General method to divide a circle into any number of equal parts

Рет қаралды 123,882

Күн бұрын

Learn a general method to split a circle into any number of equal sectors and how to construct a regular n-sided polygon inscribed in. In the example of the video we have divided the circle into 9.
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This KZfaq channel is dedicated to teaching people how to improve their technical drawing skills. It focusses on drawing figures from the geometric plane to descriptive geometry and also different systems of technical drawing representation.
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Dubbed by Frank Shaw.
Music by Antonio Fernández Ruiz. antoniofernandez.es/
#Geometry #HowtoDraw

Пікірлер: 104

@5switch09 Жыл бұрын

It's much more accurate if you don't use a compass for the division. From point Q make lines as you did for line QB with the other numbers. That will give you the points on the circle dividing it much more precisely. Every second point is your main one while the others are your mid points. I found that this method is much more reliable for me because I don't have to pinpoint the exact point with compass every time while watching if the compass moved out of alignment or not.

@5switch09 Жыл бұрын

You would have to of course make a second point Q on the other side

@wizrom3046 5 ай бұрын

Yes, BUT to only use the even numbers

@observersnt 2 ай бұрын

Beautifully done, crystal clear, superb graphics. Just thoroughly sensible. Thank you very much

@tono002 3 жыл бұрын

Appreciate your timing to make this video, just like I learned in drawing school

@eddiemorrone870 Жыл бұрын

This is a great and accurate approximation. One viewers comment pointed out that it’s only off by 2 degrees for a nonagon. Even with a spirograph you’d have to be careful of your margins for larger divisions.

@elineeugenie5224 3 жыл бұрын

Beautiful!! Using my 40 years old compass, it's pure joy

@carlaa.1617 3 жыл бұрын

This is the best explanation I have ever seen, I love drawing mandalas and sacred geometry and this is most helpful! Thank you so much!

@culmalachie 2 жыл бұрын

... thanks, and now I know what Mandalas are! these rabbit burrows on youtube ....!

@Chucktender69 Жыл бұрын

Love this so much and thanks you for posting this. I’m about to put in a spiral staircase into an uneven semi circle one piece at a time and I believe this will answer all my troubles.

@manishbairagi3006 3 жыл бұрын

Sir should i always use 2nd point as intersecting point? And why only sir 2nd point not any other?

@joefranich4933 Жыл бұрын

Well done! My nephew and I are going to make decoder rings (ROT x) and I needed to know how to do this from scratch - manually. Going to give it a go now!

@venantiusjpinto 3 жыл бұрын

Awesome, and truly appreciated!

@zwgrafakhsandrianos7784 2 жыл бұрын

I thought i was smart untill i saw this video.Thank you so much.

@GiantSnipe Жыл бұрын

This is great - no use for it currently but good to know and I will try to work out the maths behind it

@svenkamog2232 4 жыл бұрын

Very helpful thank you

@user-eb4iq2jw3y Жыл бұрын

Great video-thank you

@vulcanfeline 4 жыл бұрын

i like drawing mandalas in M$paint (since my drawing skills suck) and find your methods wonderful. a little finagling to get them to work with no compass and they're not perfect, but they're fun. thx for all your work :D

@Drew_Hurst Жыл бұрын

Try using the software called "gimp", has more tools.

@waqasarshad7904 17 күн бұрын

@@Drew_Hurst can you tell me which software the author has used in the video?

@davidbrown6295 2 жыл бұрын

This is NOT an accurate method. You get a nonagon, yes - a 9-sided polygon, but it is NOT a regular nonagaon as claimed - a 9-sided polygon with all sides equal and all angles equal. The first 8 sides are all equal in length, but the 9th is different from the rest. It is definitively proven (long ago now) which regular polygons can be constructed via Euclidean geometry and which cannot. This is the content of the Gauss-Wantzel Theorem as has already been mentioned elsewhere in these comments. If you are interested to know more about this, you can look up "constructible polygons" in wikipedia. Pulling a quote from that article: "a regular n-gon is not constructible with compass and straightedge if n = 7, 9, 11, 13, 14, 18, 19 ..." Notice that the regular nonagon is not constructible. The method presented in this video is not a neusis construction, it's just wrong. There are neusis constructions that extend Euclidean geometry sensibly and allow solutions to some classically impossible problems - that's not what's going on here. We have a false and deceptive claim that is misleading people.

