It Took 2137 Years to Solve This

[@AnotherRoof]

*COMMON COMMENTS AND CORRECTIONS!* 1. At 44:30 I say: "the next one is 257 which is one more than 256, 2^7" but of course 256 is 2^8. Terrible mistake on my part! 2. A few have asked whether I should be saying "primes of the form 2^(2^m)+1" when discussing Gauss's method. This is right but I deliberately omitted this to address it in the sequel -- I say that the method works on primes of the form 2^m+1 which is correct, it just happens that m must be a power of 2 for it to be prime. 3. 41:39 alpha_2 is incorrect: the coefficient of root(17) should be negative. 4. Regarding "transferring lengths" because the compass is supposed to "collapse" when picked up: Euclid proves (Book 1 Proposition 2) that you can move a line segment wherever you want. Originally I was going to show this, but I cut it to avoid an awkward complication so early in the video. It's proved so early in Elements that a collapsing compass can be treated as a non-collapsing one that it isn't worth worrying about! 5. Regarding the 15-gon, many have pointed out that since 2/5-1/3=1/15 we can just draw that arc and we're done. All who point this out are correct but I was presenting Euclid's proof. Like I said about the square, there are easier ways but that's how Euclid does it! 6. Regarding "2137": My patrons and I had *no idea* about the meme in Poland when we named the video! It's a fun coincidence -- the number comes from Elements being written ~300BCE and Wantzel publishing his paper in 1837. Obviously only an estimate as we don't know exactly when Elements was written!

[@samueldeandrade8535]

Ah not terrible mistake at all.

[@jeremy.N]

Isnt it actually all primes of the form 2^2^m + 1 aka the fermat primes? In the video you just say 2^m + 1

[@FDGuerin]

@@jeremy.N For 2^m + 1 to be prime, m must itself be a power of 2. So both "primes of the form 2^m + 1" and "primes of the form 2^2^m + 1" describe the set of Fermat primes.

[@samueldeandrade8535]

​@@jeremy.N if 2ⁿ+1 is prime, then n=2^k, for some k. If n had any odd factor, then 2ⁿ+1 could be factored using the generalization of x³+1 = (x+1)(x²-x+1) x⁵+1 = (x+1)(x⁴-x³+x²-x+1) etc ... So, saying "p prime, p=2ⁿ+1" is the same as "p prime, p=2^{2^k}+1"

[@pierrebaillargeon9531]

That is so entirely unacceptable that I won't unsubscribe merely only once, but 257 times, which will bring me back to being subscribed. Unless I misunderstood something....

[@KatMistberg]

It surprised me how long that problem took to solve, didn't realize you were THAT old

[@Gordy-io8sb]

What do you think about Cartesian point algebras?

[@apokalypthoapokalypsys9573]

​@@Gordy-io8sbhow does that have anything to do with OP's joke?

[@theflaggeddragon9472]

@@Gordy-io8sb nerd

[@chazcampos1258]

And that's another reason to stay active in mathematics: it keeps you young.

[@TymexComputing]

Us youtube that old already ? Some problems are unsolvable

[@SKO_EN]

2137 is a very special number indeed

[@cheeseplated]

37 appears yet again...

[@Adomas_B]

❤🇵🇱🤍

[@bogdanieczezbyszka6538]

Ah, yes. The yellow number.

[@jakubosadnik2693]

​@@cheeseplated 2137 is not about 37. It's an hour that only Polish people would understand

[@WrednyBananPL]

2137 mentioned pope summonned

[@EebstertheGreat]

So many Poles in chat, it's like the ℘-function up in here.

[@mr.duckie._.]

me when 2137

[@SuperMarioOddity]

I was gonna make a joke about |, but | realised it's called a pipe not a pole

[@salicaguillotines]

​@@SuperMarioOdditymeh close enough

[@mr.duckie._.]

@norbertnaszydowski4789 and 2763 is a bfdi fan spawner

[@dariuszjozef7654]

Frr

[@setonix9151]

JPII Moment

[@doorotabanasik1929]

Fr

[@bozydarziemniak1853]

Jean Paul Secondo GMD

[@other_paradox8437]

Ah yes, 2137. Number of the beast.

[@d3fau1thmph]

Jeszcze jak!

[@remigiusznowak7277]

O Panie

[@zyczowiek4783]

@norbertnaszydowski4789 rel

[@natan500honk]

xd

[@sigghum]

Nie ma przypadków, są tylko znaki

[@ukaszb9223]

John Paul II joined the chat

[@awesomegraczgie2131]

at 2137 he actually left the chat, RIP Juan Pablo II

[@pippicalzecorte27]

Cloning?

[@VieneLea]

Imagine my disappointment when I clicked on the video an realised the 2137 number was chosen just randomly, without acknowledging it's holiness

[@samueldeandrade8535]

How do you onoe 2137 was chosen randomly?

