All the possible polygons!

[@dannygjk]

lol I love how for 17 you have to go through a complex series of steps to be able to construct a line segment which is then used to draw the 17-agon. Insane.

[@ffggddss]

Yes, that's a heptadecagon. The rule for constructibility involves the Fermat primes, which are of the form 2^(2ᵏ) + 1 when that's a prime. The only such numbers currently known to be prime are when k = 0, 1, 2, 3, 4: {3, 5, 17, 257, 65,537} [Curiously, you can also get 2 if you're willing to set k = -∞.] If you think *that's* insane, check out the construction of the 257-gon! And, yes, there was a guy who spent ("wasted" according to the writer of the piece I read about this) 10+ years of his life composing the construction of the 65,537-gon!!!

[@dannygjk]

ffggddss lol Obviously he was insane and probably in even worse mental condition after he finished.

[@1anya7d]

Where can we find that article?

[@iCore7Gaming]

@@ffggddss Yeah no one cares

[@kvdrr]

@@iCore7Gaming stop projecting your pathetic ignorance onto everyone

[@6infinity8]

A few minutes later... Sides : +∞ * draws a red circle *

[@gabrieldavis3071]

infinity is 5 minutes! .-.

[@gabrieldavis3071]

casually skips 9 and 11

[@gabrieldavis3071]

and 13 and 14

[@ffggddss]

+ Gabriel Davis Well then, he also skipped 7, didn't he? OK, he doesn't come out and say so, but what he's covering are all the regular polygons that are *constructible* by classical methods (straightedge & compass) - up to n=51, anyway. The next one would have been 60, then 64, 68, 80, 85, 96, ... The rule for generating these possible n's is: n = {2ᵏ (for k=1, 2, ...) or 3 or 5 or 17 or 257 or 65,537} or any product of two or more of these factors - repeated *odd* factors not allowed (2ᵏ, when k > 1, is a repeated "2," but *is* allowed). So the outcasts from that list are {7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, ...} In case you're wondering where the (odd) numbers in that list of factors come from, they are what are known as Fermat Primes (see Wikipedia for a nice treatment of these), which are all numbers of the form F[k] = 2^(2ᵏ) + 1 that are prime. F[0] = 2¹ + 1 = 3 (P) F[1] = 2² + 1 = 5 (P) F[2] = 2⁴ + 1 = 17 (P) F[3] = 2⁸ + 1 = 257 (P) F[4] = 2¹⁶ + 1 = 65,537 (P) and above this point, all of them are either known to be composite, or are too big to determine whether they are. F[5] through F[11] are all known composite, as are several larger ones. F[5] = 2³² + 1 = 4,294,967,297 = 641·6,700,417 F[6] = 2⁶⁴ + 1 = 18,446,744,073,709,551,617 = 274,177·67,280,421,310,721 F[7] = 2¹²⁸ + 1 = forget it!

[@siddharthbhaiya1]

I love geometry

[@Kirkaman_hex]

*Dad* : you better go to sleep, you got school tomorrow *Me at 4 am* :

[@gamerfuntombs3169]

its 4:21 a.m so take that dad

[@erc778]

@@gamerfuntombs3169you gotta put the phone down man

[@tomjerry84]

Which alien figured out 17? I'm gonna kick this guy

[@joeyhardin5903]

It was Gauss but he's dead now

[@theviniso]

You can always kick his corpse.

[@leonebersbacher3609]

@@theviniso calm down there Satan

[@theshuman100]

@@theviniso talk about beating a dead gauss

[@martinluther123]

@@joeyhardin5903 most underrated mathematician/intellectual ever. People talk all the times about Einstein and Newton, a few people talk about Leibnitz, and Euler but in my opinion Gauss was the true mad-man.

[@someguy1478]

2:30 How the hell did Gauss figure this out

[@SmileyMPV]

Come on Euler, out of all people, you must be able to understand.

