Euclid's Big Problem - Numberphile 

exactly 7 years later and this video still bangs. What a fantastic bit of teaching

I had a professor who insisted on taking a few weeks to teach us all of this, and I really didn't get why it was such a big deal until we continued on throughout the semester. Turns out, using a straight edge and compass is a great way to not only understand geometry, but to also to become aware of just how many assumptions we never knew we were making about mathematics when we are taught it.

Message from Zsuzsanna (in the video) Hello Everyone, Everett Sass and Fenrakk101 (possibly others as well) posted correct solutions on constructing a segment of length square root 3. [Spoiler Alert! If you read them, you can discover that the idea behind both is the same: to construct a right triangle with hypotenuse 2 and one leg of length 1, so the other leg will be length sqrt(3).] This solves the puzzle of "tripling a square", that is, you can now construct a square of area 3. As some of you point out, this is not the same as doubling the cube, which comes down to constructing a segment of length cube root of 2, which is impossible by Euclidean means. Some of you mention the idea of trisecting an angle by constructing an isosceles triangle and trisecting the base. This does not work: if you look closely, the middle part will be a bigger angle than the two side parts. In other words, the three thin triangles are not congruent. As one person points out, what you'd need to do is trisect a circle arc, not the base of the triangle, and that is impossible by straight edge and compass. Of course you can trisect _some_ angles, like a right angle or a 45 degree angle, but there is no general procedure using straight edge and compass that will trisect any arbitrary angle. Sorry about staining that nice straight edge with the marker! I felt bad.

Revolutionize math. Turn 19. Die in a duel. What a life story.

Ths one is really great. She explained points very well, she created tension in the storytelling ("before I tell you the answer" and goes onto another segment) and generally was clear and fun.

12:58 Brady not only cracks this classic dad joke but then actively chooses to include it in the edit. Decisions like these are key to the channel’s success

I really enjoined this video; with my limited knowledge of math, this is one of the few videos I can fully understand without a single question. She had shown many examples and clear, understood proof. I really enjoined her!

Euclidean geometry uses a straight edge and compass in two dimensions. Origami does not allow drawing of circles within the paper (within those two dimensions) ... but ... Folding is equivalent to the use of a compass, in a third dimension! And utilizing three dimensions allows third roots. Marvelous!

"...and then people thought about it for 2000 years..." : my favorite line of the video.

Her voice is absolutely mesmerizing.

it sounds like galois was a legend. revolutionising a fundamental part of mathematics before he turned 19 and then fighting a duel. i feel so useless right now

There is something so beautiful about these simple geometric ideas. Love this content!

Good video. I was spellbound by the simple complexity and complex simplicity underlying the ruler-and-compass-constructions. Brady's questions are always to the point and he says insightful and poetic things like "so the cube root is the point you can't reach".

This video was awesome!

I always had his question. Lets say you are the first to prove something in math. What do you do after? Do you contact someone? :/

This video is odly calming and relaxing, probably from the way and rate she speaks with. And most of all, it was intresting.

what an interesting subject and such a soft spoken guest. thank you for this vid.

When lines and circles come together and intersects, points are born.

well! that smile trisected my heart

Everything Euclid couldn't. People thought about it for 2000 years.

Trisection of angles? It is certainly possible. I have discovered a truly marvelous proof of this, which this comment box allows too few characters to contain.

I absolutely love her English. omg...

I can unisect an angle.

Zsuzsannas eyes shine so brightly when explaining. I had to smile through all of the video. Awsome girl!

You can say Pierre got to the... root of the problem.

It is UNBELIEVABLE we don't learn this in school where I live. THIS is the foundation.

0:25 Oh, just like in computer science where ideal Turing machines have infinite memory and time, and in Physics where we tie frictionless masses together with massless strings. Gotta love the world of the abstract! ;)

I'm a completely halfwit when it comes to maths, yet i still do find these Numberphile videos so entertaining. I'm puzzled. But great to watch while recovering from knee surgery and way to much time indoor for the next couple of months.

Awesome video! Would have happily sat here and watched another 4 hours of this.

Wow. I've always had a piecemeal understanding of the relation between algebra and geometry. You've presented that relationship completely and elegantly. Thank you!

Amazing video. And I love Zsuzsanna's accent!

We learned these constructions in technical drawing class without knowing where they came from.....it was quite magical

There is a game called euclidea with stuff just like this

Thank you Zsuzsanna for some refresher examples.

I miss geometry class, drawing all these angles and circles and stuff was so fun

This is the best episode I've ever seen. At the very end I was even, for the first time, to begin to glimpse the relationship between Galois and these questions. What he's looking at when he's extending Fields. Good!