[@VieneLea]

@@samueldeandrade8535 I guess it's not random per se, but it just isn't related to, y'know, what the 2137 is usually connected with

[@pje_]

​@@VieneLeato the death time of JP II

[@AnotherRoof]

My patrons and I had no idea about the 2137 meme when we were drafting titles! It is kinda random but the number stems from Elements being written ~300BCE and Wantzel's paper published in 1837. Obviously we don't know the exact date for Elements and the problem likely existed before then but we thought an exact number sounded more fun than "over 2000 years" or something!

[@inthefade]

Now I'm curious

[@deldrinov]

I'm imagining Euler going back in time and explaining complex numbers to Euclid and only hearing "wow, I never thought about it this way, this is so wrong yet so intuitive"

[@LeoStaley]

Euclid would have rejected outright on philosophic basis.

[@ianmoore5502]

Would he have said "there IS a way, but it sux" or just ignored its viability altogether? Lol​@LeoStaley

[@ItsPForPea]

Knowing what Pythagoras did, I wouldn't want to go back in time and correct the ancient mathematicians.

[@eneaganh6319]

​@@ItsPForPeanot like he drowned someone for saying √2 is irrational

[@HighKingTurgon]

"so wrong but so intuitive" is, like, all math after the 17th century xD

[@thetree7403]

Jan Papież mentioned!!!

[@Blablabla-ol2tr]

I didn't expected the Pope Number in non-polish video

[@alexterra2626]

Watching this at 21:37

[@amadeosendiulo2137]

O Panie…

[@Foxy_8796]

​@@amadeosendiulo2137 to ty na mnie spojrzałeś... Dokańczajcie

[@tenkanałzdech]

Rzułty panie módl się za nami

[@mironhunia300]

Another Roof has managed to harness the power of polish memes to bring in more people to learn about math.

[@AnotherRoof]

Fun fact, my Patrons and I had no idea about the Polish meme when we named the video!

[@aykarain]

what was the meme?

[@AnotherRoof]

@@aykarain I've had to research this following the reaction to this video, and here is my understanding: Pope John Paul II was fantatically admired in Poland by the "older generation". When he died, his death was reported to have taken place at the time 21:37. The time became sacred to those who deified him, with some singing religious songs at that time. The "younger generation", tired of the obsession with John Paul II, started using the number in mockery and singing other songs at that time; it then became a meme due to internet. Don't quote me on any of this but that's what I've managed to ascertain!

[@icyrain123]

@@AnotherRoof as Polish I can confirm it. This religious song we are singing at 21:37 is "Barka" (Barge), Pope's favourite song.

[@lapiscarrot]

46:41 "You may now perform a poly-gone" that pun coming back at the end cracked me up

[@NonTwinBrothers]

damn, spoilers :(

[@NotSomeJustinWithoutAMoustache]

Nooo I got spoiled! It was my fault for reading comments before the video ended, but still, dang it.

[@Ноунеймбезгалочки-м7ч]

I JUST STARTED YOU MADMAN

[@lapiscarrot]

@@Ноунеймбезгалочки-м7ч i am chaotic evil

[@Ноунеймбезгалочки-м7ч]

you are evil evil@@lapiscarrot

[@tiagogarcia4900]

I love how elementary these videos are. Anyone could watch them, and 47 minutes is a reasonable amount in our day of 4 hour video essays.

[@samueldeandrade8535]

Brasileiro?

[@tiagogarcia4900]

@@samueldeandrade8535 Mexicano, mi padre ama Portugal.

[@samueldeandrade8535]

@@tiagogarcia4900 teu nome parece brasileiro demais. Hahahaha. Grande abraço.

[@BrianWoodruff-Jr]

Elementary? I must be preschool as I was lost after the straight edge/compass portion. What's the part "a teenager can understand"?

[@____________________________a]

@@BrianWoodruff-JrIt's pretty trivial if you've ever taken geometry in school, but other than that, this video does require some basic understanding of axioms and some general knowledge

[@tylerduncan5908]

16:34 funny to me that diophantus accepted that rational numbers exist, and we use his name to refer to equations with integer solutions.

[@gene51231356]

An important note about compass-and-straightedge construction: the compass "collapses" as soon as its fixed point is lifted, so you cannot use it to compare two distances by moving it around.

[@semicolumnn]

Note however that a collapsing compass can be used to construct anything that a non-collapsing compass can construct, and they are equivalent.

[@AnotherRoof]

@@semicolumnn Thanks for adding this -- I cut a part that deals with this because the non-collapsing compass being equivalent basically means nothing is lost by using the compass as I do in the video so it's more convenient and accessible to things this way :)

[@ingiford175]

Euclid does spend Book 1; Prop 2 proving that you can 'move' the compass around, but he did assume it was a collapsing compass, and showed that you could treat it as non collapsing

[@methatis3013]

​@@ingiford175 how would you prove that? My idea is, once you have a desired distance, and you want to translate it to a random point, you would draw a paralelogram whose vertices are 2 original ends of the segment and the 3rd being the desired point. From there, you just use the compass to get the desired length. Does Euclid's proof go similarly?

