It Took 2137 Years to Solve This
⬣ LINKS ⬣
⬡ PATREON: / anotherroof
⬡ CHANNEL: / anotherroof
⬡ WEBSITE: anotherroof.top
⬡ SUBREDDIT: / anotherroof
⬡ TWITCH: / anotherroof
⬣ ABOUT ⬣
Despite being easy to state, the problem of constructing regular polygons confounded the Ancient Greeks. It took over 2000 years to make progress, and in this video we’ll trace a path through history to learn what innovations allowed more polygons to be constructed.
⬣ TIMESTAMPS ⬣
00:00 - Introduction
01:47 - Ancient Constructions
08:14 - What the Ancient Greeks Lacked
11:20 - From Geometry to Numbers
16:28 - From Numbers to Equations
21:58 - From Equations to the Complex Plane
31:48 - Gaussian Periods
36:10 - Final Construction
⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!
⬣ REFERENCES ⬣
Euclid's constructions mentioned at 3:50:
Perpendicular lines: aleph0.clarku.edu/~djoyce/elem...
Duplicate angles: aleph0.clarku.edu/~djoyce/elem...
Alternate angles: aleph0.clarku.edu/~djoyce/java...
Parallel lines: aleph0.clarku.edu/~djoyce/elem...
Parallelogram properties: aleph0.clarku.edu/~djoyce/elem...
The Thirteen Books of Euclid’s Elements. T. L. Heath (1908)
J. Derbyshire: "Unknown Quantity: A Real and Imaginary History of Algebra" Joseph Henry Press (2006)
Al-Kamil treats irrational quantities as numbers in their own right
H. Selin, U. D'Ambrosio: "Mathematics Across Cultures: The History of Non-Western Mathematics" Springer (2000)
Al-Mahani’s definition of rational and irrational
M. Galina: "The theory of quadratic irrationals in medieval Oriental mathematics" Annals of the New York Academy of Sciences 500 (1987) 253-277.
Al-Khwaizmi quadratic equations
Al-Jabr - Al Khwarizmi
Sridhara’s method
D. E. Smith: “History of Mathematics” Vol 2 Dover (1925)
Tombstone story
C. W. Dunnington: "Carl Friedrich Gauss: Titan of Science" Hafner Publishing (1955)
⬣ CREDITS ⬣
Intro music by Tobias Voigt. Other music by Danijel Zambo and Apex Music.
Image Credits
Euclid
cdn.britannica.com/46/8446-05...
Arithmetica
upload.wikimedia.org/wikipedi...
Al-Jabr
upload.wikimedia.org/wikipedi...
Gauss
upload.wikimedia.org/wikipedi...
Heptadecagon Construction
upload.wikimedia.org/wikipedi...
Gauss Tombstone
upload.wikimedia.org/wikipedi...
*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!
Ah not terrible mistake at all.
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
@@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.
@@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"
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....
Ah yes, 2137. Number of the beast.
Jeszcze jak!
O Panie
tylko po to tu przyszłem
@@norbertnaszydowski4789 rel
xd
2137 is a very special number indeed
37 appears yet again...
❤🇵🇱🤍
Ah, yes. The yellow number.
@@cheeseplated 2137 is not about 37. It's an hour that only Polish people would understand
2137 mentioned pope summonned
It surprised me how long that problem took to solve, didn't realize you were THAT old
What do you think about Cartesian point algebras?
@@Gordy-io8sbhow does that have anything to do with OP's joke?
@@Gordy-io8sb nerd
And that's another reason to stay active in mathematics: it keeps you young.
Us youtube that old already ? Some problems are unsolvable
John Paul II joined the chat
at 2137 he actually left the chat, RIP Juan Pablo II
Cloning?
So many Poles in chat, it's like the ℘-function up in here.
me when 2137
I was gonna make a joke about |, but | realised it's called a pipe not a pole
@@SuperMarioOdditymeh close enough
2137 is a Pole spawner
@@norbertnaszydowski4789 and 2763 is a bfdi fan spawner
JPII Moment
Fr
Jean Paul Secondo GMD
Imagine my disappointment when I clicked on the video an realised the 2137 number was chosen just randomly, without acknowledging it's holiness
How do you onoe 2137 was chosen randomly?
@@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
@@VieneLeato the death time of JP II
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!
Now I'm curious
Jan Papież mentioned!!!
Watching this at 21:37
O Panie…
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"
Euclid would have rejected outright on philosophic basis.
