Why are Most Polygons Impossible to Construct?

⬣ LINKS ⬣
⬡ PATREON: / anotherroof
⬡ CHANNEL: / anotherroof
⬡ WEBSITE: anotherroof.top
⬡ SUBREDDIT: / anotherroof
⬡ TWITCH: / anotherroof
⬣ ABOUT ⬣
Only certain regular polygons are constructible with compass and straightedge. Why? And why did the first person to prove it after 2000 years get no recognition? In this introduction to Field Theory, we’ll find out, and along the way we’ll also prove other impossibilities like why cube doubling and angle trisection are impossible. Happy Birthday Pierre Wantzel!
⬣ TIMESTAMPS ⬣
00:00 - Intro
02:13 - Why ONLY the four operations and Square Roots
04:14 - Field Extensions
14:37 - Minimal Polynomials
25:54 - Cube Doubling is Impossible
31:50 - Angle Trisection is Impossible
36:35 - Constructible Polygons: The Gauss-Wantzel Theorem
50:12 - Why Pierre Wanzel got No Recognition
⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!
⬣ REFERENCES ⬣
Biographical details on Wantzel to be found in:
Jean-Claude Saint-Venant (1848). “Biographie: Wantzel”. Nouvelles Annales de Mathématiques Série 1, 7: 321-331.
Found, with English translation by Lauren Murphy, here: divisbyzero.com/wp-content/up...
Amazing treatise on possible reasons why Wantzel was never credited:
Jesper Lützen, “Why was Wantzel Overlooked for a Century? The Changing Importance of an Impossibility Result” Historia Mathematica, Vol 36 (4) 374-394, 2009.
Found here: www.sciencedirect.com/science...
Pierre Wantzel’s original paper, with proofs adapted in the video:
Wantzel, L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas", Journal de Mathématiques Pures et Appliquées, 2: 366-372.
Found here: visualiseur.bnf.fr/ConsulterEl...
⬣ CREDITS ⬣
Music by Danijel Zambo, Tobias Voigt, and Apex Music.
Image Credits
Wantzel
mathshistory.st-andrews.ac.uk...
Hamilton
upload.wikimedia.org/wikipedi...
Klein
upload.wikimedia.org/wikipedi...
De Morgan
upload.wikimedia.org/wikipedi...
Petersen
upload.wikimedia.org/wikipedi...
Pascal's Triangle
www.geogebra.org/resource/jk4...

[@Scum42]

My favorite thing, by far, about this channel is that it started with literally "what even is a number guys" and every video has built and built and built, and now we're here talking about polynomial irreducibilitly. It feels so incredibly earned, and gives an amazing sense of perspective of exactly how ALL of mathematics builds out from extremely fundamental axioms

[@mapron1]

I just got recommended this video out of nowhere and I have no idea what he talking about. I don't understand maths.

[@cartatowegs5080]

​@mapron1 I would start with the first video he made. Truly an amazing channel that explains math well. If his videos get too complicated I would recommend 3blue1brown as a start.

[@mapron1]

@@cartatowegs5080 3blue1brown is already too hard for me.

[@cartatowegs5080]

@@mapron1 question for you, what is the highest math course you understood, not necessarily the most advanced class you took.

[@xiexingwu]

@@mapron1 3b1b videos aren't always easy as there's always some level of assumed academic knowledge. Starting from the first videos in this channel, all one really needs is curiosity and the patience to think about somewhat philosophical questions such as what even is a number.

[@jamiemacleavy2482]

None of my friends appreciate when I recommend hour long videos about compass and straight edge constructions. I will not stop

[@JiMwB]

10:47 Chalkboard: (b+ch) me: well that's unfortunate

[@mrphlip]

I'm glad I'm not the only one who had to double-take at that moment...

[@AnotherRoof]

I saw it when editing and could never unsee it!

[@DumbMuscle]

The math content in this is great - but I want to specifically highlight a couple of really nice touches in here. Firstly, thank you for highlighting the spot where it's worth taking a break - for a long video like this, that's really useful. Secondly, highlighting the best place to move the subtitles to is a really nice touch (and also taught me you can move the subtitles by dragging them around the screen). Keep up the good work!

[@AnotherRoof]

I had a moment while editing where I almost deleted these thinking "hmm, maybe people don't need these...?" but viewers have been positive about it so thanks for the feedback!

[@KatMistberg]

Because you don't have enough bricks to construct all of them?