[@pdorism]

​@@methatis3013 Euclid's proof is based on a triangle because it's very early in his book. Note that the moved segment doesn't have to be parallel to the original one

[@caspermadlener4191]

I love this problem! I was obsessed with this when I was fifteen. I actually proved Wantzel's part myself, basically by inventing the Galois theory of unit roots, which is simpler than general Galois theory, since you already know all the relations, and therefore also the symmetry. I also calculated the sine of all multiples of 3° by hand. I don't know whether this is accurate, but it was a lot of effort, so here is my (fixed) list: sin(0°)=cos(90°)=0 sin(3°)=cos(87°)=(2√(5+√5)-2√(15+3√5)+√30+√10-√6-√2)/16 sin(6°)=cos(84°)=(√(30-6√5)-1-√5)/8 sin(9°)=cos(81°)=(√10+√2-2√(5-√5))/8 sin(12°)=cos(78°)=(√(10+2√5)+√3-√15)/8 sin(15°)=cos(75°)=(√6-√2)/4 sin(18°)=cos(72°)=(√5-1)/4 sin(21°)=cos(69°)=(2√(15-3√5)+2√(5-√5)-√30+√10-√6+√2)/16 sin(24°)=cos(66°)=(√15+√3-√(10-2√5))/8 sin(27°)=cos(63°)=(2√(5+√5)-√10+√2)/8 sin(30°)=cos(60°)=1/2 sin(33°)=cos(57°)=(2√(15+3√5)-2√(5+√5)+√30+√10-√6-√2)/16 sin(36°)=cos(54°)=√(10-2√5)/4 sin(39°)=cos(51°)=(2√(5-√5)-2√(15-3√5)+√2+√6+√10+√30)/16 sin(42°)=cos(48°)=(√(30+6√5)-√5+1)/8 sin(45°)=cos(45°)=√2/2 sin(48°)=cos(42°)=(√(10+2√5)-√3+√15)/8 sin(51°)=cos(39°)=(2√(15-3√5)+2√(5-√5)+√30-√10+√6-√2)/16 sin(54°)=cos(36°)=(√5+1)/4 sin(57°)=cos(33°)=(2√(5+√5)+2√(15+3√5)-√30+√10+√6-√2)/16 sin(60°)=cos(30°)=√3/2 sin(63°)=cos(27°)=(2√(5+√5)+√10-√2)/8 sin(66°)=cos(24°)=(√(30-6√5)+1+√5)/8 sin(69°)=cos(21°)=(2√(15-3√5)-2√(5-√5)+√30+√10+√6+√2)/16 sin(72°)=cos(18°)=√(10+2√5)/4 sin(75°)=cos(15°)=(√6+√2)/4 sin(78°)=cos(12°)=(√(30+6√5)+√5-1)/8 sin(81°)=cos(9°)=(2√(5-√5)+√2+√10)/8 sin(84°)=cos(6°)=(√3+√15+√(10-2√5))/8 sin(87°)=cos(3°)=(2√(15+3√5)+2√(5+√5)+√30-√10-√6+√2)/16 sin(90°)=cos(0°)=1

[@narfharder]

That list is impressive, and is surely worth a reply. I spent 5-10 minutes with notepad and Windows' calculator sanity checking these by value, and found two mere typos. This analysis was exhaustive, there are no more mistakes. # an extra ) at the end sin(27°)=cos(63°)=(2√(5+√5)-√10+√2) } /8 # a missing ) after 6√5 sin(78°)=cos(12°)=(√(30+6√5 } +√5-1)/8 I wonder if there is some way to derive a single formula, with various √3 √5 √15 etc throughout, where you can just plug in the angle in degrees and it reduces to one on this list.

[@pauselab5569]

you actually calculated all that? I tried to do the same with roots of unity got to 11, lost patience with 13 and stopped because I knew that it could be done with a computer anyways...

[@samueldeandrade8535]

Oh my Euler ... this is insane ... insanely awesome.

[@samueldeandrade8535]

​@@narfharder double "oh my Euler"! One person makes a list of sines of multiples of 3° and someone else checks it? Who are you two? Math Batman and Math Superman? What's going on here?

[@jacksonsmith2955]

Couldn't you also use the triple angle formula to get sin and cos of all integer degrees from this?

[@НейтХиггер]

Pan kiedyś stanął nad brzegiem Szukał ludzi gotowych pójść za Nim By łowić serca słów Bożych prawdą O Panie, to Ty na mnie spojrzałeś Twoje usta dziś wyrzekły me imię Swoją barkę pozostawiam na brzegu Razem z Tobą nowy zacznę dziś łów Jestem ubogim człowiekiem Moim skarbem są ręce gotowe Do pracy z Tobą i czyste serce O Panie, to Ty na mnie spojrzałeś Twoje usta dziś wyrzekły me imię Swoją barkę pozostawiam na brzegu Razem z Tobą nowy zacznę dziś łów Dziś wyjedziemy już razem Łowić serca na morzach dusz ludzkich Twej prawdy siecią i słowem życia O Panie, to Ty na mnie spojrzałeś Twoje usta dziś wyrzekły me imię Swoją barkę pozostawiam na brzegu Razem z Tobą nowy zacznę dziś łów

[@marekwnek5797]

OOO Paaanieeeeee! To ty na mnie spojrzaaaaaałeeeś!