@andreash.6175 2 жыл бұрын

What a pity ! But Arthur is wrong with this construction ! First i thought i was messing arround with my compass, but then i tested it with a CAD program and now i understand, thats not my fault ;-)

@davidbrown6295 2 жыл бұрын

@@andreash.6175 Not your fault at all. I applaud your use of a CAD program to check this out more carefully. It is one of the perrenial flaws in geometric "arguments" to rely upon the accuracy of manual drawings - the accuracy of such drawings degrades very quickly with the increasing number of drawing steps. It is this kind of flaw that allows people to believe that they have achieved something that they have not. CAD is far better, but every CAD "drawing" has its limitations also. There is no substitute for the rigor of mathematical proof, which Arthur's construction is not. As I noted already, it is difinitively proven which regular polygons can be constructed in Euclidean geometry and which cannot. The ones of many sides that are "interesting" have constructions that are quite complex. They are beautiful in their own way, but not simple and not achieved by any "general method". An interesting subject, but quite non-trivial.

@Alkis05 2 жыл бұрын

It is an approximation and it has it's uses. It has reasonable precision, since the error is that it devides the circle in 40.3º instead of 40º, which is just as good as one can do with a protractor. And if want to be precise in the langugage as well as the drawing, although the method is accurate, it is not perfectly precise.

@andreash.6175 2 жыл бұрын

@@Alkis05 Yes, but it should be mentioned, thats only an approximation. IF using this way here, it is advisable not to use the compass measuring a side length around the circle, but the lines throu the diameter where the cros the circle... By the way: you always can also use the formula a = 2 * r * sin(180/n) by your pocket calculator and to measured and transferred as side length with your compass. This would be exact....

@Alkis05 2 жыл бұрын

@@andreash.6175It wouldn't be exact because "sin" is a transcedental function. Shouldn't you mention that? No you shouldn't, because the people that don't know about it don't care for that degree of precision and the people that would need to know probably already does. Besides, this requires a calculator. There are some situations where you can't use one, most notably when you just don't want to. Also, you would need some kind of protractor too, which you might not have, at least one that is appropriate to the scale of the drawing.

@divyam3.1415 4 ай бұрын

Absolutely amazing. Loved it dude !!!!!!!

@Simon_Jakle__almost_real_name 6 ай бұрын

Carl Friedrich Gauss (en.wikipedia.org/wiki/Carl_Friedrich_Gauss ) was so proud of having found a way to divide a circle into 17 (equal) parts (as in en.wikipedia.org/wiki/Heptadecagon ), "claiming" this to be his greatest discovery in the year 1801 as far as it's told, so he might used calculus, but this must have been a rather/mostly analog(ue) way.

@kentmumar2901 3 жыл бұрын

thank you so much sir

@davidmcgrath9581 7 ай бұрын

Brilliant!!!☘

@harryjega5643 Жыл бұрын

Very good video.

@the_luminary 3 жыл бұрын

So whats the point of drawing ALL 9 parallel lines after drawing the 2nd line which intersects "Q"?

@joshuaschiffman624 2 жыл бұрын

Did you ever find out?

@Ani-yt4nf 2 жыл бұрын

It must have been to divide the straight line into n parts equally. After getting the length from the intersect point to B, the parallel lines are no longer needed. They were just used to equally slice the line into 9 parts

@M05tly 2 жыл бұрын

I also wonder what the purpose of this is. The needed points seem to come from the center line, with the preceeding line not needed? Is this the most accurate way to divide the circumference of a circle with only a straight edge and a compass?

@borislum1998 Жыл бұрын

@@M05tly that is a geometric trick to make equi-distance divisions on a line. in this case, just trust the geometry and you really only need that location of the second division on the vertical diameter. then follow the steps as described. so if you want to draw a clock, ie twelve division. use this trick to mark out 1/6th of the vertical diameter line. draw the line from Q and this point to the edge of the circle. use that compass trick to mark out the other 10 divisions to get 12 lines of the clock. easy peasy

@M05tly Жыл бұрын

@@borislum1998 I don't understand on first reading but looking at my comment it's been over a year since I watched the video. I'll rewatch it and let you know when it's clicked. Thanks for the info.

@johnchristian7788 Жыл бұрын

If you search "set square" in KZfaq, you will find videos on how to draw parallel lines (which is useful for finishing this diagram.)

@daljeet247 3 жыл бұрын

Wow man ,

@cyclingroanoke4837 3 жыл бұрын

1:50 this doesn't seem to jive. What is the second reference point when connecting the diagonal lines to centerline? Especially if done by hand these could be all over the place. And the placement of the second diagonal line seems critical. 3:07

@constablecomfortable 2 жыл бұрын

Omg. That's just blown my mind box. Is this how they did the 56 Aubrey Holes at Stonehenge perhaps?

@paulss8503 Жыл бұрын

I am stuck, I’ve been trying to use this method and I’m always slightly out ie when I measure around using the compass it doesn’t return to the origin it’s off -I’m measuring and remeasuring in fact it’s more accurate if I just measure 36 with a protractor (I’m dividing the circle into 10 equal parts) I’ve even tried drawing from QB every 2nd point instead of a compass and it’s still slightly out!!!