[@1.4142]

it's the goofy conlang guy

[@creepad667]

My man do not spoil the video without warnings

[@lobsterfork]

Being able to watch a video with a chalkboard and not feel disturbed by the sound is so refreshing. Thank you for that edit =)

[@jimbobago]

17:05 I love the "derivation" of "monic".

[@isobarkley]

absolutely love the production quality of your vids. you deserve a lot more recognition than you have currently, but for the time being, i am quite appreciative of and thankful for your content

[@AnotherRoof]

Aw comments like this make my day! I'm still trying to grow my channel but I'm glad you enjoy the videos :)

[@Wolfiyeethegranddukecerberus17]

Bro made math interesting 😭, like I'm actually staying up to watch this and I'm about to clip it to watch tomorrow

[@pierrebaillargeon9531]

To avoid the Wantzel curse, the next video will show which haircuts can be constructed with only scissors and a straight blade.

[@mathfincoding]

Happy birthday, Wantzel!!!

[@TheQuicksilver115]

Ooooooh yes!! I freaking love this channel man

[@fredg8328]

I didn't know a video like that was possible on youtube. And you did it without even sweating.

[@tejing2001]

You really made field theory hang together for me a lot better. Prime power galois fields, in particular, make so much more sense to me now. I could work with them before and not get things wrong, but it was all not quite as well founded. Now it's so obvious what's going on there. And in my book, if it's not obvious to you, you haven't truly understood it yet. Thanks so much for connecting all this together on a subject I never got the chance to work my way through alone.

[@JeSuisNerd]

You're such a fantastic educator, the way your enthusiasm comes through even with a carefully scripted video is always engaging! Lots of educational content can be hard to absorb for those of us with ADD, but you've turned what could be boring lectures into my favorite math youtube channel

[@mranonymous5268]

Throwback to my field theory course two years ago. Amazing how even though I hated that course with a passion, this video is able to get me excited again about the subject!

[@robharwood3538]

Really well made video. You've put a lot of time and effort into this, which is amazing! Congrats to you, and Happy Birthday to Wantzel!

[@pietrocelano23]

Thank you so much for these videos! My Galois Theory professor tasked me with doing a seminar on this exact topic. Glad to find an easy digestible source that isnt embedded in textbooks or papers.

[@GhostyOcean]

Beautiful explanation! I love the bricks, what a good idea to use them since the beginning.

[@petevenuti7355]

I remember the clumsy folded paper ones in the beginning😂

[@andrewkarsten5268]

Good job giving a very basic introduction to the fundamental idea behind inconstructible numbers. I remember taking an introductory ring a field theory class and learning everything up to a basic introductory idea behind gallois theory, and you did a good job of getting the main ideas across without getting too bogged down in all the (important, but tedious) details.

[@funktorial]

this video is really good at helping me remember the parts of galois theory i forgot/missed the first time. like so many things in math, it makes so much more sense now

[@funktorial]

also I think there's a tiny typo at 33:36 (forgot a prime on the f, so it should read: p divides (df) i.e. p divides every term in f')

[@universallanguageproject]

I love the way you expand on all of the specifics to do a complete explanation of things. Iff most of your viewers understand the use of logic, I'm pleasantly surprised. Beautiful belated callout for an amazing mathematician 🎉

[@hughobyrne2588]

Gasp! How can you say that most polygons are nonconstructable, when you can pair each nonconstructable one with a constructable one, and still have an infinite number of constructable polygons unpaired?!

[@AnotherRoof]

My patrons and I were discussing this while we were drafting titles and how someone would point this out! You're right of course -- but we justified it by saying that the natural density of constructible polygons must be less than 1/2 :P

[@programmingpi314]

Because there are only countably infinite polygons, the only way to show a property holds for "most" cases using cardinality is if there are only finitely many counterexamples. Therefore, you need density or a similar measure to meaningfully define "most" in this context.

[@angelmendez-rivera351]

The asymptotic density of the ratio being described is less than 1/2.

[@arkaprovodas4920]

Happy birthday Wantzel you will not be forgotten.

[@ishtaraletheia9804]

This was incredible and beautiful, thank you so much for making it! This is what math often feels like to me at it's best, crunchy and sweet.

[@invisibules]

Thank you for such a detailed tour of the maths without feeling obliged to hide the technicalities. Bravo!

[@kaustubhpandey1395]

Such a great video! A great teaser for field theory❤

[@puyaa3000]

What a consequence, today I read on regular polygons, chapter 17th of the book "Galois theory" by Ian Stewart.

[@zugzwangelist]

This man is an amazing teacher.

[@Macieks300]

This might be the best math video I've seen on KZbin this year.