[@Grzmichuj2137]

OOOOO PAAAANIEEEEE

[@amadeosendiulo2137]

TO TY NA MNIE SPOJRZAŁEŚ

[@tenkanałzdech]

twoje usta

[@marekwnek5797]

dziś wyrzekły me imię

[@luisemiliocastilloncaracas8447]

Only 12K views for a video with this quality of content is outrageous, great work.

[@MarcelGeba-t9p]

It's been 12 hours bro give it some time, I do gotta agree that this KZbinr is really slept on

[@AnotherRoof]

@@MarcelGeba-t9p Tell your friends!

[@ssl3546]

This is one of the best undergrad-level math channels I've found. The issue a lot run into is the presenter goes too slow or goes on lengthy tangents and then I stop paying attention and then 30 seconds later I have no idea what's going on. Or the presenter lacks dynamicism. You do a fine job.

[@TheOriginalSnial]

hmmm, but this is a geometry video, he's supposed to go off on a tangent ;-) !

[@salicaguillotines]

​@@TheOriginalSnialdo we at least get to eat cos law?

[@Wielorybkek]

jan paweł drugi konstruował małe wielokąty

[@maklovitz]

Po maturze chodziliśmy mierzyć kąty

[@zecuse]

7:45 More simply, since the regular triangle and regular pentagon share a vertex on the circle they will necessarily share all of their own vertices with the 15-gon that shares a vertex with both shapes. So, the distance between the triangle's 2 other vertices and their nearest pentagon vertices will be 1/15 of the circumference of the circle. This construction works for any 2 distinct primes. The opposite edge of the smaller prime polygon from the shared vertex will have those 2 vertices closest to 2 vertices of the larger prime polygon. They're closest to the vertices that go towards the opposite point on the circle (180°) of the shared vertex. No need to subtract.

[@cogwheel42]

8:00 - The bisection seems unnecessary. The arc from the base of the triangle to the base of the pentagon is already (2/5 - 1/3) = (6/15 - 5/15) = 1/15

[@SKO_EN]

That's what I thought too!

[@vytah]

In fact, picking any arc between vertices is unnecessary. Just take the 1/3 arc from the triangle and draw it from every vertex of the pentagon, and by Chinese Remainder Theorem you'll hit every vertex of the 15-gon.

[@AnotherRoof]

It's like I said about the square -- there are simpler ways but I was presenting how Euclid did it!

[@lucahermann3040]

1:45 Actually, duplicating lengths isn't something you're allowed to do additionally, but something you're already able to do by following the other rules, drawing exactly six circles and two straight lines (apart from the ones you already have and the one you want). let's say you have three points •a, •b, •c, and you want to copy length a-b. You can draw a circle C1 around •a trough •c and circle C2 around •c through •a. Then you draw a straight line L1 through a •a and •c and a straight line L2 through the two points where your circles C1 and C2 meet. Now the point •m where the two straight lines meet is in the middle between •a and •c. Then you draw a circle C3 around •m through •a and •c. Now you only need three more circles: First one circle C4 around •a through •b, which meets the straight line L1 in two points. Draw a circle C5 around •m through one of those two points. C5 also meets L1 in another point •d. Now you can draw a circle C6 around •c through •d. C6 and C4 have the same radius a-b, and there you have it.

[@foley2663]

toż to papieska liczba!

[@ThisIsX2_0]

Anyone from Poland? ;p

[@Adomas_B]

PRAWDA JEST TYLKO JEDNA 📢 ‼❗ 💪🇵🇱💪POLSKA GUROM💪🇵🇱💪 P O L A N D B A L L 🇲🇨🇵🇱 ‼ 🦅 ORZEŁ JEST POLSKI 🦅 ‼ ✝ JAN PAWEŁ 2 JEDYNY PAPIEŻ ✝ POLSKA CHRYSTUSEM NARODÓW ✝ 🇵🇱🌍 🚔JP🚔JP🚔JP🚔 🤍 LWÓW JEST POLSKI 🇺🇦🇵🇱 WILNO JEST POLSKIE 🇱🇹🇵🇱 MIŃSK JEST POLSKI 🇧🇾🇵🇱 MOSKWA JEST POLSKA 🇷🇺🇵🇱 ‼ 🇵🇱MIĘDZYMORZE🇵🇱 ‼❗🟥⬜ 303 🟥⬜ JESZCZE POLSKA NIE ZGINĘŁA 🟥⬜ POLAND IS NOT YET LOST 🟥⬜ NIE BRAŁA UDZIAŁU W KONFLIKCIE W CZECHOSŁOWACJI ❌🇨🇿🇸🇰❌ 🟥⬜ 500+ 🟥⬜ TYLKO POLSKI WĘGIEL 🟥⬜ ❤🇵🇱🤍

[@Secretgeek2012]

Yes, there's lots of people from Poland, it's quite a big country. 👍

[@Piooreck]

Me

[@3Max]

Thank you so much for this video! Loved every bit of it. This is the first time I've seen constructible numbers in a way that clicked for me, and it's so fascinating! I also really appreciate how your videos leave some of the imperfections with correction overlays, it makes them feel more human and approachable. Also the "algebra autopilot" on the blackboard was a great effect. P.S. Is it a coincidence that Gauss was born in "17"77?