That seemed like a lot of extra steps to construct a square with length root 2. Once you have the perpendicular lines, all you need to do is draw a circle with radius 1 around the intersection of the perpendicular lines and those 4 points are your vertices for a square with side length root 2. It works because the distance between the points on the same line is going to be the diameter of the circle: 2. this is going to be the diagonal of the new square, so if you have a square with diagonal length 2, you necessarily have a square with side length root 2.

I remember an old Mathematical Games article by Martin Gardner where he described a device for trisecting angles. It was something like a compass, with two more legs between the outer ones. As you expanded the outer legs to a specific angle, the two inside legs always maintained a division of 1/3 - 1/3 - 1/3 of the distance (or angle, actually) between the outer legs.

I think the secret to Brady's success is the paper. Lots of mathematicians in different Numberphile videos have asked if they could have more of the "wonderful paper". This video brings back nostalgic memories of doing geometry in primary school, with compasses and straight edges and protractors, long before I took high school geometry and learned about postulates and theorems and etc. The Greeks were really on to something when they thought everything boiled down to geometry, that geometry was pure and everything else revolved around it. The Fundamental Theorem of Algebra video showed how algebra is connected to geometry.

This video made me nostalgic because half of the things done in this video were things that I learned how to do when getting my certificate in mechanical/architectural engineering: bisecting and abritrary angle, finding the perpendicular bisector of an arbitrary line segment, finding parallel lines, trisecting an arbitrary line segment. However, I never was taught that you could double an arbitrary square. And one thing that was not talked about in the video was finding the center of an arbitrary triangle. Boy, I used the word "arbitrary" a lot in this comment.

She is great! Super understandable. I we get to see more of her in future videos. ^_^

Drawing right now a heart only with a ruler and a compass to Zsuzsanna!

Another great video! I am loving these, is anyone else too?

I don’t know why, but geometry has always been the ‘prettiest’ branch of maths to me. I like a formula or some interesting arithmetic, but there’s something so satisfying and aesthetic about things like this video

He died at 19 in a duel LOL ... manliest mathematician ever

This was lovely 😍. I find myself exploring videos of classic Math that can be relevant today, and this is one of them.

YOU DEFILED THAT STRAIGHTEDGE WITH A MARKER. Edit: I now see from +Numberphile's comment that Zsuzsanna apologized for this. XD

I was just about to say you were wrong as I had learnt how to trisect an angle using Origami, then I watched the last part and saw that you had beaten me to it lol. Interesting video thank you.

14:34 But… I didn’t mean to cause you trouble… _sobs and leaves_

Trisecting any angle is actually possible. Based on work I did 30 years ago, and a snippet I got from this video, I finally figured a way. I plan on completing the proof and submitting it for publication in the new year. Happy Holidays..

This was a very relaxing video to watch.

Nice to see a hungarian mathematician on Numberphile. She speaks english like everyone else here in Hungary. Funny to hear the hungarian accent your video. Amúgy ha olvasod ezt Zsuzsa grat hogy kijutottál Amerikába és most ott kutatsz! Pacsi!

the most thing that i like in Numberphile is that almost every matematician its from a different country

As for trisecting the angle... If you draw a circle with the centre at the meeting point of both lines. Then you draw a line through both points where the circle intersects with the lines of the angle. Next you trisect this line. Now you have the two points you are looking for to trisect the angle.

1:10 Wait a second! So now we're talking about modern compasses instead of Euklidian compasses?

Sorry that my comment is irrelevant but her voice so calming and has one of the best accents I've ever heard. I think I'm gonna go to sleep listening her.

I was reading about classical constructions the other day, and there's something I don't understand. One of the restrictions of compass-and-straightedge constructions is that the compass is assumed to collapse when lifted from the plane, making it impossible to directly transfer lengths. But the compass equivalence theorem means that this is ultimately an immaterial restriction, since lengths can still be transferred indirectly (albeit in a complicated fashion). So what I don't get is, why does that restriction even exist if it doesn't make any practical difference?

What a beautiful and relaxing video. A young woman with a lovely accent reciting ancient questions and answers.

poor ruler at 2:57 hurts to watch

I don't know what it is, but this woman's eyes and demeanour make me melt

Beautiful talk. Beautiful teacher,Thank you very much

This geometry makes sense. Numbers are a human construct of measure, so the Eucilean seams more of a anchoring reality in how we understand our surrounding. This is rather amazing, and I am shocked I never learned this basic method to geometry....sure some overlap, but not based in. This should be in schools from perhaps 2nd or 3rd grade on and into the complexity in high school and university.

men i nschool , this is what they also should have shown us.... might be really usefull for CAD programs

This is great! Zsuzsanna. Dancso is a wonderful teacher.

Well if you were in a 4D world and had a 3D surface to mess about on cube roots should be possible right?

Euclid's compass was a collapsing compass. You can not use it to transfer a distance from one starting point in the plain to another. As soon as you lift the compass from the plain, it collapses! If you want to construct a 20 cm line from a given 1 cm line you must use Book I Proposition 2 of Euclid's Elements.