[@iamtraditi4075]

This was phenomenal! Thank you :)

[@ruilopes6638]

Once again I feel the need to thank you for such an amazing video. As stated on the last video that thorem holds quite a special place for me But that is not all that makes all of your videos great, I simply love how you take your time to explain what is behind all of this math, the history, the characters, the mtivations, that makes even the most abstract problem compeling. (and once again something that I hold dear) So thanks for being such an incredible educator, one that I deeply admire and as a math teacher myself, am inspired by.

[@_ajweir]

What's the angle on this topic then.. 😎

[@Adomas_B]

You better square up and find out 😎

[@jizert]

@@Adomas_B lets not get obtuse about this! (sorry if that pun was only *tangentially* related)

[@pierreabbat6157]

It's one third of the given angle.

[@projecteuclidv2582]

Masterful work!

[@dschaegkarthur1093]

What an amazing Video!! A lot of love from Germany. Every of your videos is just amazing. Sadly this one came about a year too late since Back then I Had a Algebra course myself and the video would helped me a Lot in the field theory Part of the lecture. Keep Up the great Work, you are a huge inspiration for me and you Feed my motivation to keep on studying maths ❤️

[@petrosthegoober]

BRICKS. I'm in love with the bricks!

[@elshadshirinov1633]

I find it sad, that geometry and algebra are taught very separately at school/highschool. While there is this amazingly deep connection between the core foundations of both fields. Construction with ruler/compass for geometry and finding roots of equations for algebra.

[@honkynel]

Bizarre. I'd just watched the other vid yesterday..So that is a happy coincidence. I'm assuming this is going to take me a few watches to get my head round it. Thank you.

[@jonahunderhill]

As I'm watching it, I keep wanting to give more likes but it only lets me give the one. Really well explained! Thanks!

[@Tinybabyfishy]

I had the first part of this video on my to-watch list for ages before finally watching it today, without even having realised that the second part was out. Happy coincidence! This leads me to wonder where the "2-ness" of our constructible numbers and shapes comes from, in a deep sense. If we add another tool to our compass and straightedge, how does that change what we can construct? What if we were fourth-dimensional beings, playing our game of geometry with a hypercompass which traces a sphere and not a circle, and a flatplane instead of a straightedge? What do the shapes that we can construct start to look like? Is that even a meaningful set of rules to try to adopt? Wantzel taught us how far Euclid's rules can take us, but what if we add new rules to Euclid's game?

[@tejing2001]

The 2-ness comes from the squaring in the pythagorean theorem, basically.

[@Tinybabyfishy]

@@tejing2001 oh, so that makes all the other stuff I mentioned totally unrelated. Then it seems like no matter which rules we're playing by, we can never construct a cube root.

[@tejing2001]

@@Tinybabyfishy Yup.

[@AdrianCruz_]

Happy Birthday to Pierre Wanzel

[@MathPro0]

This is called high quality video , discussing Maths , I think with no beautiful animations this video is still at the level of 3b1b or greater than it ... Thanks for the video broo , keep making more Also I made a video about a new calculus, "discrete calculus" Can you make a video on it in your style ?

[@isobarkley]

5:38 that straight line was hella impressive

[@thephysicistcuber175]

This is going to be interesting.

[@padaii]

The use of tangible items in this video is very engaging.

[@AnotherRoof]

Thanks, I try to use physical props in most of my videos!

[@TrimutiusToo]

Origami construction teased?

[@Hinotori_joj]

I'd like to see that video on the "twist" that you mention. Just before you got to that, when you said that the result was based on the construction axioms, it made me wonder how you could change those axioms, then you said that exact thing. More generally though, it makes me wonder how you could change other axioms of math to make impossibility results now possible.

[@MathHunter]

Yay my favorite math explainer 🎉

[@PMA_ReginaldBoscoG]

23:15 ah yes "indedendent"😂

[@japedr]

2:30 looking at the equations, wouldn't it be simpler to cancel out the 1/4 of Gamma with the 4's it is multiplied by? Or is it written like that because Gamma, defined like that, has a particular meaning that is not apparent to me?

[@TopRob1]

and subscribed, very interesting and entertaining

[@wormius51]

Nicely delivered explanation but this way above my level. I have no clue what's going on.

[@zanti4132]

It would be interesting to see a video that shows which additional polygons are possible if you have an angle trisector. I've seen in other sources that every regular polygon with 20 sides or less would be constructible *except* the 11-gon, but I can't claim to understand the math. I also know cube roots would be constructible with an angle trisector, hence it would be possible to do the doubling of a cube construction that is impossible with just compass and straightedge.