[@DiegoTuzzolo]

nice job on explaining ring theory without so much technicality!! loved it well done

[@zakolache4490]

I hope Editing Alex & Future Matt can get together to have a drink and complain about their present-time versions of themselves sometime!

[@helhel9753]

21:37

[@chinesegovernment4395]

You should play "barka" as background music and eat kremówki

[@tenkanałzdech]

Swoją baarkę pozostawiam na brzeeegu

[@MarlexBlank]

Your videos are so well made. Great topic, great explanation. Thanks

[@ddichny]

That was a magnificent video. At first I thought a 47-minute math video would be plodding or needlessly complex, but it was paced perfectly and covered an amazing amount of material clearly and without glossing over anything nor making any unnecessary side tangents. Bravo.

[@allieindigo]

See you on the 5th of June 😢

[@OakQueso]

That’s my birthday

[@Zosso-1618]

I think I might just read Wantzel himself instead of wait haha

[@justghostie4948]

I don't usually comment much, but oh my god dude this channel is seriously underrated. I was stunned to see only 51K subs! The clarity in explanation is perfect and the humor is just right! You'll make it big one day, I can see you among the ranks of 2b1b and standupmaths

[@AnotherRoof]

Thanks so much! Comments like this make my day. I don't think I'll ever be that big but I'm still eager to grow the channel so please share my videos if you can :D

[@justghostie4948]

@@AnotherRoof You'll make it dude! Just keep at it. Your embrace of long form content fills a gap that the bigger channels don't come close to. Remember me when the algorithm inevitably works in your favor 🙏🏻

[@MrSubstanz]

Not fully comprehending every single thing you're doing, but this is the most rigorous math class I had in decades and I enjoyed it!

[@mallow4715]

its kinda funny that the first thing we did in the "use a compass and straight edge (not a ruler)" game was create a ruler

[@vytah]

You cannot make an actual full-fledged ruler (neusis) with only a compass and straight edge. Neusis constructions unlock many more constructive numbers, you can do cube roots and construct any regular polygon up to 22 sides.

[@JalebJay]

Just happen to run into this video after my Abstract class covered it only a week ago. Good to see an edited version of it to rewatch.

[@DjVortex-w]

Fun fact: If we allow folding the paper onto which we are drawing with the straightedge and compass, it actually enlarges the set of constructs that can be constructed with these three tools (ie. adding paper folding to the other two allows constructing mathematical shapes that are not possible with straightedge and compass alone). Folding would have been available to Euclid, but I suppose he didn't think of it.

[@arden-chan]

I find it quite demeaning when mathematicians and theoreticians say “I leave it as a simple exercise to the reader”.

[@Geek37664]

I’ve never understood why angle trisection fell out of favor after the Greek golden age. Archimedes discovered a simple method of trisection and we laud him as much as Euclid, if not more. That simple deviation from the rule, marking the straightedge allows for the nonagon to be constructed. There are many other examples made by other mathematicians from that period, but that severe reluctance to deviate from the compass and unmarked straightedge really robbed math students of a richer education for millennia.

[@TheLuckySpades]

Gauss was a madman

[@mateuszszurpicki6931]

PAPIEŻ POLAK MENTIONED

[@tinkeringtim7999]

This presentation is absolutely brilliant. I think this is more like how geometry and numbers should be taught in school.

[@astrovation3281]

Actually really appreciate the suggestion for a break, I'm not such a great mathematician, as my experience thusfar is highschool mathematics and some specific deeper ventures. Sometimes with these videos I lose track with what is happening like midway through and just stare at my screen for the rest of it pretty much, this helped with letting it process a bit more.

[@Kaneeren]

Yep, it's always nice to give yourself some time to "digest" the content. It has happened to me so many times spending hours trying to understand a specific topic, taking a break, and then understanding it almost instantly

[@kayleighlehrman9566]

Regular pentagon is absolutely my favourite straight edge and compass construction. Something seemingly so simple, and yet simultaneously not immediately almost obvious.

[@matiasgarciacasas558]

Great video! My favourite so far I think.

[@obiwanpez]

8:00 - Or, draw a regular triangle through each of the five vertices of the pentagon. Since the LCM of 3 & 5 is 15, we will have 15 evenly spaced points.