[@brienmaybe.4415]

Sit around with a pen paper a compass and a circle and think. Now go my child and find another infinite! There's infinite more to find!

[@dandanthedandan7558]

@@brienmaybe.4415 *draws a circle* I HAVE MADE INFINITEGON!!!

[@theM4R4T]

Reverse engineering maybe

[@flamematerial02]

@@theM4R4T no haha

[@nigovorilo]

3:08 so u done all of this to draw a line? Edit: i know that this line is needed to draw a perfect shape, it is just a joke.

[@Klblaz]

A line with exact length for this to work.

[@lordlix6483]

That line was a work of Gauss...

[@Zimodo]

A very precise line

[@forloop7713]

A curvy line

[@idigger3545]

Ze D

[@SamirHajili]

17. easy to memorize :)

[@mwwve]

I’m reply

[@sadcribby-RePENT]

Who's mama?

[@Nikos-iy7hi]

MaThS aRe EaZy!

[@denyraw]

Can't believe how gauss managed to pull that off

[@diceLibrarian]

Yay, Gauss

[@krinkovakwarfare]

The method to make the 17 sided polygon is pretty cool. All those lines and connection to make just one particular line segment in order to draw with compass for all the 17 side is really radical.

[@skiggywiggy8386]

It’s always darn seven and it’s impossible primeness. I mean they got 17 for crying out loud!

[@Jkirek_]

11 feels the same way about it

[@mrpellagra2730]

Theres also the ducking 256-gon and 65537-gon

[@nilsschenkel7149]

There is a proximation method credited to Albrecht Dürer you can find online

[@davinchristino]

7, the number which ruins geometry in every way possible

[@asheep7797]

@@mrpellagra2730257-gon.

[@aldoaldoz]

Go to the italian version of wikipedia, and look for "poligono". There is a table with the number of sides and italian polygon names - click on those names: you'll find an animation that explains how to draw all the polygons up to 20 sides. (of course, 7, 9, 11, 13, 14, 18 and 19 sides polygons are not exact)

[@CraftingTableMC]

Ok

[@chrisbova9686]

You still around? Whats the name of this program? Nice work. I'm very impressed.

[@aldoaldoz]

@@chrisbova9686 These frames have been individually created by old programs I wrote myself - so ancient they used screen 12, so can't run any more.

[@chrisbova9686]

@@aldoaldoz Im looking to have someone help me do just that, create a program that intersects different radii. Have you continued to do more with geometry?

[@69420guyhaha]

@@aldoaldoz hey how to make the 17-gon (heptadecagon) it was like humanly impossible

[@gavinmcginness3423]

The process for 10 seems more simple than 5. Wouldn’t you be able to use the 10 method to create a pentagon by connecting every other point?

[@leonebersbacher3609]

Technically yes, it would be more practical in reality but the the method shown flr 5 is the one with the least amount of steps

[@marksabol8758]

For the square, why not just connect the four intersections of the radii and the cicle?

[@aldoaldoz]

The way you suggest gives a perfect construction too - but it liked to draw a square with the sides parallel to the axes.

[@rosiefay7283]

In that case, just bisect one of the 90-degree angles to get a 45-degree diagonal. That locates one of the vertices of the square you want. You already have the side-length. Alternatively, turn the paper 45 degrees.

[@tinybabybread]

@@rosiefay7283 That's exactly what the video did. It bisected the upper right angle first and the rest followed. In order to bisect accurately you'd need to do what it did.

[@MeiZhang-q5k]

Though the octagon was off-axis? Wow

[@zazinjozaza6193]

Aye bruh, how come this video isn't infinately long?