[@AnotherRoof]

As I said I might revisit this topic in the future but, briefly: angle trisection allows us to solve cubics to form field extensions. So to see if you can make an p-gon, subtract one from it, and if the resulting number is only made of 2s and 3s then it's constructible. A 7-gon is now possible as 6=2*3, similar for 19-gon as 18=2*3*3, but not an 11-gon as 10=2*5 which would require us to solve a quintic. Hope that helps!

[@Jojiho]

Amazing video! Compass and straight edge constructibility was always an interesting topic for me but I wasn't brave enough to dive into the details until now. This is a very nice introduction to Galois theory. I have a few questions about some parts of the video. You showed around 35:10 that the third degree polynomial whose cos(20) is a root is irreducible (over Q). With a few extra steps, you conclude that you cannot possibly construct cos(20) with a compass and straight edge because the degree of the corresponding field would be a multiple of three, compared to the fields of degree of powers of two that we can make with the basic compass and straight edge operations. However, wouldn't this require to show that the degree 3 polynomial is irreducible over any quadratic extension of Q (not just Q)? I.e cos(20) is not a root of any degree two polynomial whose coefficients are in Q(sqrt(r1),sqrt(r2),sqrt(r3)...,sqrt(rn)) ? I feel like this is a central argument to the reasoning, but I might be missing something. Otherwise, it is possible that cos(20) would be the solution to a degree 2 polynomial with coefficients in Q(sqrt(r1),sqrt(r2),sqrt(r3)...,sqrt(rn)), making the extension degree 2 and preventing us from drawing a conclusion on the degree alone. Similarly, when finding which n-gons are constructible, you demonstrate that some polynomial is the minimal polynomial of the nth-root of unity (again, over Q). Wouldn't this also require showing that these polynomials are irreducible over any quadratic extension of Q? But then you wouldn't be able to use Eisenstein criterion or Gauss Lemma (which work for polynomials in Q or Z). I hope this makes sense.

[@AnotherRoof]

You're right about showing the degree is 3 over *any* Q extended by square roots -- that's why we use the Tower Law. Maybe rewatch the section where we show exactly why the cube root of 2 isn't constructible. Hope that helps!

[@JM-us3fr]

Excellent proof! I wasn’t aware of the original proof; I had always seen it proven with Galois theory using the Galois group. And hear’s to Wanzel who died too young! 🍻

[@kennethgee2004]

slight correction on the trisecting an angle. it is possible to trisect some angles like 30 60 90 triangles. it is not possible to trisect an arbitrary angle using only compass and straight edge.

[@msclrhd]

Does this mean that the minimal extension to a ruler and protractor would be a device that constructs the p-th root of a number, where p is a prime? The reasoning is that the other roots could be constructed with combinations of prime roots. E.g. for the 4th root you can take the square root twice, for the 6th root you can take the square root then the cube root.

[@stanleydodds9]

For cube roots this is true (if by minimal, you mean the minimal degree algebraic field extension we could include), but after this, for degree 5 and higher, there are many many more algebraic numbers which are not otherwise constructible. This is because in general, a degree 5 or higher polynomial cannot be solved with just the field operations and nth roots. So for example, after adding the "take 5th roots" operation, you could add the "take the solution x of x^5 + x = N" operation. This is also a degree 5 extension for "most" N, just as the 5th root of N is a degree 5 extension for most N. But importantly, the algebraic numbers given by this extension are not covered by any larger nth roots, so you really do "need" this extension, or something equivalent, for completeness. And it is the smallest degree extension needed after all square roots and cube roots, alongside 5th roots and any other degree 5 extensions.

[@Ashbakhaaz]

Really really dense video! That was a tough but awesome watch. That being said this video inspired me a few unanswered questions about field extensions (and, more essentially, about the nature and families of irrationals)... It was shown cosine 20's minimal polynomial is degree 3, and that building it therefore requires a degree 3 extension of Q, just like building cube roots. But what does this similarity between cosine 20 and cube roots mean exactly? Let us assume we were working in Q + cube roots, then would cos 20 be constructible? In other words, does cos 20 belong to the same family of irrationals as cube roots? is cos 20 somehow related to cube roots? Or, in the contrary, are there several separate families of irrationals which all require DIFFERENT degree 3 extensions of Q? I'm very interested in all these questions, and more generally in what kinds of families of irrationals exist, as well as which minimal additional axioms could be added to make them constructible without making ALL algebraic numbers constructible at once?