[@Tsudico]

I wonder if there is an easier way? The second point of the pentagon going clockwise from the top is 144° around the circle and the triangle's first point is 120° around the circle with the difference being 24° which is 1/15th a complete circle. So is it always the case that if you plot two shapes with a given number of sides that the smallest difference between two of their points would equal the angle for the polygon that their sides multiply to make? If it was a square instead of a triangle, the closest points would be at 90° and 72° with a difference of 18° which is 1/20th a circle.

[@vytah]

@@Tsudico If and only if they're coprime. Then (assuming a p-gon and a q-gon) picking the closest vertices is like solving the equation mp-nq=1 modulo pq, which by Chinese Remainder Theorem is always solvable if and only if p and q are coprime.

[@petrosthegoober]

I love the stack of axiom bricks propping up everything so so much.

[@isobarkley]

ive never heard a youtube educator say "okay, time for a break!" and honestly? i really appreciate it!!!! i never really stop and ponder unless i am going to write a comment. thank you

[@norude]

30:45 You can actually get a simple, mathematically sound proof from the rotational symmetry: I've learned it in the context of vectors, so: If O is the midpoint of a regular n-gon and A_i are the vertices, consider the vector X=A_1+A_2+...A_n Now rotate the whole picture around O in such a way, that A_0 goes to A_1, A_1 goes to A_2 and so on. The image hasn't changed, and that means, that if we rotate X by some angle, we get X. Thus X is the zero-vector

[@Kaneeren]

wow, so simple but so clever at the same time

[@Danylux]

im taking a course on field theory and galois theory and this video was really good explaining all the stuff i have learned so far

[@nosy-cat]

Thanks for another great video! And on a topic I was already interested in. I hope you don't feel bad about the mistakes, they're entertaining and relatable.

[@rudyj8948]

13:14 There is such an interesting parallel between constructing numbers out of geometry and the construction of numbers from set theory like one does in real analysis

[@gonzalovegassanchez-ferrer6712]

Wow. This is a fantastic work! So much explained in a totally accessible way. Congratulations!

[@WeyounSix]

Though I'm not very good at math myself, I think it's so cool how it's DIRECTLY built upon THOUSANDS of years of collaborative work, and problems that last that long as well. Its so cool

[@JTolmar]

29:28 more like Gausskeeping

[@JeraWolfe]

You just blew my mind... I love your channel. I fell in love with geometry all over again... Thank you for making these videos. Keep it up! Really, watershed life moment... Eureka moment. Thank you for that.

[@AnotherRoof]

Welcome!

[@ThierryLalinne]

Fantastic! Crystal clear explanations as always. Thank you for all the work you do. 👍

[@tails55]

Getting a bit confused around 9:20-9:48. Fourth root of 2 (or sqrt(sqrt(2))) *is* constructible because you can construct square roots of numbers using the geometric mean theorem and sqrt(2) is obviously constructible. Is the definition of rationality mentioned here tied specifically to unit squares or to what we now call constructibility in general?

[@AnotherRoof]

I glossed over this because it isn't important to understanding the video; I just included it for historical context. If you're interested, Euclid defines lengths as "commensurable" if there is a third length which can measure both. E.g. 5 and 6 are commensurable as 1 can measure both. Similarly, squares are commensurable if a third square can measure both. So squares of area 5 and 6 are commensurable as a square of area 1 can measure both. Euclid then defines rational by saying: declare some length to be "rational". Then any length which is commensurable with the chosen length, or whose square is commensurable with the chosen length's square, is also rational. So say our chosen length is 1. Then take length x. x is called "rational" if either x or x^2 is commensurable with 1. It's a nightmare and I'm glad the definition shifted across history! Hope that helps and thanks for watching!

[@Ma_X64]

It's interesting that in English the word "compass" means also a tool to draw circles. In Russian we call it circule (lat.circulus).

[@lagomoof]

It's an abbreviation of "pair of compasses". Technically each leg is a compass, which point in their own direction, just like the arrow on a magnetic compass. There was a time that a student would be told off or punished by their teacher for calling the device "a compass", but these days, the teacher generally offers a weary correction or doesn't bother. It is a very minor thing to be angry about, after all.

[@Ma_X64]

@@lagomoofThanks for your reply. Interesting historical background.

[@Legion19999]

In polish, it's "cyrkiel"

[@joelproko]

I heard that if you add folding (origami-style) to compass and straightedge, cube roots become constructible. Do you plan to do a video on that too? Also, does it also make heptagons constructible?

[@SangaPommike]

At 37:36 you seem to have missed an x in the middle term. Edit: Same at 38:48 for both (though you seem to have noticed those)

[@cecilponsaing2749]

Fantastic detail and clarity of presentation. I just subscribed.

[@AnotherRoof]

Welcome!

[@modolief]

0:59 "From the Greek 'poly' meaning 'many' and 'gone' meaning 'leave' a 'polygon' describes the common audience reaction to a mathematician telling jokes." Subscribed.

[@nowonda1984]

Cool video, informative and entertaining. One small slip - the primes appearing in the product @45:39 are Fermat primes, which are of the form 2^(2^m)+1, instead of just 2^m+1. Apparently there's even a theorem that 2^m+1 is prime if and only if m itself is a power of 2. I looked up more about constructible polygons after watching your video and noticed the mistake. "Coincidentally", 3 and 5 are also Fermat primes.