[@kirbs0001]

Because this is only the polygons that can be drawn with a line segment and a compass

[@jakebrowning2373]

@DRAGEN RAID eh it was a weak r/woooosh

[@fensti7917]

@@kirbs0001 well you can split all of them into more and more halfs so yk

[@thatoneguy9582]

@@fensti7917 non-trivial polygons, i guess

[@Altinget]

Maybe because it gets so complicated that math science haven't found out more so far. 🤓

[@MAGNETO-i1i]

2:30 - when you invite friends for pizza

[@Taib-Atte]

Underrated

[@VayBeeqqwdqwd]

most underrated comment I've ever seen

[@QuotientGD]

Actually, we call it a 'straight edge' instead of a 'ruler'. The main point of this video is to construct right polygons WITHOUT markings on the ruler, aka straight edge.

[@Taffytyrann]

More ridiculous is, that Carl Friedrich Gauss invented this very method (to contruct a regular heptadecagon using only a straightedge and a compass) in 1796, at the age of 19. I'm 19 this year, and still struggling with high-school algebra...

[@aldoaldoz]

The heptagon can't be EXACTLY drawn. Nevertheless there is a simple method that gives an error of about 1/10° on the central angle

[@Landis963]

And what of the nonagon? The triskaidecagon? The other skipped polygons?

[@aldoaldoz]

@@Landis963 7-gon and 9-gon are missing because these polygons CAN'T be exactly drawn by means of compass and strightedge only. Here only exact polygon constructions are shown.

[@Landis963]

@@aldoaldoz And the 13-gon has the same problem, I take it? And by extension, do all polygons with prime numbers of sides have that difficulty?

[@kohwenxu]

@@Landis963No! 257 and 65537 gons are possible to do with compass and straightedge, though I doubt anyone would want to do them.

[@aldoaldoz]

You can also start from a given point (let's call it the origin): draw a line passing through it; with a compass determine two points on this line, at the same distance from the origin; build an equilateral triangle that has these two points as a base, and connect the third vertex with the origin: you get the perpendicular line.

[@brenki]

Nice work, thank you! Please note that aproximate poligons: 7, 9, 11, 13-gon can also be constructed by using only a straightedge and a compass.

[@azimovwatts6425]

how

[@ras662]

7: commons.m.wikimedia.org/wiki/File:Approximated_Heptagon_Inscribed_in_a_Circle.gif 9: commons.m.wikimedia.org/wiki/File:Approximated_Nonagon_Inscribed_in_a_Circle.gif 11: commons.m.wikimedia.org/wiki/File:Approximated_Hendecagon_Inscribed_in_a_Circle.gif 13: commons.m.wikimedia.org/wiki/File:Approximated_Tridecagon_Inscribed_in_a_Circle.gif

[@MatheMagiX]

If you read above, you will see those are approximated. It's actually IMPOSSIBLE to construct a regular 7, 9, 11, 13-gon using a straight edge and compass.

[@samisezgin]

After 10 years, i found the best practical video on KZbin. Thanks for your animation. Cheers!

[@GewelReal]

Needs some smooth jazz

[@four40four8]

Clear straight forward explanation of a complex issue. Best video on this subject I have seen. The amount of work that went into this is awesome. Thank you.

[@adamxue6096]

Oh nice, somehow youtube felt like I need to watch this now. That said, this was incredibly well done, especially that 17gon... Christ... that was SOMETHING...

[@arthurreitz9540]

I hate when it happens

[@therealzilch]

Very nice, especially 17! Is there an elegant mathematical reason that 17 is the highest prime polygon possible? Grazie! cheers from rainy Vienna, Scott

[@bernardz2002]

Scott Wallace 65537 gon is constructible as it is a fermat prime. Search for it (Especially Numberphile.)

[@therealzilch]

Wow, very cool. Thanks, Bernard! One reason I asked is that I do polymetric music, and 17 is one of my main rhythms. Check it out if you're interested- soundcloud.com/scott-wallace-189088488/lydia-ventures-into-the-jungle Cheers from cold Vienna, Scott

[@sage5296]

I think someone found the method for 257 but I don’t think the method for the 6553$ is known lel

[@asheep7797]

@@sage5296it is known, and people have constructed the 65537-gon, but only digitally.

