Drawing a square, pentagon, decagon, 20 sided regular polygon

Discussion in 'Physics & Math' started by GodLied, Aug 7, 2003.

  1. GodLied Registered Senior Member

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    436
    In another thread one was puzzled with drawing a pentagon with compass and straight edge. The fundamental problem with some geometric approaches is that the method was not something most people would say: this is the way I will arrive at that answer. One poster even suggested it took many years to puzzle out drawing a regular pentagon with straight edge and compass.

    With that problem in mind I looked on the web and found a puzzling method to draw a regular pentagon. Although it works, the rationale behind it is not necessarily lucid. So, I have considered the problem and found a comprehensible solution.

    To draw a regular pentagon, first draw a straight line. On that line make 11 equidistant marks. Draw a circle of radius 5 marks centered at the center mark. Recenter the compass at the first mark on the left edge of the circle. Draw arcs originating from each mark on the line to the edge of the circle. The arcs intersect the circle at 18 points, 9 on top and 9 on bottom. Along with the intersection points of the line and circle, there are 20 total equidistant points on the circle. Use those 20 points to draw a square, pentagon, decagon or even a 20 sided regular polygon.

    May my solution might help those puzzled by geometry and the Egyptian methods other people pose: papyrus was found where instructions were given to compute a quantity but no reason for the method was on the same papyrus.

    Any other queries that puzzle people may or may not have a solution. Pose it. For a fee, I'll give you an answer.

    JMG.
     
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  3. Pete It's not rocket surgery Moderator

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    Very nice!

    But unfortunately, the marks made on the circle will not be equidistant

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    (Try it and see)
     
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  5. GodLied Registered Senior Member

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    How then would you split a circle into twenty equal arcs with a straight edge and a compass?

    JMG.
     
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  7. GodLied Registered Senior Member

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    One could draw two circles. One with diameter D and the other with diameter 5D. The circumference of the lesser circle is one fifth the circumference of the greater circle. Using the circumference of the lesser circle as a measure, one can divide the larger circle into five parts to draw a pentagon.

    How would you split a circle into five parts?

    JMG.
     
  8. Pete It's not rocket surgery Moderator

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    I don't have a method. But your neither of your methods works (sorry).

    Regarding your second method:
    Please tell me how you unroll the circumference of the smaller circle onto the circumference of the larger circle using a compass and straightedge?
     
  9. Merlijn curious cat Registered Senior Member

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    1,014
    Great!

    HI all!
    This is great fun. Some month ago I set myself to find a way to construct the pentagon myself.
    I wrote a small article about it. It's in Dutch, unfortunately for many. But the illustrations will be easy to understand and tell the same story as the rest. I may translate the piece later.

    It was a difficult job to find the answer, it took me several days. But I got it! And I had lots of fun, frustrations and -afterwards- a great feeling of satisfaction.

    Here is the address: http://home.wanadoo.nl/rmboers/wiskunde/pentagram.html

    After one has split the circle in 5 equal pieces, it's easy to split it into 10, 20, 40, etc. pieces as well.
     
  10. Merlijn curious cat Registered Senior Member

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    Dank je wel Andre. (Thanks Andre)
     
  11. Crisp Gone 4ever Registered Senior Member

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    Mijn god, al die Nederlanders hier...

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    (My god, all those Dutch people here)

    Bye!

    Crisp
     
  12. GodLied Registered Senior Member

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    The first method is a false method. The second method requires creativity to roll a circle onto the other. Here is a third method.

    Consider a regular n-gon inscribed within a circle of radius R. Lesser circles of radius r may be drawn centered at each vertex of the n-gon which are already touching the larger circle. When r = R sin(180/n) where 180 is degrees and n is the number of sides of the n-gon, the centers of the lesser circles are equidistant points from which the n-gon may be drawn.

    A person with a compass and straight edge need only make a measuring triangle to measure R and r. Some texts just cite values of trig functions to four significant digits. The accuracy is up to the artist: the higher the accurracy the more tick marks to make on the hypoteneuse of the measuring right triangle for translation to the base of the measuring triangle.

    The equation relating r and R comes from Marks' Standard Handbook for Mechanical Engineers. It came from the section on drawing an annulus. Drawing an annulus involves dividing a circle into n parts by drawing n contiguous circles of radius r between two other circles. For additional reading visit a library or try to find the information on the internet.

    JMG.
     
  13. GodLied Registered Senior Member

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    436
    Re: Great!

    Your solution is the least complicated, most practical method of all. In addition, no measuring. Only straight lines and circles. Brilliant.

    JMG.
     
  14. Pete It's not rocket surgery Moderator

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    Godlied,
    If you're after practical method not restricted to a compass and straightedge, why not just mark the required angles using a protractor?

    Marlijn's method is good. Unlike your methods, it validly constructs a pentagon using only compass and straightedge.

    Here is another method, perhaps slightly more elegant:
    http://mathcentral.uregina.ca/qq/database/QQ.09.02/mary1.html

    (Note that the original problem is more restricted, requiring the construction of a regular pentagon starting with a given side).
     
  15. Pete It's not rocket surgery Moderator

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  16. Merlijn curious cat Registered Senior Member

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    Cool!
     
  17. Pete It's not rocket surgery Moderator

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    If only I had the brains and inclination to discover such constructions myself...

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  18. dpwe Registered Member

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    Another method to draw a pentagon

    I've found several ways on the web. This one (due to Charles Lederer, 1913, though perhaps not his invention) seems quite robust; you set the compass only once, to the length of the pentagon side:

    It's part 8 of learn-how-to-draw-now dot com, which appears to reproduce Lederer's 1913 book. Googling the exact phrase "draw parts of three circles" takes you there, and to a Google Books version.

    But I can't easily prove that it's a regular pentagon.

    DAn.
     
  19. dpwe Registered Member

    Messages:
    2
    Another method to draw a pentagon

    Re: the Lederer technique (three equal-sized circles and two intersects).

    By my trigonometry, the inside angle is 108.366 instead of 108 degrees,
    so it's not exact - but a quick way for a visibly adequate version.

    DAn.
     

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