@yatendrakumar6909 2 жыл бұрын

✌👌

@c848 3 жыл бұрын

I am trying dividing the circle into 11 equal parts with this method and I am not succesfull. One division is much smaller. Why is not working?

@etyrnal 2 жыл бұрын

is this an approximation? is there any 'imperceptible warpage'?

@jamilrahman5960 Ай бұрын

if we needn360 divisions then?

@spudnickuk 2 жыл бұрын

looks good to what you have done, but how did you come up with using the number 2 and what are all the other numbers for so why is it needed to draw al the other divisions Please explain :)

@sunflowerbadger 2 жыл бұрын

I think this method relies a lot on being really accurate. I tried this for 13 points and the first time ended up with exactly 12! The second time I got 13 but the final gap was smaller than the rest. I shall persevere, but any hints on doing this by hand would be appreciated. My ultimate goal is to make a massive circular calendar with 365 or 366 divisions... has anyone tried this before? I wanted something about 1 meter diameter.

@zarau01 2 жыл бұрын

I had the same results when trying to divide into 13.

@AngelobOfficial 3 жыл бұрын

Great video!! So well explained! Could I ask you for a formula, or a written source like if n*6 x(distance)=… . My question is coming for a musical purpose. From Geometry and music theorised my EUCLID, in music we call them euclidian beats or patterns. The idea is, no matter the number of beats/hits they are always spread evenly around a circle. So I was looking around but never found a formula to put it in code form. Thanks in advance!

@elleeo1495 2 жыл бұрын

You would need to do that mathematically. It is not possible to divide a circle by any given number. Look up "constructible polygons" for proof.

@johnrobertson93 2 жыл бұрын

@@elleeo1495 Yes. It simply can't be done for anything other than 2^n number of arcs (i.e., 2, 4, 8, 16, etc.) or 3 x 2^n number of arcs (i.e. 3, 6, 12, 24, 48, etc.), and certain other specific numbers including 5 x 2^n, 15x 2^n, 17 x 2^n, etc. or p x 2^n where p is the base number of sides of known constructible polygons.

@pivocopii8254 Жыл бұрын

Using this for building a roulette

@mark2tech Жыл бұрын

Whoever figured this out first had to be a genius

@mglenadel 6 ай бұрын

For some reason I can't find it anywhere but in Portuguese-language sites and videos, but the method is known as the "Rinaldini Method", and I believe this to be the name of the guy who invented it. Link in Portuguese Language Wikipedia: pt.wikipedia.org/wiki/Divisão_da_circunferência_em_partes_iguais_(processo_geral) I have found references to the Bion method, and the Tempier method, which seem to be the same thing, somehow.

@alexzuma2024. Жыл бұрын

mark the inclined line as many times as you want divisions

@user-yo1jn6uv8h 2 жыл бұрын

Great Video!But i still don't now how it work😃

@Co-uy4mi 3 жыл бұрын

Hello, I have a question Is there a formula to divide a circle into n equal parts? Like using pi or sin or cos Something like that?

@abdellahbellafqi3357 2 жыл бұрын

No there is not

@Alkis05 2 жыл бұрын

Yes there is: ___________________ ╱ ╱ ⎛ ⎛2 ⋅ π⎞⎞ L = R ⋅ ╱ 2 ⎜1 - cos ⎜───⎟⎟ ╲╱ ⎝ ⎝ n ⎠⎠ L is the side of the n-polygon inscribed in a circle with radius R

@teeeff5179 7 ай бұрын

actually there is (sort of) : "chord length" = 2 x R x Sin (theta / 2) where 'R' is the radius of the circle and 'theta' is the angle subtended at the center of the circle by the chord. This is useful when you are dividing a circle into N equal segments: the formula simplifies to "chord length" = D x Sin (180 / N). So here you'd set your compass to "chord length" in order to mark off N equal segments of a circle with diameter = D.

@teekaesthete 3 жыл бұрын

I tried to divide 13 parts. This method did not work there...

@stephenroberts7459 9 ай бұрын

Errors accumulate! ---- on a 13 division exercise, a 0.5 mm error on the first division leads to a 5mm error on the last one!

@botondnagy2699 Жыл бұрын

Is there a proof to this somewhere? I am really interested

@keithwiley7122 4 жыл бұрын

It's clear that this method is only approximate, since it's famously impossible to construct a regular 7-gon using only a compass and straightedge. However, I can't seem to find a resource that discusses this approximation in more detail (e.g. how approximate is it? When is it no longer wise to use it?). Is there a particular name for this approximation that I can search?

@MrStudent1978 4 жыл бұрын

It's not an approximate method, however the accuracy does depend upon how accurately you draw points, arcs and lines etc. The method is based on "Thales theorem ".

@keithwiley7122 4 жыл бұрын

Gurpreet Singh but if it weren’t an approximation, then all polygons would be constructible polygons, and that would contradict the Gauss-Wantzel theorem, no?