[@m8sonmiller]

But will it blend?

[@Jackcabbit]

After fiddling around it seems I've found a way to do 11 sides: Make the bottom edge of that equilateral triangle as done at :20. Set the compass to the distance between the top of the circle and that line. Make arcs to each side using that distance. Where those arcs intersect the main circle is where you put the turning point of the next compass pass. Repeat until you get 11 sides.

[@kattejuice]

17 was sick!!!

[@dannystoll84]

That about the 257- and 65537-gons?

[@jaoreir]

Well y'know. You can just draw a circle.

[@atrumluminarium]

The more the sides the more negligable is the difference from the polygon to a circle so there's really no point unless you have a paper the size of a building :/

[@dannystoll84]

Fair enough, but these constructions were never studied because they were useful (we'd use a protractor then). People studied them because they were mathematically interesting, and the breakthrough that proved the constructibility of the 257- and 65537-gons also laid the ground for huge advancements in algebra - ultimately leading to proofs of the unsolvability of the quintic polynomial and the three-body problem.

[@tetraspacewest]

Danny Are those the only two provably constructable polygons that large?

[@tetraspacewest]

Thomas Jones *that aren't just trivial 2^n-tuples of other polygons

[@nagysamuel4963]

dude u have no idea how much did u help me am in don bosco institute of technology and i have a geometry exam next Thursday i looked every where and i couldnt find a video as clear as yours

[@nibbletrinnal2289]

What you studied: Hexagon Whats actually on the test: 17-gon

[@TunaBear64]

Is a Heptadecagon

[@dannygjk]

@@TunaBear64 ok what is the term for a 2d shape with 97 sides? Also is there a system by which I can correctly name any number of sides without a lot of memorization?

[@Rando2101]

​@@dannygjkThere's a system, but I don't really remember it

[@TaranVaranYT]

@@dannygjk Enneacontakaiheptagon is a name for a 97 sided polygon, I don't think there is a unique system out there though.

[@dannygjk]

@@TaranVaranYT Ridiculously complicated. Why don't we just use numbers for the number of sides?

[@learnerlearns]

Very interesting and a good demonstration! I had never seen the 17 sided version before!

[@aldoaldoz]

Actually I wondered to start from a circle, without knowing its center. But the way to determine the center if a circle is too complicated: so I started from a line and a circle centered on a "undeclared" point of the line (see below); I find the vertices of two equilateral triangles (one above and the other below the line) and connect them with a line, which is perpendicular to the initial one. The intersection of the the two lines is the center of the circle.

[@2t22tornadosiren]

i wish we did this in my geometry class

[@tsoneberry]

2:30 You can see visually how crazy and ingenious Gauss is.

[@PekaCheeki]

you don't know why you're watching this, you're not interested and you have better things to do. stop procrastinating.

[@sylicone6952]

So what are the better things that I have to do?

[@PekaCheeki]

@@sylicone6952 figuring that out is probably the first thing you have to do

[@katetoolboxbishop6676]

Get out of my mind

[@bernd8608]

Sometimes we have to relax.

[@esotericbeep5923]

Needed to hear this

[@tbone28steak]

Im not gonna lie. Im more than confused but thats a cool way of creating polygons and increasing faces

[@alexin6647]

It hurts my brain

[@imtrunchbullicious1862]

When the art teacher says we decide what to draw and the nerd kid is there

[@metamochibear]

that's... that's a lot of work just for a shape

[@MAGNETO-i1i]

I just google circle.jpng

[@FireyDeath4]

At some point the corners get so blunt and hard to distinguish from apeirogonal circulation that you just stop caring I wonder why it is that people care, really. Probably just the completionism and challenges

[@notnotandrew]

There technically may be an infinite number of constructible polygons, even excluding the trivial ones that are the result of combinations of previous ones. We won't really know until someone solves the question of whether or not there are infinitely many fermat primes. I guess I'd better get to work on that.

