General method for drawing any regular polygon given the measurement of one side
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- Опубліковано 12 лип 2024
- How to draw a any-sided regular polygon given the length of one side.
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I've read through all these comment, yapping on about "It's wrong!", but no one seems to be fascinated by the fact that such extremely high accuracy is reached by such an extremely low effort method.
For instance: How many division in "n" parts of lines and angles are suddenly possible after the "Origami Axioms" come into play?
I've not yet seen Mr Galois or any of his students tackle the -morphisms from Geometry including Origami axioms to their rings and fields and their other headache-inducing, albeit fascinating, contraptions!
Anyways, late to the party, but YT Algorithm should've known better and recommended this channel years ago... So, a belated, cheers on an awesome channel!
And secondly, if anyone can tell me who the originator of this method is, I'd love to Delve into that Rabbit Hole!
the hexagon drawn here is perfect
because the construction establishes
an equilateral triangle ABO
and replicates it 5 more times around the circle.
an equilateral hexagon is indeed comprised of
6 identical equilateral triangles meeting at a common vertex.
however - any construction purporting to draw equilateral polygons
of successive numbers of sides - based on the notion that
the respective circumscribing circles have centers equally spaced
along the central vertical line is simply WRONG -
as a trigonometric analysis will clearly show.
at best -
this method will yield good approximations of equilateral polygons
whose number of sides is "quite close" to the original number of sides -
in this case 6 - thus 7 8 and 9 as well as 5 4 and 3.
Sir this is very good useful easy method important lecture thank you
It's an approximation method ... the accuracy is 1.9 degrees ... try to draw it in CAD ...
l understanding it thanks for this good work
Well done . Applied mathematics for machine trades ... Divideing head math . 👍
If you love this geometry,
then you might love this higher geometry:
mathworld.wolfram.com/QuadratrixofHippias.html
Yes, the heptagon thing is just an approximation because for it to be perfect 7sin(180/7) must be exactly 3. But that is not the case as 7sin(180/7) = 3.037... or 1.24% more than 3.
The sector is a vintage drawing instrument. It uses the same method to lay out any regular n-gon.
useful for beginners
Thank u alot ave really appreciated from Kenya😊
Thanks from algerian 😀
extremely helpful. Thank you!!
Nice job dude.,
Really helpfull
nice
This 9-sided polygon is not regular, neither the others (probably), except indeed the hexagon !
Neat
what is the angle of the T line??
Any.
But this is not a exact method. It is approximately method.
This is an exact method if you already have constructed a line segment with the same length as the side of the regular polygon you want to construct.
Nameci2718, you're just flat-out wrong. I would hope the creator didn't intend for this method to be seen as an always-exact method (one doesn't exist for general regular polygons with compass-and-straightedge only). I would assume that either he knows this is an approximation and didn't think that was worth saying, or he was duped as well.
Rndm Dud, while what you've said is true, you're missing the point of what Online is saying. What Online is saying is that if you COULD use "perfect" methods perfectly, without any errors, you would get a perfect regular polygon, but if you likewise could use this method perfectly, without any errors, the resulting polygon would unavoidably be NOT regular.
True, the error may be small enough for most practical purposes, and may easily be overshadowed by other factors when doing an actual physical construction - but again, that was not the point.
I'm not entirely sure about that. Your expansion seems to be more along the lines of "No method can be perfect in the physical world" which is completely true. To be fair, I don't know exactly what Online's point was, but I took it to be "Even in a perfect world without physical limitations, the method is flawed." That is, dealing with it in a purely mathematical sense. Assuming that is what he meant, your point about physical limitations, while true in our universe, isn't really relevant for the purely mathematical one.
As I'm typing this, I realize we may very well have the same opinions, just different focal points - your focus being on the physical world, mine being on the pure mathematics. I assume you would accept that, say, Euclid's construction of a regular pentagon is "perfect" in the pure mathematical sense, though it's impossible to perfectly replicate it in the physical world?
Probably, yeah, the only difference is what we emphasize.