[@AnotherRoof]

Yeah, you've got it right about "families" of irrationals! If we had a means of constructing cube roots, then we could essentially solve any cubic and construct degree-3 extensions, which opens the door for any irrational that can be obtained as a root of a cubic (like cos20). Thanks for watching and hope this helps!

[@Ashbakhaaz]

That's fascinating! Thanks a lot for the answer!

[@RigoVids]

14:30 my guess is that you intend to show that 2^k dne 3^k for any nonzero k, so having the ability to create any square roots means it’s impossible to have cube roots.

[@bronteman]

This deserves to be watched by Euclid..... and all the mathematicians that followed :)

[@user-ny5hh9wv3l]

On 8:27 , why is the cube root of 4 included in the field, but not cube root of 16, or 256, or 256^2 and so on?

[@AnotherRoof]

The cube root of 16 is just 2x(cube root of 2) and so on. They can all be written as a multiple of the cube roots of 2 and 4. Hope that helps!

[@rebokfleetfoot]

@@AnotherRoof the problem with the polygons is that they are almost all imperfect :)

[@kiro9291]

I now see why this took 2000 years

[@aoay]

Wow, this is such a fascinating (and engaging) explanation of complicated subject but it puts me in mind of a question I've had since high school -- What is so important about the straightedge and compass? Did the ancient Greeks think that rulers (i.e. straightedges that you can put markings on) were too practical or unclean or something? I get that understanding the limits of the set of constraints you are working under is important but how long has it been since the straightedge was at the cutting edge of technology? We have slide rules now; heck, we even have origami (which CAN be used to trisect angles and solve cubic equations, apparently). It seems strange to me that so much attention is paid to describing the capabilities of the straightedge and compass in this day and age. Is it just because its so simple and thus easy to use as an example? Also... "the equation of a line can always be written in the form y = mx + c" -- even vertical lines?

[@AnotherRoof]

Watch my first polygons video where I briefly discuss the origin of these types of construction (it's my latest video before this one). Just omit the y and the m for a vertical line to get 0 = x + c.

[@erikhaag4250]

I'm guessing the extension of Euclid's axioms you mentioned at the end are *complex* linkages?

[@ChristianTamblyn]

Here we go!!!

[@programmingpi314]

Introducing... (x+1)^(p-1)+(x+1)^(p-2)+...+(x+1)+1! Thank you for confirming that this was possible.

[@kordellcurl7559]

Another way would be to prove “What is the smallest angle that you can construct?” For example if you have 2 parallel lines divided into equal parts what would be the smallest angle if the lines go to infinity.

[@jadetermig2085]

You can bisect an arbitrary angle. Thus you can construct infinitely small angles through repeated bisection so I don't think there is a "smallest angle that you can construct".

[@Mark8v29]

I became lost very early on, but as I think I've commented before I enjoy basking in what I am unable to follow or understand. I find I quickly loose interest in things I believe I understand. So thank again for your amazing contribution to youtube!

[@AnotherRoof]

This is one of my densest videos yet. I wrote it in the hope that, even if people get lost, they appreciate the gist of why these constructions are impossible. And if people want to pause and make notes along the way then all the information is in the video for those sorts of completionists!

[@Mark8v29]

@@AnotherRoof Thanks. I very much enjoy listening to mathematicians talk about maths even if I do not follow much of it. It's a 'vibe' I enjoy. Yes, indeed, if maths were my sole interest I'd really get into attempts to understand each step of a proof. The most advanced I got was mathematics for a Physics degree. I found number theory too difficult at A-level but if I had been able to 'master' it I might have done mathematics at University. Proving sqrt 2 is irrational and a few other basic proofs was as far as I got but I did enjoy doing number theory until it crushed me.

[@petevenuti7355]

I've asked more than once before, about the existence and nature of a trinary operation. Like where an absolute value is a unitary operation, multiplication addition and such are binary operations. In my question, I was asking if there was any operation that took three values that couldn't be reduced to one made of components that used one and or two values... Though I didn't fully understand this video, I think it has helped me understand one of my questions. If a trinary operation existed, it's results could only be members of type degree 3 correct? As in don't exist in the number line as ones that could be made of degrees 1 and 2. Are there symbols, or generic transformations that are operations for working only with numbers that are constructions of degree 3 that don't exist in the set of degrees 1 and 2?? Or do there only exist formulas? Does there exist some kinematic device that operates on degree three numbers? Such as compass and ruler work on degree 2&4? Perhaps some tetrahedral monstrosity?