[@AnotherRoof]

Thanks for watching, and well spotted! It's actually not a mistake -- Gauss's method works for p prime where p is of the form 2^m+1. It just so happens that 2^m+1 is prime *only if* m is also a power of 2. But it's "only if", not "if and only if", as 2^32 + 1 is not prime. I'm saving this discussion for the sequel video though! However I did misspeak at 44:30 where I say that 257 is one more than 2^7, because of course it's one more than 2^8 >_

[@angeldude101]

@@AnotherRoof Well 32 = 2^5, which certainly _isn't_ 2^2^m, so that explains pretty clearly why 2^32 + 1 isn't prime if, to be prime, it needs to be 2^2^m + 1 rather than just 2^m + 1.

[@joeybf]

​@angeldude101 32 isn't of the form 2^2^m, but 2^32 is. So we wouldn't expect 32+1 to be prime, but it would be reasonable to expect 2^32+1 to be

[@angeldude101]

@@joeybf Oh. Never mind. (Then again, 2^32 itself is so large - about 4 billion - that I didn't even consider that it's what we'd actually be talking about.)

[@samueldeandrade8535]

"... and noticed the mistake". Not a mistake.

[@QuantenMagier]

8:00 I did it differently; I saw there was already a small difference between 2/5th and 1/3rd and therefore calculated 2/5-1/3=1/15 which directly gives the right distance; no halving steps needed.

[@f1r3fox235]

At 7:50 we could just take the distance between 1/3 and 2/5 which gives 6/15 - 5/15 = 1/15, which is already there, so we don't need to bisect the part between 1/5 and 1/3

[@ryforg]

I can’t believe they needed an entire book on how to draw a triangle 2000 years sgo

[@samueldeandrade8535]

Hahahaha.

[@mjmeans7983]

Since it only works for primes, can this method be used to find arbitrary length primes?

[@keithwinget6521]

Wow, I really like how you explain this stuff. Brings me back to first learning much of it in high school. I use it all the time in my game development, since I deal with physics, targeting, procedural animation, etc... It's just really good to get a refresher of how it all used to be done (and is hopefully still taught in classrooms).

[@e1woqf]

5:22 How do you construct a triangle like this?

[@AnotherRoof]

It's here but it's quite messy and technical! aleph0.clarku.edu/~djoyce/elements/bookIV/propIV10.html

[@e1woqf]

@@AnotherRoof Thanks!

[@harrymoschops]

Liked & subbed! Fantastic job working us through the beautiful history of mathematics

[@AnotherRoof]

Welcome!

[@michaelniederer2831]

I'm going to watch this again, and try to follow along, again. Great video! Thanks!

[@trejkaz]

11:17 it's amazing how much one line can hurt, if someone hearing it is in the appropriate context for it to hurt.

[@elf835]

Amazing video can’t wait for the next part

[@catcatcatcatcatcatcatcatcatca]

I know you did include the references, but I really wish you had given some links for reading about the compass and ruler techniques. Maybe do a short video of them? After all, replicating angles at arbitury location is a great life skill everyone should learn.

[@AnotherRoof]

Thanks for watching! I've just updated the description with the exact links to the constructions that I gloss over. The links are to David Joyce's adaptation of Elements which is freely available.

[@STEAMerBear]

8:21 When you bisected 2/15 you introduced some unnecessary construction. Isn’t it better to recognize the adjacent angle (below the equilateral triangle, CCW to the 2/15 you began with) is already a perfect 1/15 and just sitting there? (2/5 - 1/3 = 6/15 - 5/15 = 1/15) Did you do it this way because it’s what Euclid did? (Asking for a friend 😉)

[@carlosgaspar8447]

at 27:00 with a circle radius 1, wouldn't the imaginary axis be labelled i*2 and -i*2...?

[@AnotherRoof]

The number i is a distance of 1 from the origin, and so is -i, so they are the numbers on a circle of radius 1. The numbers 2i and -2i would be on a circle of radius 2. Hope that helps and thanks for watching!

[@carlosgaspar8447]

@@AnotherRoof sorry, but i meant i-squared.

[@AnotherRoof]

@@carlosgaspar8447 I see! Then no because i^2 = -1 is a real number and so belongs on the horizontal real axis. The imaginary axis has strictly imaginary numbers.

[@carlosgaspar8447]

@@AnotherRoof yet, it is one real unit along that circle. i'll have difficulty getting my brain acknowledge that.

[@AnotherRoof]

@@carlosgaspar8447 I think of imaginary numbers as just a different "direction", just as we think of negative numbers as the real numbers but increasing in a different direction. We have the positive real direction, the negative real, the positive imaginary, and the negative imaginary. Maybe that helps?

[@seneca983]

31:30 One more way you can see this is that sum of the roots of a polynomial is equal to the coefficient of the second highest term multiplied by -1.