[@harryandruschak2843]

+Andrew Becker only five Fermet Primes are known. To date, no one has been able to find a sixth one, or prove that a sixth one cannot exist.

[@notnotandrew]

Yes, exactly.

[@rtxagent6303]

You forgot the Hexecontapentakischiliapentakosiotriakontaheptagon (Or the 65537-gon) This polygon is constructable and has a prime number of sides.

[@aldoaldoz]

I created an animated gif showing the first step of the construction, look at: it.wikipedia.org/wiki/65537-gono#Idea_della_costruzione In addition, here is the complete construction of the 257-gon: it.wikipedia.org/wiki/257-gono#Costruzione

[@blinkstanks]

2:28 How to split a pizza between friends

[@GodMineptas]

Who hás 17 friends?

[@aldoaldoz]

@Mrhellotinfish: the radius of that circle is not important, since it is used to find some angles needed to draw the fist black stright line.

[@forg7864]

0:11 to 0:25 *illuminati confirmed*

[@KnakuanaRka]

Some info from Wikipedia on constructible polygons: When doing a straightedge and compasses construction, you’re effectively drawing lines and circles based off other points in the constructions and seeing where they intersect; describing these lines and circles in terms of equations gives quadratic at most, so the results come from solving a bunch of quadratic-at-worst equations. That means any constructive numbers must be writable in terms of integers, +-x/, and square roots. For constructing polygons with n sides, proving it constructible just requires showing that cos(2pi/n) is constructible (for example, for a pentagon, cos(2pi/5) is (r5-1)/4); this is the distance from one corner of the polygon to the line between the center and an adjacent corner, which lets you get one side, and repeat to make the rest. It was proven that this only happens if n is the product of any number of 2s and distinct Fermat primes (primes of the form 2^(2^k) + 1). As it turns out, the only Fermat primes known are for k=0-4, giving 3, 5, 17, 257, and 65537. Each of these is constructible (the first three are shown, the latter 2 even more complex), and you can construct a product of them by drawing the two and using the distance between corners (for example, the construction for 15 shown here is simpler, but you can guarantee you can make it by drawing a triangle and pentagon with a shared corner in a circle, then the distance between a corner on each is a multiple of 1/15 the circumference you can use to build it out). And powers of 2 can be added to any of these (or the degenerate 2-gon, a straight line) by a simple angle bisection.

[@stylesheetra9411]

2:38 how did he choose the compass' angle?

[@derekbernhard9191]

That had been bothering me for way to long, I rewatched that part and thought about it for way to long before checking the comments. Still confused

[@artashgyolecyan8498]

The angle isn't important. Just instead of drawing whole circle he drew an arc of it for simplicity.

[@danieleckert5008]

he just drew a little more than he needed. the exakt point to do the next line or circle is found by different constructions

[@dede6giu]

it's the size of that line he did right before

[@aliseifelnasr672]

02:39 where did that circle come from?

[@dumtalentlessboi]

3:25 cant fight the homestuck :(

[@oyegorge7605]

La mera neta no sé por qué me lo recomendó KZbin, pero al final me quedé fascinado con este vídeo, como unos trazos pueden predecir los lados de todos los polígonos que existen, me gustó el vídeo.

[@estebangalarza9953]

El círculo del minuto 2:39 con que medida de referencia lo hace?

[@aldoaldoz]

Cualquier rayo está bien: este círculo se usa para encontrar solo ángulos, no distancias.