Yeeey a fellow Hungarian :D She has the same family name as one of the most famous comdeian in Hungary :D

doing a bit of oscillation, and mid way through, had an epiphany, on how to triple the square! love this difficulty of question for the viewer to do! Feed me more problems

12:13 impossible in Greek is: αδύνατον, not άδυνατον. In ancient Greek it would be: ἀδύνατον. I'm just correcting something that I saw it's s wrong. Generally I really liked this video a lot.

As a big fan of geometry, this is one of my favourite numberphile videos!

Sweet, pretty, nice and mathematician..what a combination..!

My son showed me how to trisect an angle with unmarked tile and compass. Yes teally it took a whole wall and finally I understood and did it. I am a 30 year math teacher. It worked. Very exotic technique! He did it and finally I did it for a random angle. There is a mathematical proof of Weimpossibility.Been done.

At around 3:10, shouldn´t it have said "perpEndicular"? I´m really unsure now.

Finally someone else who writes their 1’s like that!

but.... if it's possible to divide a line segment in 3, couldn't this be done to trisect an angle: cut the 2 lines from the angle in same length; close an isosceles triagle; divide this new segment in 3; connect the dividing points to the original corner... ?

This kind of thing is maybe why the Greek saw maths (which was for them mainly geometry) as so closely related to music and architecture.

Is that really called "a compas" in english? It's really strange, because I think of "a compass" every time I hear that word. In Danish, they are called (If translated directly) "a school-follower". It's a really strange name, but I guess it's because it helps you follow school? Or something? o.O

As for trisecting an angle, I'm not sure if this is allowed, but it's mathematically valid... - Use the compass to draw an arc connecting the two lines. - Move and rotate a markable edge (straight edge, piece of paper, etc) from one end of the arc to the other, marking it's arc length as a straight line (this can be interpreted as 1 unit, if it makes you any happier). - Trisect the straight line that is the arc length. - Retrace the arc length with your trisected line, marking 3 equal arc segments. - Connect the vertex to the marks on the arc. The problem, of course, is knowing how precise the measurement and retrace of the arc length and segments are. If it were a free body as opposed to a drawing, you could just roll the arc over a straight line and measure that way.

Isn't it entirely possible to make a root-3 segment from a triangle? Take your unit length, double it, use the new line as the base of an equilateral triangle. If you draw a line from an angle to a side (bisecting that side) the length of that line would be root 3.

If people are interested in straightedge and compass constructions, there is an app called Euclidea that basically poses these construction problems as puzzles. If you like thinking about this stuff you will love the app, it can be really tricky and fun to find the most efficient solutions. For example, an especially tough one: given a circle and a point on that circle, can you construct a tangent line through that point using only 3 lines/circles?

So you have to go one dimension up to crack cube roots. What about fourth roots? O_o; Do we go up one more dimension? :P

"I hope they also taught you not to have duels." [mathematician laughs nervously]

She's Hungarian!!!

I love Numberphile. Period.

PLEASE HELP!! I don't understand why this is difficult...! Can't you just - Draw a circle at angle A, making points B and C equidistant from A; - Connect BC, trisecting the new line at D and E - Connect AD and AE, and then you're done?? I mean, I understand from a recent video that all triangles can be represented as an Equilateral triangle viewed in 3d space

This is a wonderful video! It brings back many memories.

That seems so wrong being unable to trisect an angle with Euclidian tools. It just seems too benign to be impossible. I'm going to waste so much time now trying to get it to work even though I was just told it's been proven impossible.

wow, i had never truly seen geometry like this, this is enlightening! it is way more interesting than what i previously thought! extraordinary!

Even a slaveboy could figure out how to double a square! -Socrates haha....Plato reference.

My favorite numberphile video. Mind boggling for sure

Adidas = Trisector :0

This kind of pure geometry is really interesting coming from a modern algebra-centric perspective :)

>>> HEY BRADY!!

Hey, in th '50s we could have done with a charming and fine teacher like this. Well done, lass!

Hungarians unite!

if my math teachers had been so kind and friendly, I might have learned to love math... I'm certainly in love with math now that I found this channel...

Brady's big problem: Getting Zsuzsanna into his bed

I love how we study this in school (7-8 grade) Like how to construct the basics (perpendicular bisector, angle bisector, and the Thales theorem) It's fun :)

all this is 10th grade

You can use a cube to construct a square whose area is three times the area of the square that defines one of the six faces of that cube: Just as the hypotenuse of a right triangle whose sides are equal in length to one another is equal to the sqrt-2 times the length of either side, the diagonal within a cube that connects the two most-remote vertexes (four such paired vertexes exist on a given cube -- all equal in length to one another) is of a length which measures exactly sqrt-3 times the side of the square that defines one of the faces of the cube.