You're welcome; I agree that too often UA-cam comments can be insulting and childish, and while I have a bad habit of sometimes being too blunt, I do try my best to not descend to quite that level. (I just hope I didn't go too far in my comment directed to Nameci...)
So I can in theory I can construct my own protractor 360 degree polygon? Sounds like a fun challenge.
This guy sounds like the one from Organic chemistry tutor channel. 😅
It was proven *CENTURIES* ago that not all polygons can be constructed with a straightedge and compass. See en.wikipedia.org/wiki/Constructible_polygon . For example, a 7-sided polygon is not constructible.
It's very easy to prove that this method isn't correct and only produces approximately correct results in certain cases. Try using the C3 for a 3-sided polygon and it won't even work.
A seven-sided polygon (a heptagon) is in fact able to be constructed.
ua-cam.com/video/cErccoHui9g/v-deo.html
@@kingfuff402 Its close but not 100%
@Murtaza Nek: That's right. In particular, the actual centers for those polygons would *not* be equally spaced.
The constructible regular polygons are those whose side count, n, is a product of distinct Fermat ("Fair-mah" - it's French) primes, and any number of factors of 2 (including none).
The known Fermat primes are 3, 5, 17, 257, 65537.
[To generate these numbers, start with 2, square each term to get the next, then add 1 to all of the terms.]
It's unknown whether there are any more (that are prime), but if there are, the next one is larger than an astronomically huge lower bound, and therefore, of absolutely no practical importance. For that matter, neither is 65537.
The n-values up to 100 that are and aren't constructible, are:
Yes: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96
No: 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100
But using an accurate ruler and some trig (or a good protractor, for that matter!), you can draw any regular polygon with a given side length.
With side length s, the height, h, of the center (h is called the apothem of the regular polygon), is
h = s/[2 tan(180º/n)]
The method shown in this video is exact only for n=6, which is what it started with, anyway.
Fred
Straightedge yes, impossible, but marked rulers do make all other polygons constructible.
And marked rulers render primitive straightedges obsolete.
Beware that this is only a really good approximation, if you want to draw 1-gons, you might run into some problems.
And, the limit step is 1/pi, so by doing an inversion over a unit circle, pi can be obtained, which is impossible, so it also is an approximation going out to infinity, and will slowly become less and less accurate.
A 7-gon cannot be constructed using a compass and straightedge.
If it could only be created from a side, it could be "transported" to a circle using parallelograms.
this method is not give accuracy which have
Thank you for your contribution.
This is an approximation method for drawing by hand, not for an accurate mathematical solution.
Which method have you got ? Please share it.
@@ArthurGeometry You are correct, sir, this is an approximation. Actually, it's plenty good enough for many "pencil" drawings. As an example, using your method to inscribe a tridecagon (13 sided polygon) with sides 2 units in length, gives a relative error of about 0.2% larger than the "true" circle should be. This is just fine for dividers that use pencil lead. However, with carefully sharpened machinist's dividers, working on metal, this error will be obvious. Similarly, the error will be quite obvious if using a CAD package with precision set smaller than 0.01 units.
I'm sure you know this, but for the benefit of other readers, if machinist's/CAD accuracy is required, then a tiny bit of trigonometry will give a precisely accurate solution. As an example, take the above tridecagon with side 2 units long. The radius of the inscribing circle is 1/sin(180/13) = 8.373136 units. In general, the radius is (s/2)/sin(180/n), where s is the length of each side and n is the number of sides.
Best regards,
Gottfried
that is not what I am looking for
This method is not accurate enough .All sides are not equal .Please rectify that
construct yours and post for us to watch.
There is no method with the classical tools - straightedge and compass - that will work for most numbers of sides, n; but with a ruler and some trig calculation (or a protractor), it can be done. As accurately as can be measured with those tools, anyway.
With straightedge and compass, the scorecard reads as follows...
Constructible: n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ...
Not constructible: n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, ...
Fred
I don't like this one
So there is not need at all to draw the silly hexagon and the smaller circle?
Total waste of time all you need are the points A B and O which is just 2 arcs. And to divide A B by 2 so you can get T (that division is a weak point of this system).