[@Neko_Void]

Y que se fumo gaus

[@estebangalarza9953]

Si ya lo sé, pero por ejemplo, en todas las lineas que hace tiene una referencia, por ejemplo la mitad de el círculo o la intersección de dos lineas, pero en esta solo hace un semicirculo si ningún tipo de referencia

[@ArielGoV]

@@estebangalarza9953 te lo ha dicho, no usas nada de referencia, solo quieres ese circulo como guia en si mismo para poder ir encontrando los angulos (si te fijas, lo que hace despues de ese circulo son todas bisectrices, y estas solo dependen del angulo entre 2 lineas)

[@ochentaycincoalbricias]

You can do the heptagon, try this: 1. divide a segment in 7 parts 2 make a circle with the center un the midpoint of the segment 3. pick the compass and, with the measure of the segment, draw a circle with the center of the first point 4. do the step 2 again but with the last point of the segment 5. make a line which starts in the point where the 2 circles made in steps 3 and 4 intersect and passes trough the point when finishes the 2/7 of the segment and finish when It touches with the first circle 6 .with the compass, pick the measure between the first point in the segment and the point we made in step 5 and we draw arcs in the circle until we reached the start point.

[@kat-oh3hx]

i believe the problem is "divide a segment in 7 parts"

[@prismarinestars7471]

There is a way to divide a segment with a strait edge and compass. I learned it in geometry class, it involves copying angles.

[@entercherpfhalckhontralyty3542]

@@kat-oh3hx Phales' theorem, as we call it in Russia, could do.

[@aldoaldoz]

I used methods that are more than 2000 (two thousand!) years old. They are mainly based on Euclid's Elements: in my opinion, the geometry heaven! Only the 17-gon is much more recent, since it has been found by Gauss on the late XVIII century.

[@itryen7632]

You have to respect that they fully animated the compass.

[@Centaur-xs8ge]

Here, i learned that circles have infinite corners.........

[@Sergeant.Choccymilk6568]

Welcome to the "Why is this recommended to me" show. Episode 230.

[@meanbob088]

17 was ridiculous...

[@CoolyanEmoji]

Wow, you didn't forget the digon!

[@Xardy5806]

360:17 = 21.17647058823529411764705882352941 ??? lol ??? i want to know who did it first and how ...

[@cyxpanek5302]

Gauss

[@Rando2101]

Well apparently sin(2π/17) can be written with the 4 basic operations and square roots, so it's possible to construct it

[@rabit1998g]

-please youtube can i go sleep -hold on i gotta make you watch something

[@jaimeeoww]

i know 7 is not a "regular" polygon, but you missed it.

[@aldoaldoz]

+jeremy hansen a 7-gon can be a regular polygon. The problem is it can't be exactly drawn using strightedge and compass, and this is the reason why I messed it.

[@jaimeeoww]

+aldoaldoz oh. that makes me really sad then; 7 is my favorite number. the rest of your video is really good though. I like the simple and fast animations. they get straight to the point.

[@ffggddss]

+ jeremy "that makes me really sad then; 7 is my favorite number." Well then you can take comfort in its distinction as the smallest integer for which a regular polygon cannot be constructed by classical methods, with straightedge and compass. "they get straight to the point." Nice double-pun!

[@jaimeeoww]

lol i didn't even realize i made a pun! thanks!

[@BudderB0y2222]

fuck you

[@tzovgo]

sad to see there isn't a construction of the regular 65537-gon here

[@aldoaldoz]

I described the construction of 257 and 65537 sides regular polygons on the italian version of wikipedia. Here is the complete construction of the 257-gon (explanation and animated gif are mine): it.wikipedia.org/wiki/257-gono#Costruzione and here is the first step of the 65537-gon: it.wikipedia.org/wiki/65537-gono#Idea_della_costruzione

[@Diamondblade2008]

@@aldoaldoz Given that the 65537-gon would most likely look an ordinary circle, I wouldn't think it mattered how the 65537-gon is constructed given the sheer effort that would be needed. It would be just easier to draw an ordinary circle with a compass! (No offence or anything; I do appreciate your knowlege you have demonstrated in your videos).

