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 20th, 2013, 12:28 PM   #21
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Hi atharvjoshi!

Your drawing is not accurate - the angle is close to 19°, not 20°
There are many methods leading to angle much closer to 20° (of course not exactly 20°)
For example, the very simple drawing below give 20.003° , from page 9 of the paper "Drawing of any angle using compass and straightedge": http://www.scribd.com/JJacquelin/documents
Attached Images
 20 degree.JPG (25.4 KB, 671 views)

Last edited by skipjack; February 18th, 2017 at 04:26 AM.

 April 20th, 2013, 05:35 PM #22 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O Gosub viewtopic.php?f=13&t=18374&start=15#p163586 Return That's digital-to-analog fakery. The HP-35 was Hewlett-Packard's first pocket calculator and the world's first scientific pocket calculator[1] (a calculator with trigonometric and exponential functions). . . Introduced at US\$395, the HP-35 was available from 1972 to 1975. - Wikipedia. CORDIC (for COordinate Rotation DIgital Computer), also known as the digit-by-digit method and Volder's algorithm, is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions . . .The modern CORDIC algorithm was first described in 1959 by Jack E. Volder. - Wikipedia. Digital methods are only invented in the late 20th century. You're not learning anything doing it this way. You might as well punch in the tangent and use a ruler to measure X and Y at 90 degrees. Compare that with real analog technology. http://www.flickr.com/photos/85937466@N ... hotostream Proof http://www.flickr.com/photos/85937466@N ... 074235305/ We can draw a 60 degrees and trisect it. "This method was known to Hippocrates more than 2400 years ago." How much more elegant it is. How durable it is. It's been around for thousands of years, not decades. What can we learn from Archimedes method of angle tri-section? Everything. From this method, called neusis, I invented "on the spot" a way to solve a quadratic equation, just like that, without remembering anything. I even had to look up the algebraic solution to a quadratic equation on Wikipedia. But I invented a method of solving a quadratic equation by understanding the analog angle tri-section. viewtopic.php?f=13&t=39806#p163400 What can you learn from knowing that inverse tangent 5/8 = 32.0053 degrees? Nothing, except calculators sure are damn fast and accurate and let's hope we have oil forever to make our modern plastics and stuff, otherwise it's back to an agrarian society for us. Watch Gone with the Wind if you don't know what an agrarian society is, on your DVD cause they don't have analog VHS anymore!
April 20th, 2013, 05:52 PM   #23
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
Originally Posted by long_quach
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 reinvent the protractor?

viewtopic.php?f=13&t=39895#p163585
We can't even get 2° using unmarked straightedge and compass (collapsible)

Think about the following reasoning (I'm not sure 100% so I'm asking other MMF members to help fix the logic)

1) If 2° could be constructed then sin2° could be constructed

2) Then we would be able to get exact value of sin(18 + 2) = sin(20) in terms of simple square roots of positive integers

3) This would have as a consequence that a 20° angle is constructible and that would mean a 60° angle can be trisected. We have reached a contradiction so sin2° is not constructible and 2° is not constructible.

I'm not 100% sure 3) follows from 2) so I'm asking for help.

Another way to see this,

1) In order to construct angles of 1° or 2° , regular polygons of 360° and 180° must be constructible.

2) http://oeis.org/A003401

3) The numbers 360 and 180 are NOT in the above sequence therefore angles of 1° and 2° are NOT constructible.

Last edited by skipjack; February 18th, 2017 at 04:29 AM.

 April 20th, 2013, 06:32 PM #24 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O I wonder if there is a genius to this. We have been walking around with a fake protractor for thousands of years. I wonder if it is intentional. I mean if I want to write a message to the future "don't forget about neusis" in addition to classical stuffs (straight edge and compass), I would encode it in a 360 degree circle. Because that would be genius! I can guess by re-constructing history in my imagination. Neusis was frowned upon because you cannot for certain "dial" into a correct length. http://www.flickr.com/photos/85937466@N ... hotostream But our modern experience tells us that we can dial into a correct length. We can dial into a radio station, dial into a tv station (back in the days), dialing the focus of a camera (back in the days), we tune musical instruments by dialing the tensions on the strings. I would blow my mind away if that was the intention of the 360 fake protractor, as a message to the future to remember neusis.
 April 20th, 2013, 11:18 PM #25 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: Constructing a 20 Degree Angle using Compass and Ruler O Hey atharvjoshi why are you so obsessed by the 20° angle? Why not the 1 radian angle? It should be useful as well, maybe more. Have a good time to construct a 1 radian angle with compass and straightedge. Last edited by skipjack; February 18th, 2017 at 04:30 AM.
April 21st, 2013, 05:06 AM   #26
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by JJacquelin Why not the 1 radian angle ?
I had to look this one up.

http://www.teacherschoice.com.au/maths_ ... angles.htm

"One radian is the angle of an arc created by wrapping the radius of a circle around its circumference."

How the heck can you "wrap" a straight line around a circumference? Besides rolling a length on a circle.

What's this radian business? 2?? How do you get ?? Rolling a circle on a ruler? That's more digital dependency.

April 21st, 2013, 05:12 AM   #27
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by JJacquelin Hey atharvjoshi why are you so obsessed by the 20° angle ?
Atharvjoshi is very clever to notice something out of the ordinary. We look at a protractor and it looks neat and simple. Nice divisible 360 on a decimal system. Who would ever suspect that those nice round numbers 10, 20, 30 . . . are fake?! Though I have my suspicion, I haven't really looked at it until now.

I have my suspicion when trying to construct a 12 degree angle, a classical problem invented by Pythagoreans who invented the pentagram. I learned that from "Donald Duck in MathemagicLand".

http://www.flickr.com/photos/85937466@N ... 639808715/

Last edited by skipjack; February 18th, 2017 at 04:31 AM.

 April 21st, 2013, 05:51 AM #28 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O I found the answer. http://mathforum.org/library/drmath/view/54183.html In the reply to a question regarding constructing an angle of one degree it was stated that an angle of one degree cannot be constructed using just a straight edge and a compass because the sine and cosine of one degree both require cube roots and only square roots can be constructed (http://mathforum.org/dr.math/problems/c ... 25.00.html ).
 April 21st, 2013, 06:08 AM #29 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O Which leads us the the question: Why does tri-secting an angle involve cube roots? http://www.flickr.com/photos/85937466@N ... 074235305/ I intuitively understand it. But no way can I explain it.
 April 21st, 2013, 06:16 AM #30 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Re: Constructing a 20 Degree Angle using Compass and Ruler O I'm reading this. This is an algebraic explanation. It will take me awhile to understand it. Still, I prefer a geometric explanation. Page 10. http://www.math.tamu.edu/~mpilant/math6 ... idterm.pdf

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