Would he have said "there IS a way, but it sux" or just ignored its viability altogether? Lol@LeoStaley
Knowing what Pythagoras did, I wouldn't want to go back in time and correct the ancient mathematicians.
@@ItsPForPeanot like he drowned someone for saying √2 is irrational
"so wrong but so intuitive" is, like, all math after the 17th century xD
I didn't expected the Pope Number in non-polish video
You should play "barka" as background music and eat kremówki
Swoją baarkę pozostawiam na brzeeegu
Another Roof has managed to harness the power of polish memes to bring in more people to learn about math.
Fun fact, my Patrons and I had no idea about the Polish meme when we named the video!
what was the meme?
@@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!
@@AnotherRoof as Polish I can confirm it. This religious song we are singing at 21:37 is "Barka" (Barge), Pope's favourite song.
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
OOO Paaanieeeeee! To ty na mnie spojrzaaaaaałeeeś!
OOOOO PAAAANIEEEEE
TO TY NA MNIE SPOJRZAŁEŚ
twoje usta
dziś wyrzekły me imię
toż to papieska liczba!
16:34 funny to me that diophantus accepted that rational numbers exist, and we use his name to refer to equations with integer solutions.
46:41 "You may now perform a poly-gone" that pun coming back at the end cracked me up
damn, spoilers :(
Nooo I got spoiled! It was my fault for reading comments before the video ended, but still, dang it.
Only 12K views for a video with this quality of content is outrageous, great work.
It's been 12 hours bro give it some time, I do gotta agree that this KZheadr is really slept on
@@user-hy8ju1yn5g Tell your friends!
Anyone from Poland? ;p
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 🟥⬜ ❤🇵🇱🤍
Yes, there's lots of people from Poland, it's quite a big country. 👍
Me
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.
Note however that a collapsing compass can be used to construct anything that a non-collapsing compass can construct, and they are equivalent.
@@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 :)
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
@@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?
@@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
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.
Brasileiro?
@@samueldeandrade8535 Mexicano, mi padre ama Portugal.
@@tiagogarcia4900 teu nome parece brasileiro demais. Hahahaha. Grande abraço.
Elementary? I must be preschool as I was lost after the straight edge/compass portion. What's the part "a teenager can understand"?
@@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
2137 hehe
See you on the 5th of June 😢
That’s my birthday
I think I might just read Wantzel himself instead of wait haha
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
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.
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...
Oh my Euler ... this is insane ... insanely awesome.
@@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?
Couldn't you also use the triple angle formula to get sin and cos of all integer degrees from this?
PAPIEŻ POLAK MENTIONED
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.
Gauss was a madman
21:37
jan paweł drugi konstruował małe wielokąty
Po maturze chodziliśmy mierzyć kąty
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.
hmmm, but this is a geometry video, he's supposed to go off on a tangent ;-) !
@@TheOriginalSnialdo we at least get to eat cos law?
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?
Your videos are so well made. Great topic, great explanation. Thanks
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
nice job on explaining ring theory without so much technicality!! loved it well done
Fantastic! Crystal clear explanations as always. Thank you for all the work you do. 👍
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.
29:28 more like Gausskeeping
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.
Liked & subbed! Fantastic job working us through the beautiful history of mathematics
Welcome!
I like adding another operation, folding. Even papyri can be folded.
I hope Editing Alex & Future Matt can get together to have a drink and complain about their present-time versions of themselves sometime!
2137 🇵🇱🇵🇱🇵🇱
Please explain the 2137, Poland, and JP II connection.
@@bethhentges21:37 is the hour when pope john Paul the second died, john Paul the second was polish.
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
That's what I thought too!
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.
It's like I said about the square -- there are simpler ways but I was presenting how Euclid did it!
It's interesting that in English the word "compass" means also a tool to draw circles. In Russian we call it circule (lat.circulus).
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.
@@lagomoofThanks for your reply. Interesting historical background.
In polish, it's "cyrkiel"
Wow. This is a fantastic work! So much explained in a totally accessible way. Congratulations!
Great video! My favourite so far I think.
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.
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.
Fantastic detail and clarity of presentation. I just subscribed.
Welcome!
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.
I'm going to watch this again, and try to follow along, again. Great video! Thanks!
Pan Kiedyś stanął nad brzegiem
Szukał ludzi Gotowych pójść za Nim
By łowić serca Słów bożych prawdą
O panie To ty na mnie spojrzałeś
Twooooje uóóóóósta dziś wyyyrzeeekły meiiiiiiiiimieee
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.
