).

45° = 360°/8

8/3 = 2.666..., so 45° is trisectible.

36° = 360°/10

10/3 = 3.333..., so trisectible

20° = 360°/18

18/3 = 6, so not trisectible.

@Grizzly01 Cool, thanks for explanation , hadn't come across that before

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!

A joke: you see doubling cubes all the time in backgammon.

Oh my gods!

I can't believe it!

I actually made it through the video!

I mean, I had to repeat some parts, a couple times, but I made it through!

Although...

I did have to just take your word for it on a couple of things because, unfortunately, I didn't learn *all* the math things in high school, and I never came across some of them in college (I don't exactly dream of taking trig, lol).

But for the most part, I was right there with you! And honestly that surprises me! (I've never been that good at higher-level maths)

you need to start selling those shirts.

Or give us a link to them!

I got nerd-sniped by that t-shirt. It's pretty easy to see the pattern that produces that shape, but it is not obvious that the series converges; although, if it divereged, I suppose the entire t-shirt would be red. After doing some working out, and then some googling, I found out that it is a quite famous problem.

My brain cant keep up to this! 😭 Why is it im so slow at this field 😭😭😭

Stick to it, and you will understand it bit by bit.

Just take it one nibble at a time.

♡

At least he gave several warnings. His regular maths videos tend to get quite deep (for non-mathematicians), so when he says this one is going to be tough...brace yourself.

@Stephen Holtom Not only for non-mathematicians. These videos are a revelation for everyone. It's just a different perspective. You will never see these kinds of solutions in normal textbooks nor college classes. We are blessed to be able to access to the deepest possible ideas around our mind. Enjoy, my friend.

I gave up after 23 minutes.

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

:)

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.

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.

The video was absolutely fantastic! But one issue that bothers me is that the image of Pierre Wantzel that you showed is, according to Wikipedia, the image of Gaspard-Gustave Coriolis (of Coriolis effect fame). Please clarify this.

Peter Bočan a parabola can be “constructed” without a straight edge and compass, using only a piece of paper! Wax paper works best. (You possibly already know this).

@M1lkweed 761 yeah, that was what i was thinking (straightedge vs ruler)

@fynes leigh Why do you bother with YouTube. As for never needing mathematics, kindly never involve yourself in the design or construction of any building, bridge, vehicle, or other device that I will ever use.

I made it through!

Unfortunately after 30 minutes i sometimes had to take you on your words because im 14 and havent had everything in school yet...

But i really enjoyed the video!

will intuitively i always knew that a circle is no square .

28:26 I don't feel "rooted," more like "stumped."

"Here's this result which you're going to prove in your homework. Now with that result we can do all these cool things on the board"

You should have said:

"When in doubt, work it out."

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

14:35

Mathloger: use pythagoras

Me : similar triangle

Also how to solve it using pythagoras

Don't expect to get these videos the first time around. Esecially if you don't take notes and practice doing it yourself. It's an unrealistic expectation. There's no good reason to limit yourself to one viewing if you enjoy a topic. Besides, you have your whole life to learn this stuff. You don't need to rush through it. =) Cheers.

It took me three viewings to understand the gist of it. And yes, do take notes during a viewing, it really helps.

Michael Lubin that comment makes me want Noah to get the boat

@The unknown Let O be the center of the circle of radius (A+1)/2. Let P be the intersection of the circle with the perpendicular line to OA passing through A. Triangle OAP is rectangle and has hypotenuse=OP=radius=(A+1)/2 and sides x=AP(pink segment) and OA=A-(A+1)/2=(A-1)/2. Now use pythagoras to solve for x: x²+((A-1)/2)²=((A+1)/2)², therefore x²=(A²+2A+1-A²-1+2A)/4=(4A)/4=A, and then x=sqrt(A)

Thanks! I particularly liked that the demonstration highlighted the consequences of the symmetry of polynomials. This was overlooked in the book I read on the subject that used the language of abstract algebra. A standard approach, I assume, and certainly more powerful and general one. But definitely a tough cookie for this poor engineer.

I can construct point cube root of 2 away from origin.

If I have infinite time.

Me: "Oh cool, another Mathologer video"

Burkard: "In this short 40 minute video"

Me: "oh fuck yeah..."

that is one of the coolest shirts i have seen. anyone know where i could get one?

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!

I would love to see your presentation of the heptadecagon construction, including the algebraic part of the proof :)

8:52 - looks like a representation of earth and its magnetic fields.

Thanks for this amazing effort! Just 200 hours to make this (..) seems time well spent, given how many viewers enjoy the results. Keep 'm coming!

Squaring a circle is easy, have you ever seen a boxing ring?

@k1w1 area . not circumference!

@parabot2 wow that's... very interesting, got more? I'm fascinated

No.I can't

If you include MMA rings you can also hexagonize a circle.

Welcome to the Squared Circle.

I feel like am in primary school. This video gives me the same vibes I had when listening to my teachers, sitting on those tiny desks. I even remember how the room smelled like

I mostly of the time was philosophing about his shirt xP

looking forward to the continuation this was pretty cool

I like how with each level came increasing brevity.

Level 7 - just a tease of it's existance!

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

I love this channel now, took me some time but you've won

Great video! It's been fun to remember things I've learned back in my Math bachelor courses. :)

Amazing video, thank you! You made the Pierre Wantzels proof much more understandable than how it was presented to me when I was studying.

BTW, made all the challenges while watching this video except 7:50 cuz I'm too lazy to do it rn lol

Great, mission accomplished as far as you are concerned then :)

When I see a new video from Mathologer, I first like it and then start viewing. Always high high high quality stuff. I had gained familiarity of the rules of the ruler-and-compass game after playing Euclidea on my phone (https://www.euclidea.xyz).

You know when you said at Level 5, "you're still here"? Well, no I wasn't!

Thank you for the video! All of you friends are super awesome! Oh, moments in this video are sad.

I'm imagine Euclidea (the game) to add these three for its next early April 'update for bonus challenges'.

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

I consider your username an Easter Egg as someone who's watched Samurai Jack, Extra Thicc.

I dont get it?

@Mathologer You mean https://www.youtube.com/watch?v=yce0ztxNIDI&t=3m35s

Nathanial Reed that’s super cute

This video is why I love youtube. Thank you for your contribution!

Макс Ф. - 2020-09-18

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."

breinnarn - 2020-11-27

That happened all the time when I last attended math class almost 20 years ago. Students would make deals with one another to write several names on the attendance list, because too low attendance = no grades.

Julie Hovar - 2021-01-14

You are a good human my next clienr will be here. WHETHER us discuss math or earth, be Well. Be nice. Till next time.