@keithwiley7122 4 жыл бұрын

For those also interested in this question, this method seems to be referred to as "the Bion method", and is mentioned in Chapter 1, Use 17 of 'The Construction and Principal Uses of Mathematical Instruments' by Nicolas Bion, though he refers there to Johann Christophorus Sturm's 'Mathesis Juvenilis' as a source for the method. He also discusses the (very small) error fundamental to the method.

@MrStudent1978 4 жыл бұрын

@@keithwiley7122 thanks Keith! I intended to say that theoretically this method is not an approximation but as the number of sides will increase, the manual error accumulated may become appreciable enough.

@samisiddiqi5411 3 жыл бұрын

I always thought it was called the Neusis construction, and that's what the Wikipedia on this topic is called but even then I'm asking the same question about the method...

@waqasarshad7904 17 күн бұрын

Can anyone guess what software tool has been used in making such videos?

@brentbuckley285 Жыл бұрын

Why did we divide AB line by the number of spaces we want and not use it? Also, Thales's theorem requires a 90-degree angle if I'm not mistaken. Why did you use cm's? I've tried this to make a 26-part circle 5 times, and it hasn't worked. It seems like you're leaving something out.

@Jedidiah07 11 ай бұрын

How to solve across flat if only diameter is available?

@waygonedon 3 жыл бұрын

What am I doing wrong? My hash marks don't meet up at the apex of the circle, and they are skewed by a few degrees.

@lukasfilipsky31 3 жыл бұрын

You expect correct results while using incorrect method. That is what you are doing wrong. Use trigonometry, you will see.

@borislum1998 Жыл бұрын

you have to be very accurate with hand drawn compass lines. here is a tip. draw with a bigger circle. your error in the angles will be smaller. good luck. or use a CAD or computer aided drawing program

@Goofayball 2 ай бұрын

Why use 2 if you want 9 segments?

@modernworld3685 2 жыл бұрын

YES THIS GOOD, BECAUSE, STUDENT WISE FOR KNOWLEDGE NEED TO HAVE THAOUGHT, HOW TO MAKE DRAW CIRCLE AT ANY ANGLE OF DIGREE , LIKE 5 DEGREES , LIKE 10 DEGREES, LIKE 45 DEGREES, LIKE 90 DEGREES. IS THIS GOOD GIVEN KNOWLEDGE.

@chotafallen8390 4 ай бұрын

Remember that 40 years ago in school. ^^ but why do you use Nr2 not 3 or 1? explaination is missing

@BariumCobaltNitrog3n Жыл бұрын

Random width parallel lines, yeah good luck wit dat.

@ArthurGeometry Жыл бұрын

Better use your 45º and 30 set-squares

@celticcreations4856 2 жыл бұрын

If you only used the second marker, what was the point of drawing the other segments?

@veritas5008 4 ай бұрын

Because the angle of the parallel lines is determined by the angle of the last point intersecting Point P.

@erickarmigos1486 2 жыл бұрын

@jaghook 2 жыл бұрын

HELP. HELP HELP,,,,A normal circle has 360 degrees,,,,,,how can i create a circle with 147 equal arcs of degree. Or to put it another way, How can i create a 360 degree circle, with a 260 mm diameter, divided into 147 segments of arc, each with an angle of 2.4489795 of a degree. Any help will stop me breaking down, regards Jim

@Alkis05 2 жыл бұрын

With the following formula _________________ ╱ ╱ ⎛ ⎛2 ⋅ π⎞⎞ L = R ⋅ ╱ 2 ⎜1 - cos ⎜───⎟⎟ ╲╱ ⎝ ⎝ n ⎠⎠ L is the side of the n-polygon inscribed in a circle with radius R. That being said, unless you are using a computer, it is very hard to make a polygon with so many sides by hand.

@derekwilkinson7939 Жыл бұрын

Why did I get ten divisions?

@ArthurGeometry Жыл бұрын

kzfaq.info/get/bejne/rMiUm7OIlsfYpWg.html

@kksphilomath384 3 жыл бұрын

Why we have to join Q to 2 only, why not any other point ???

@mr.jayeshrangwani 3 жыл бұрын

Same question

@mr.jayeshrangwani 3 жыл бұрын

Any idea? Or any ratio is this? Can this possible for other polygons?

@waltontyson3996 3 жыл бұрын

I believe 2 is always used because the person drawing already knows the first point as 'A'. Since they're trying to find the second point on the circumference, they walk the line from Q through the second point on diameter to the circumference to find where the next point is for even sides. After 1 is 2 so they always pass the line through 2. Hope this helps.

@etyrnal 2 жыл бұрын

because the metered line represents the WHOLE circumference... but your mark is only for half that circumference... as you are projecting the points out only to one side of the circle