[@user-xy5yg6se1k]

37:39 where does the x for B2 come from?

[@AnotherRoof]

@@user-xy5yg6se1k There isn't an x there...?

[@user-xy5yg6se1k]

@@AnotherRoof sorry i meant B1 😅 but still x² and B2 stay the same but B1 gets an x for the quadratic equation which wasn't there before maybe you just forgot to put it? (considering 38:54 XD)

[@AnotherRoof]

@@user-xy5yg6se1k Ahhh I see. Yeah I make the same mistake with b1 as I did later! Apologies

[@user-xy5yg6se1k]

@@AnotherRoof no problem, love your channel, you deserve way more attention

[@jursamaj]

7:55 Easier to do 2/5 - 1/3 = 1/15. That's the little arc near the bottom of the circle, and doesn't need bisecting.

[@TheArtOfBeingANerd]

Yay I found another high quality math channel I can binge until 4 am and then not have any more math videos to watch until I find another

[@AnotherRoof]

Welcome! Enjoy the binge :)

[@ontheballcity71]

That was superb; very enjoyable.

[@RFC3514]

11:46 - Well, Collins Dictionary agrees with you: *_constructable_* _in British English_ _adjective_ _a variant spelling of constructible_ So do Wiktionary, Webster's, etc..

[@DocKobryn]

Cool video. You actually made me look up Pierre Wantzel to find out when the next video is coming out. 😎 And no. I'm not telling! Looking forward to it!

[@Heisenberg2097]

This is a great video in more than one way! 1. You put so much dedication into it 2. It showed how much I really don't care too much about math beyond entertainment 3. The real wonders of the universe don't come in numbers. Numbers just sometimes match to fit a subset.

[@DirtShaker]

35:30 did you mean "sum" instead of "product" when you said "so this product is just -1"?

[@AnotherRoof]

I meant product because I'm saying that -1 is the product of beta_1 and beta_3. Thanks for watching!

[@ИванСкворцов-п7о]

Great video! There also is a next (and in a way the final) step in this problem (called Galois theory) and it finally gives a way to prove inability to construct something. As you have said multiple times -- the only constructable numbers have such form (built up from basic operations and square roots), and it it relatively easy to prove (starting with Q, each constructed point lies in a quadratic field extension, which is to say it is a root of a quadratic equation with coefs in previous field). The issue is to show that numbers like cubic root of 2 can't be written in such form and it wasn't clear before Galois. (if it has such form than it lies in some extension K over Q. Build by the series of the quadratic extensions it's degree ([K / Q]) has to be some power of 2. Our assumption is that Q(2^(1/3)) is a subextension so 2^n = [K / Q] = [K / Q(2^(1/3))] * [Q(2^(1/3)) / Q] = 3 * [K / Q(2^(1/3))] which leads to a contradiction) This exact theory was used to show that doubling the cube and trisecting the angle can't be solved and that the general polynomial of degree 5 or greater can't be solved in radicals. Though it is much more complicated than Gaussian construction and in a way leading to the basic algebraic geometry

[@blango-san]

38:07 damn, where does Alex get this neat fluorescent chalk?

[@DeclanMBrennan]

Oops I left my parrot's cage open ... This was a fantastic video. I knew about Gauss's 17gon but the nitty gritty of why was fascinating. Would love to see your take on regular polyhedra perhaps involving quaternions? I quite like Gauss's suggestion for calling *i* the "lateral unit". Or maybe the orthogonal unit would work. No chance of changing it now, so we can only imagine.

[@DMSG1981]

@15:08 Note to Editing Alex: Presenting Alex started measuring at about 1mm, so 26mm seems to me like an accurate reading

[@Patrik6920]

One shall never under estimate humanitys ability to seemingly find meaning and pattern in random occurances...

[@ghildiyalsanjay]

You explain it wonderfully. good job bro..

[@joeeeee8738]

Excellently explained, as usual !!

[@darthrainbows]

When I first took a geometry course as a kid, the "you can't trisect an angle with a compass and straight edge" fact was handed on down, with no explanation for why (which makes sense in retrospect, there's no way any of us [barring any prodigies out there] would have been capable of comprehending the proof at that age). But I was a stubborn kid who liked nothing more than doing what I was told I could not, so I wasted countless hours trying to trisect angles. Sadly, I was not able to overturn proven mathematics.

[@GamingRN001]

Bruh.. I can't stop laughing when he put down the board which briefly showed a few bricks and said "Let me brick it down."😂😂

[@Kavukamari]

I'm curious what tool you could add to the compass and straight edge that would add the most utility while still being just as simple as those two, I've always wondered about things like folding the paper, or what other techniques we could use to construct more shapes

[@Kavukamari]

and i don't mean just a computer, i mean a physical tool akin to the others

[@patriciageo1618]

Thank you!! Practicing this stuff may help me in my search for a continued fraction that exactly equals a cube root!

[@willjohnston2959]

en.wikipedia.org/wiki/Hermite%27s_problem