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!
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
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.
Welcome!
the best I have seen in a long time. Thank you sooo much
Regular pentagon is absolutely my favourite straight edge and compass construction. Something seemingly so simple, and yet simultaneously not immediately almost obvious.
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
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
@@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 🙏🏻
Fantastic work !! Love it!!
I love the stack of axiom bricks propping up everything so so much.
Excellently explained, as usual !!
This presentation is absolutely brilliant. I think this is more like how geometry and numbers should be taught in school.
Amazing video can’t wait for the next part
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.
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
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!
That was superb; very enjoyable.
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
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
wow, so simple but so clever at the same time
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).
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.
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.
@@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.
great video as always!!!
Thank you once again Alex for the amazing video. Gauss-Wantzel theorem might be my all time favorite theorem. I always loved constructing with straight edge and compass, only side of geometry that I find really interesting, and because of that and it’s nice connection to algebra and number theory, I’ve known the statement of the theorem by heart. That leads to a funny story where I was asked on a geometry test whether the angles of 2 and 3 degrees were constructible. We haven’t seen gauss-wantzel in class, but that was my way out of it (2º is not because the 180-gon isn’t , as 3 is because the 120-gon is , 120 being 8*3*5). As we haven’t seen the theorem in class the teacher assigned me the mark given I made a presentation to the class on it. Which I did and loved it. But all the explanations I found online relied on Galois theory, only saying briefly that Gauss used some other method relying on Gaussian periods, which I didn’t have enough time on my hands to understand properly (neither Galois theory 😅, but being and advanced topic the teacher oversaw that ) Understanding Gauss method gave me the most profound joy and I’m so thankful for that On a side note : in Brazil we call the quadratic formula Bhaskara’s formula, which is another ancient Indian mathematician. Surprised to see that not even in India the formula is known by that name. As far as I know we call it that way because in the early XX century there were really few elementary math textbooks and the one that was used across the country called it so
Ok... so next video is June 5th?
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)
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.
Some amazing constructions. Never knew about the square root one.
Excellent visuals like always 👌
Fantastic video, thanks!!
Fascinating. I think it would take me many days or weeks or longer to be able to fully understand this in order to reproduce this. It's strange that whereas I think nothing of forgetting a simple fact such as the name of someone or a word for something, I feel anxiety over the fact I have forgotten virtually all the maths and science I learned at school and university by the use it or loose it principle. Alas the human mind, or my mind, is not capable of retaining things it does not regularly use! And yet I still retain a fascination for what I have forgotten and what I never knew. Thanks for the video.
Superb video, thank you!
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.
Yooo this actually went quite in depth and I could follow it relatively smoothly! I love some in depth KZhead mathematics!
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?
Honestly this is a wonderful video - thanks so much
Wow, that was a story! Almost have a poly-headache 😂 My compliments: world class quality!
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.
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 >_
@@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.
@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
@@joeybeauvais-feisthauer3137 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.)
"... and noticed the mistake". Not a mistake.
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
I love these Videos so much!!!!!
Need a Short version of this
Super cool :))) Thanks for making z-transform prime time!
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.
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
2/5 - 1/3 = 1/15, so you already have a 1/15 arc between the third vertex of the pentagon and second vertex of the triangle (assuming the shared point is the first).
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.
Great video :) For anyone into compass and straight-edge construction, there's an awesome mobile puzzle game called Euclidea which involves exactly that.
Someone showed me the pentagon version of this heptadecagon argument about 10 years ago and it immediately became one of my favorite pieces of math. The pentagon one is much simpler -- you only have two Gaussian periods of length 2, so you end up with the quadratic (x - z - z^4)*(x - z^2 - z^3) = x^2 + x - 1 = 0, from which you can show that cos(2π/5) = (sqrt(5)-1)/4. I experimented with higher numbers a little and found out that by pairing 7th roots of unity with their conjugates, you can construct a cubic equation with rational coefficients that has cos(2π/7) as a root. But I didn't realize I was so close to the heptadecagon proof!
It's lovely -- my original script had the pentagon version as a "trial run" but I cut it for time. I also experimented with 7 and 9 to get a feel for why it fails in those cases! My sequel video explored this and will be released in about 12 days
29:27 or better, Gauskeeping...
Thanks for the introduction joke. You’ve had me there.🤣