[@aldoaldoz]

@@Diamondblade2008 Of course the 65537-gon looks like a circle. But even 257-gon does. The interesting thing is the discovery by Gauss, after more than 2000 years in which geometry was stuck at the pentagon and pentadecagon, that it was possible to exactly construct polygons of 17, 257 and 65537 sides. It was a sensational announcement. And he was only 18 years old!

[@JubJubAc]

This is why I didn't take geometry in highschool

[@iasimov5960]

Obviously you're not an electrician.

[@JubJubAc]

@@iasimov5960 ironically, I wanted to be an electrician, but I dropped the electronics course in high school to make room for some mandatory grad planning class...

[@Orphiwn]

thank you for the GREAT JOB here and also in Wikipedia. Without you Wikipedia in all languages would be very poor in all polygons' entries and your exceptional drawings not discovered fully yet.

[@theswanp1199]

Am i the only one wondering wtf happened to 7

[@Neko_Void]

It's imposible to make the 7

[@mateussouza3979]

Cheesy Par 7 isn’t constructible with compass and straightedge.

[@keterpatrol7527]

for doubling shapes above ten sides, you are basically turning them into circles

[@ChrisWilliamsRMWpigeon]

A regular tridecagon is not constructible with compass and straightedge. However, it is constructible using a Neusis construction

[@AJBallistic]

rest in piece any poor soul who needed to construct a 51 sided shape by first constructing a 17 sided shape

[@aldoaldoz]

Look for "257-gono" in the italian version of wikipedia. There is the construction section I wrote myself - unfortunately I hadn't the time to translate it into english

[@codytrim5402]

Student: “Teacher this problem doesn’t make sense.” Teacher: “Just follow the instructions.” The instructions: 2:30

[@thurlmusic]

nobody insterested on 7 , 11 (on progress) , 13 , and (probably) 29 ? btw, i found cuberoots inside seventh roots in cos (2kπ/29) so constructing regular 29 gons might require angle septisections and angle trisections

[@aldoaldoz]

I worked also on 257 and 65537 regular sides polygons, which in theory can be constructed with a strightedge and a compass. I did also some tries with all the regular polugons up to 20-gon. You can find all of them in this italian page of wikipedia, where the linked animated gifs are all my works: it.wikipedia.org/wiki/Poligono_regolare#Tabella_riepilogativa

[@JoeBrowning-n9k]

What about the Nonagon?

[@aldoaldoz]

7-gon and 9-gon are missing because these polygons CAN'T be exactly drawn by means of compass and strightedge only. Here only exact polygon constructions are shown.

[@Diamondblade2008]

You are starting to encounter diminishing returns at 3:29 onwards because the shapes are starting to look more and more like circles instead of polygons.

[@mambazo5]

Excellent video. I found #4 amusing, as the circle is already divided into 4 but at a 45 degree angle (like a diamond).

[@hex-automata]

Pleasing to the eyes & brain. Awesomeness.

[@Diamondblade2008]

I tried drawing the heptadecagon using the method shown in this video. One thing I don't understand is how to draw that arc at 2:39 (which is shown after drawing that line from the centre of the 'eye' to the right hand edge of the circle). I know that the heptadecagon is constructible but it seems simply too messy to draw with all those steps. I'll just find a picture of a heptadecagon online and print that off instead.

[@aldoaldoz]

The arc at 2:39 can be drawn whatever radius you want, as it is only needed to define some angles (not lengths).

[@Diamondblade2008]

aldoaldoz Good afternoon and thank you for your reply. After reading your helpful reply I tried to draw the heptadecagon again step by step and managed to get far as drawing the 'egg' at 3:05. However when it came to constructing the segments (3:05 onwards) my sheet of paper just became a total mess and everything got muddled. I don't know how the Ancient Greek mathematicians pulled it off!

[@aldoaldoz]

Well, this drawing method is not so ancient! It cames after Gauss: he (at the age of 17!) understood some polygons could be drawn in addition to the classic series of polygons, those with an odd number of sides (triangle, pentagon, 15-gon). The "new entries" were 17-gon, 257-gon and 65537-gon. Look at wikipedia: you'll find many explanations, as well as my own animated gifs.

