My Math Forum Constructing a 20 Degree Angle using Compass and Ruler Only
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 View Poll Results: Is this a useful construction? Yes 1 9.09% No 0 0% There are other ways 3 27.27% This method is wrong 7 63.64% Voters: 11. You may not vote on this poll

April 12th, 2012, 03:03 PM   #11
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
Originally Posted by Denis
Quote:
 Originally Posted by mrtwhs On the other hand, if you mean that you constructed an angle of exactly 20 degrees using the classic Greek tools of a collapsible compass and unmarked straightedge, then save yourself and your teacher the humiliation. It can't be done.
Sure it can: construct a regular 18-gon; draw 2 straight lines from center to 2 consecutive vertices
My first attempt to construct a 20 degree angle was to use a square. I was hoping it would be a foregone conclusion.

April 12th, 2012, 09:09 PM   #12
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by atharvjoshi Q - Is it Possible to Construct a 20 Degree Angle using Compass and Straightedge only?
A much simpler and more accurate construction:
Figure on page 2 of the paper "Tracé d'un angle quelconque" :
http://www.scribd.com/JJacquelin/documents

.

Last edited by skipjack; February 18th, 2017 at 04:17 AM.

April 16th, 2013, 11:48 AM   #13
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
Originally Posted by mrtwhs
Quote:
Originally Posted by Denis
Quote:
 Originally Posted by mrtwhs On the other hand, if you mean that you constructed an angle of exactly 20 degrees using the classic Greek tools of a collapsible compass and unmarked straightedge, then save yourself and your teacher the humiliation. It can't be done.
Sure it can: construct a regular 18-gon; draw 2 straight lines from center to 2 consecutive vertices
My first attempt to construct a 20 degree angle was to use a square. I was hoping it would be a foregone conclusion.

i have constructed a perfect 20 degree by jst compass and ruler..... and i checked the construction accuracy on autocad too

 April 16th, 2013, 05:45 PM #14 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Constructing a 20 Degree Angle using Compass and Ruler O "2. Step 2:Keeping your compass wide open with the same length..." You violate the rules in your step 2, the compass must collapse when lifted off the page.
 April 19th, 2013, 06:00 AM #15 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O It can't be done. As an amateur origami artist, I know a little something about geometry. The most you can get down to is 12 degrees. http://www.flickr.com/photos/85937466@N ... 2639808715 I posted my \$100 challenge on the Origami Forum for a 12 degree angle, and only 1 person came up with the solution. http://www.orime.de/downloads/origami-3 ... degree.pdf http://www.flickr.com/photos/85937466@N ... 2639808715 Once you have 12 degrees, you can bisect it. To 6, to 3. And that's it for whole numbers. You can't get down to 1 without neusis. To construct a 20 degree angle you have to trisect a 60 degrees angle, which can't be done without neusis. http://www.flickr.com/photos/85937466@N ... 3074235305 Tom Hull's origami method of angle trisection. Performed by me. http://www.flickr.com/photos/85937466@N ... 3074235305 https://www.math.lsu.edu/~verrill/origami/trisect/
April 19th, 2013, 07:41 AM   #16
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by JJacquelin A much simpler and more accurate construction: Figure on page 2 of the paper "Tracé d'un angle quelconque" : http://www.scribd.com/JJacquelin/documents .
That's digital-to-analog fakery. How do you know that the the inverse of tangent 5/8 = 32.0053...degrees without digital methods? i.e. a calculator?

Imprecision from analog-to-digital is perfectly acceptable to me, because the method is correct, the imprecision is in the limitation of the definition of the instrument. Digital-to-analog is fake, and you don't learn anything, you don't have any insights into the relationships of things.

I call analog methods "The Definition of Definition".

An example of an analog technique, angle tri-section.

http://www.flickr.com/photos/85937466@N ... hotostream

On my challenge:

viewtopic.php?f=13&t=39806#p163169

I specifically put in a no digital-to-analog fakery clause.

Last edited by skipjack; February 18th, 2017 at 04:20 AM.

 April 19th, 2013, 03:01 PM #17 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Constructing a 20 Degree Angle using Compass and Ruler O Anyone interested in deriving exact values of sines and cosines of special angles of the form 3k for integer k should look at the instructional video below. http://www.youtube.com/watch?v=kb_HyCThxE0 The whole video is extremely valuable but I would like to highlight a few things pertinent to finding exact values of special angles that are integer multiples of 3. 1) At time index 12:30 Ptolemy's Theorem is nicely derived. 2) At time index 22:00 exact values of some special angles are derived using triangles and addition/subtraction formulas for sines and cosines. Examples: 75, 60, 45, 30, 15. 3) Shortly after that, Ptolemy's Theorem is used to derive exact value of sin(18°) and once you have that sin(3) can be derived using subtraction formula sin(18 - 15) = sin(3). I strongly recommend looking at the entire instructional video, 30 minutes well spent. Question for [color=#0000FF]long_quach[/color], have you ever used Ptolemy's Theorem in your origami constructions? Last edited by skipjack; February 18th, 2017 at 04:22 AM.
April 20th, 2013, 05:41 AM   #18
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by agentredlum Question for [color=#0000FF]long_quach[/color], have you ever used Ptolemy's Theorem in your origami constructions?
No I haven't. but I've used just about everything else on my Flickr page.

Excellent video by the way.

April 20th, 2013, 05:44 AM   #19
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by agentredlum 3) Shortly after that Ptolemy's Theorem is used to derive exact value of sin(18°) and once you have that sin(3) can be derived using subtraction formula sin(18 - 15) = sin(3).
I can get down to 3 degrees too. But how do we get down to 1, to re-invent the protractor?

viewtopic.php?f=13&t=39895#p163585

Last edited by skipjack; February 18th, 2017 at 04:24 AM.

April 20th, 2013, 10:38 AM   #20
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by atharvjoshi Q - Is it Possible to Construct a 20 Degree Angle using Compass and Straightedge only? Atharv Joshi
I have re-created your drawing in GeoGrebra. It is 19.107 degrees.

http://www.flickr.com/photos/85937466@N ... otostream/

How old are you atharvjoshi? Telling us something about yourself tells us about the question you pose.

Don't take this "failure" the wrong way. Genius is NOT coming up with the solution. Genius is coming up with the question. How did you realize that 20 degrees is not so easy to construct? You attempt didn't work, but your question led to the question of the validity of the the protractor. How the heck was it constructed? We take it for granted that a protractor is easily constructed, but it's not. Don't let failed solutions discourage you. Genius is coming up with the questions, in noticing things out of the ordinary.

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