[@gustavonomegrande]

You can "construct" any polygon with a compass and straight edge, it's more of an approximation tho, Ian Mallet managed to get a method do do it, you need to mark a lenght from the circunference to the radius that measures to a [13/(2n+1)], n being the number of sides of your desired polygon, and use that lenght to mark arcs on the circunference, then you can just connect those new intersections, iIrc, the error is around 0.07 degrees.

[@Ashbakhaaz]

That's not a mathematical construction if it's an approximation. We're doing maths here, not engineering ;)

[@Ashbakhaaz]

@@azimuth4850 Yes, you are: mathematics is about the ideas and indeed uses an IDEAL ruler and an IDEAL compass to form IDEAL lines and dots. Mathematical lines and dots do not exist in real life, because real life lines and dots have a thickness, while mathematical ones do not. For the same reason, real life circles and triangles are NOT mathematical circles and triangles. When you are doing mathematics, you have to look beyond the appearances (imperfect lines, dots, circles, triangles) to reach for the immaterial ideas that make the thing interesting. If you don't do that, you miss the whole point.

[@Ashbakhaaz]

@@azimuth4850 The ruler and compass are equal here, and they are taken as axioms in this exercise because they were the simplest tools available to the ancient Greek, upon which the axioms of Euclid are based; it is amazing how far the Greeks were able to go while basing themselves solely on these two simple ideas - that of the ruler and that of the compass. This is an example of mathematical beauty. But it doesn't mean the protractor or other ideal tools are inferior in any way. You could just as well take them for axioms to get another form of mathematics.

[@Ashbakhaaz]

@@azimuth4850 No worry, and likewise! You CAN use a protractor, it would just be a different game with different rules!

[@Temari_Virus]

​@@azimuth4850 With an infinitely precise ruler, you can construct any real number by just measuring it. Similarly, with an infinitely precise protractor, you can construct the sine of any real angle and then scale it to get any real number. I think the main reason why we still study compass and straight edge constructions today is because the limitations result in some very interesting non-trivial results, whereas with rulers and protractors construction becomes trivial. I'd say those tools belong better in analytical geometry

[@djsmeguk]

Linear algebra truly is god tier

[@jumbopopcorn8979]

Thanks for creating such an informative and helpful video! I have a small question because I don’t understand one of the last conclusions. You said that if m and n gons are constructable, then an m*n gon is also constructable. Doesn’t this imply that if a p gon is constructable, then a p^2 gon must also be?

[@stanleydodds9]

No, he said that if m and n are *coprime*, and m and n gons are constructible, then the mn gon is constructible. Notice that he uses Bezout's identity, which only holds for coprime numbers. if you use p and p, they are not coprime, so Bezout doesn't hold (the smallest linear combination you can make is the gcd, which is p, not 1). Therefore overlapping a p-gon with another p-gon doesn't give you any new lengths. I mean, just think about it. If you line up the vertex of an equilateral triangle with another equilateral triangle inscribed in a circle, do you get a 9-gon? No, you just get 2 sets of overlapping vertices, nothing interesting.

[@codyhansen]

That statement had the qualification that m and n be coprime (i.e. don't share any prime factors). If your two factors are the same prime p, they share a prime factor, namely p, so a p^2-gon is not constructable.

[@jumbopopcorn8979]

Ahhh okay, thank you both so much!

[@volkerswille]

Are there any generic, yet still simple tools (like ruler and compass) that can be used to open up new possibilities?

[@AnotherRoof]

Check out the numberphile video on origami!

[@bartekabuz855]

We know how many constructable polygons there are, aleph0

[@jan-pi-ala-suli]

let’s go!

[@Jaylooker]

The minimal polynomial of the p^k-th roots of unity derived at 45:15 are Frobenius endomorphisms

[@2712animefreak]

So, what about squaring the circle? Are you going to tackle that one, too, or is it too far and we have to do more work before we can tackle it? My intuition tells me that it's related to the proof that pi is a transcendent number and therefore not a root of any polynomial.

[@gisopolis77]

I'm confused about one point in your arguments involving field extensions - what about constructible numbers with nested square roots in their expressions like sqrt(1 + sqrt(2))? It seems like these were skipped over when you were talking about how to determine the degree of a field extension

[@AnotherRoof]

Hello! This is counted when I talk about degree-4 extensions. Build a rational number r, then root it, so we have a degree 2 extension Q(sqrt(r)). Now build a number s *in this set*, then root it, and now I have a degree-4 extension of Q called Q(sqrt(r), sqrt(s)). (That is, assuming sqrt(s) isn't already a member of Q(sqrt(r)).) This s is any member of Q(sqrt(r)) so could be a rational or could be something of the form a+b.sqrt(r). That's how your get a nested root as you described. Hope that helps!