[@JOELwindows7]

This is your daily dose of Recommendation Possible polygons

[@Invalid571]

We know that the more vertices a polygon has the better it approaches the shape of a circle. Also when n --> infinity an n-gon (with finite area) will become a circle. Therefore the vertices of any polygon are points of the circle the polygon is inscribed. So if we find the equation of the circle the polygon is inscribed we can easily find the coordinates of its vertices. Generic circle equations: A circle with its center at (0,0) has the following equation: x^2 + y^2 = R^2 and its parametric form is: (x,y) = (Rcosθ,Rsinθ) , where θ = the angle of the point. A circle with its center at an arbitrary point (a,b) has the following equation: (x-a)^2 + (y-b)^2 = R^2 and its parametric form is: (x-a,y-b) = (Rcosθ,Rsinθ) --> (x,y) = (a+Rcosθ,b+Rsinθ), where θ = the angle of the point. Now we have all we need to find the vertices of any polygon.

[@denelson83]

How do we even know these work? How were these constructions first devised?

[@aldoaldoz]

3 sides (and 6, 12, 24...) and 4 sides (and 8, 16...) constructions were demonstrated by ancient greeks. 5 sides too, though the construction showed comes from Ptolemy. 17-gon was demonstrated by Gauss.

[@Jaehaeron]

I don't know why this is on my recommended list, but I like it.

[@the_luna_lily6234]

So you use two perpendicular lines crossing at the mid point to make two more perpendicular lines crossing at the mid point to make a square instead of just using the original two perpendicular lines crossing at the mid point

[@spretcher]

So, now we know the egg comes before 17. Slowly working our way to the chicken.

[@aldoaldoz]

Well, on the web you can find many articles on this polygon... just do a search for "heptadecagon"!

[@QuotientGD]

Legends say that the construction of the 17-gon produced 99% of the comments on this video

[@ailurophile4341]

This video is 8 years old! Very cool.

[@Filomatia]

Great video! You should say that you are not constructing the regular polygons, but also inscribing then in a circumference.

[@javulicraft2228]

I FINALLY WATCHED IT. ARE YOU HAPPY NOW, KZbin?

[@bpdolesdominoes4]

Okay I just want to know who was able to come up with this complex thing to get a 17-sided polygon like what

[@mpboomslang]

But the sin and the cosine of pi/7 (in radians) can be found in simplest radical form.

[@Rando2101]

Only square root works.

[@epsilonthedragon1249]

How does someone figure out the process for seventeen?

[@Mrhellotinfish]

how to draw 17? when you do the...first 'blue' in colour circle where is the crossing point of this circle?

[@HerrXenon_]

2:30 Well that escalated quickly

[@sushamagirish]

nice thing but I like to give a mathematical explanation to the whole process. Is it based on the length or angle?

[@Paleoint]

So forgive me, but what was the trick for achieving the two orthogonal lines to begin with? Seems like you'd make an arc from any edge point through the center and where the edges of the arc cross the perimeter, you'd do it again? Any simpler way? Of course you can always use the Pythagorean Theorem, but wondering if there is a quicker approach. Thanks.

[@KevinKurzsartdisplay]

Now I know how to draw polygons, this will take my art to the next level.

[@iqbalconan21]

Legend says he still drawing this polygon until indefinite time

[@rk2045]

Excellent method how did you got the idea of making such polygons. Thanks i will tell my friends about the methods .

[@SHIN2025_official]

Oh my gosh! You put this on KZbin before I was born (Jul 25, 2011) !

[@-liquid-7548]

Why on 17 is there so many steps just to make a single line

[@aldoaldoz]

Of course yes! Feel free to use the same animated gifs I uploaded to wiki commons. In addition, look at "65537-gono", is another article of mine (there is only the first construction step)