[@gisopolis77]

​@@AnotherRoof I see, for some reason I got confused that the section on extensions was only referring to taking the square root of rationals. Thank you for the clarification!

[@finnboltz]

2:29 Is that Heron's formula?

[@msolec2000]

17 is weird... It's also the number of different wallpapers that can exist (when boiled down to what symmetries they have)

[@publiconions6313]

Question for you guys out there who know more than me (im amateur to the extreme) -- say i could construct in 3D space with a 3D version of a compass (maybe like an inflatable sphere tool) could i then construct cube roots and associated trig stuff? If so, does that mean for every new prime number dimension, i would need a new tool (sorta like needing a new algebraic operation)? Aaaaand if all that, what does that mean?

[@willjohnston2959]

Nope -- The 3-d distance formula still simply involves a square root: sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).

[@sallylauper8222]

I must say, I've never thought of "constructing" a compass.

[@publiconions6313]

@@willjohnston2959 ohh!!.. yup, that makes sense. Cool

[@codahighland]

I have one really big question: How can you discuss the impossibility of doubling the cube in the framework of plane geometry? It seems like a strange non sequitur that it's part of the standard discussion of the subject.

[@codahighland]

EDIT: My previous post is still valid but I made a mistake in this one. Like... If you're allowing that you can construct a cube in the first place, then you can draw a line between the opposite vertices of a cube and thereby construct the cube root of 2.

[@willjohnston2959]

To construct cube with volume 2, what is needed is the ability to construct ³√2, which could act as the side length of the cube. That's what's impossible.

[@willjohnston2959]

The line joining opposite vertices of a unit cube can't be used, because it is ²√3 whereas we need ³√2. They are not the same numbers.

[@rebokfleetfoot]

@@willjohnston2959 agree

[@codahighland]

@@willjohnston2959 Oh right, that's a mistake on my part, but the core question still stands: How does 3D geometry even get into the discussion?

[@CaedmonOS]

Crazy thought, but what would happen if you allowed compass and straight edge to go into three dimensions?😅

[@sanctifiedcustoms]

bisect a given angle into trysect a angle bisect the angle 3 times giving you quarters bisect the angle between 1/4 and 1/2 giving you the 1/3 point of the angle long way around but can be done with a compass and a ruler

[@willjohnston2959]

That creates 3/8 of the original angle, not 1/3.

[@NitFlickwick]

As a late diagnosed autistic, Wadsel’s experience shares a lot of similarities with mine pre-diagnosis: shining brightly early, burning myself out, using something to take my mind off reality. I have no doubt I would have met my end in an opium den if I lived then. Not saying he was autistic, but autistic people existed before the 1900s.

[@AnotherRoof]

I feel that. I actually got quite emotional reading about him!

[@thumper8684]

Take an inverted prism. Fill that with water to a depth of one. Pour the contents into a jug. Fill the prism again to a depth of one. Add the water from the jug. The depth of water in the prism will equal the cube root of two. The ancient Greeks could have done that with ease.

[@debblez]

Wait when did we rule out that cbrt(2) can be written using nested square roots like a + sqrt(b+ sqrt(c))? Is there some obvious reason why ruling out linear combinations of square roots is sufficient that I’m not seeing? I dont see how we answered the question from 3:17

[@debblez]

ok it seems somebody else had this question i read your reply that makes sense I was completely misunderstanding the proof

[@AnotherRoof]

@@debblez Ah, glad you saw the other reply! I don't think I made this entirely clear so I think I'll make a pinned comment about this. Hope you enjoy the rest of the video!

[@diribigal]

8:38 West Side!

[@sallylauper8222]

Is doubling a cube possible in origomi?

[@japedr]

23:00 What happens for alpha equals pi, e or ln(2)? That is irrational but presumably does not have a minimal polynomial in Q, is Q(pi) considered of infinite degree perhaps?

[@stanleydodds9]

These are called transcendental extensions, while the finite degree extensions are called algebraic extensions. Note that in transcendental extensions, each number is expressed as some rational function of the transcendental number being added, rather than simply a polynomial. This is because in algebraic extensions, we can find the inverse in terms of higher powers using the minimal polynomial, but this is not the case for transcendental extensions which have no minimal polynomial. In general, any transcendental extension of F by a single trancendental alpha, that is, F(alpha), will be isomorphic to the field of fractions of F[X] (the polynomial ring of F), which is the field of rational functions over F. This field is denoted F(X), using X as the formal variable. Essentially, transcendental extensions are a little bit "boring" but also "not nice" if all you can see is their algebraic properties. They have no special algebraic relations; that's what it means to be transcendental. And therefore you can't distinguish between different transcendental extensions with algrebraic structure alone, like a field.

[@japedr]

@@stanleydodds9 Very interesting, thanks for writing the comment. I think it should have been more emphasized in the video the fact that inverses can be dealt with root rationalisation, something that cannot be done with trascendentals, as you point out.

[@theeyeguysBR]

Is this video colour corrected?

[@gedstrom]

I believe that you can't construct a cube root or trisect an angle because people much smarter than me have said so. But I have never been able to follow ANY of the proofs and I got an 'A' in geometry in school!

[@marshmellominiapple]

my brain hurt

[@maxsilvester1327]

I think you forgot sqrt(r/s) and sqrt(s/r) in the example at 7:15

[@Jojiho]

sqrt(r/s) is equal to sqrt(rs)/s (multiply both numerator and denominator by sqrt(s), which is of the form a*sqrt(rs) with a in Q so I think it's already covered by the product term

[@jamiemacleavy2482]

Could anyone explain why cube roots can't be built from rational combinations of 4th roots or higher? The proof uses field extensions of square roots but you can square root a square root etc

[@willjohnston2959]

No power of 2 is a multiple of 3.

[@rebokfleetfoot]

@@willjohnston2959 i really want to argue against that, but i can't

[@hassenwesleti5300]

Hi you have brillant presentation . Could you relate this to Galoi theory and explain it from skrach thanks

[@AnotherRoof]

Just wait till you watch the video!

[@InvaderMik]

y=mx +… c? Not b? Oh god it’s equivalent but my mind refuses to allow this!

[@diribigal]

Why can't we say that 8x^3-6x-1=0 is irreducible because of the following? "If it factored, the degrees would be 2+1 or 1+1+1, and so there would be a linear factor. But by the rational root theorem, there can't be one."

[@stanleydodds9]

Yes, this would be simpler, same as with the cube root of 2 proof (he could have used the rational root theorem on the linear factor). But I think using Eisenstein's criterion here sets up the general cases where it is important, and where the rational root theorem is not useful, for polynomials of degree higher than 3 (which don't necessarily have a linear factor).

[@AntonDiachuk]

Technically you can trisect an angle. First you have to bisect it and make a perpendicular to the bisector. Then build a right triangle using that perpendicular as a hypotenuse. We can build a 30 degrees angle so we can divide the right angle into 3 30 degrees angles. and then we draw lines from our angle to the points of intersections of the perpendicular and lines that divide the right angle. I know that I explain badly and maybe I'm wrong.

[@AnotherRoof]

Thanks for watching! Unfortunately it cannot be done and I show why in the video.

[@willjohnston2959]

The impossibility is about trisecting any arbitrary angle. There are definitely many specific angles that can be trisected, such as 180°, 135°, 90°.

[@AntonDiachuk]

@willjohnston2959 If you can trisect 90° angle, that means you can trisect a hypotenuse of the right triangle. So we can build an isosceles triangle with the given angle. The base of that isosceles triangle is the hypotenuse of another right triangle that can be trisected. So if we can trisect the hypotenuse of the right triangle, we can trisect every angle

[@AnotherRoof]

@@AntonDiachuk Unfortunately, it doesn't work. Trisecting the right angle in a right triangle doesn't actually trisect the hypotenuse. Even if you could trisect the hypotenuse, it wouldn't trisect the original angle. Lengths don't work that way, and school level trigonometry can be used to demonstrate this. If you're still unconvinced, get a ruler and protractor out and do it yourself. If you're *still* convinced, use this Geogebra page. The green angle is the one we want to trisect. Use the blue dots to change the angle. I built an isosceles triangle and adjoined a right triangle as you described, then trisected the right angle (which is possible). Notice that this doesn't actually trisect the hypotenuse and doesn't split the green angle into three equal parts: www.geogebra.org/calculator/cvhwydhb Trisecting the hypotenuse doesn't work either. If you're interested in why it is impossible to trisect an angle, I made a whole video about it that look hundreds of hours of my time -- I hope you enjoy it!

[@AntonDiachuk]

@AnotherRoof I see your point. The last thing I want to check when I get a chance to get to my computer if trisectors divide a base of isosceles triangles with same proportion or different. If different, then indeed it's impossible to trisect an angle

[@rawsugarrage]

Another Roof video? More like Another pRoof by contradiction video (I love you your drawings sustain me)