Can You Divide a Circle into 360 Equal Sections Without Special Instruments?

[mintparasol]

Hi forum,

Here is a little challenge that I came up with that some of you may find interesting:-

Using only a compass, ruler and pencil, can you draw a circle, of any diameter you wish, and divide its circumference exactly into 360 equal sections?

It seems to me that one should be able to accomplish this using only high school (Euclidean) geometry, but I haven't achieved it yet myself.

Bear in mind that I'm not a mathematician. I'm sure that many of you could do this quite easily.ad

 

[disregardthat]

It isn't possible. The reason is that one would be able to construct a 360-polygon out of that, but 360 is not a product of a power of 2 and distinct fermat primes, and thus not a constructible polygon.

 

[mintparasol]

How was it done in antiquity?

Or even comparatively recently?

 

[AlephZero]

The Greeks knew how to construct a regular pentagon.
See http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV11.html
Click the links on the right hand side to see how to do each step - the first step is not at all "obvious", you will probably need to go back two or thee "levels" to understand it.

They also knew how to trisect an angle using a marked straight-edge (i.e. a ruler) and compasses. See http://en.wikipedia.org/wiki/Angle_trisection

Bisecting an angle is easy.
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI10.html

The pentagon gives you an angle of 72 degrees. Trisectiing it twice gives 24 and then 8 degrees. Bisecting that three times gives 4, 2, and 1 degrees.

The "recent" easy way is use trigonometry to find the length of the side of a 360-sided polygon. Mathematicians had figured out how to make accurate tables of trig functions hundreds of years before electronic calculators and computers were available.

For machine tools, there were "dividing plates", which look like they were invented by Rube Goldberg, but are actually very accurate measuring devices:
http://www.atmsite.org/contrib/JSAPP/divide/divhead.html

 

[256bits]

Here is a description of trisecting an angle a little easier to comprehend.
http://www.uwgb.edu/dutchs/PSEUDOSC/Trisect0.HTM

Of course in a circle, the common angles in degrees of 90, 60, 45, 30, 15 are easily deduced.

You may be interested in some other unsolvable problems mentioned on the site:
1. construct a square the same area of a circle. ie if the circle has radius 1, the square has to have sides of length pi
2. construct a cube double the volume of another cube - this involves cube roots of lengths

I am sure there are many more.

 

[mintparasol]

Thanks Aleph and 256!

With my layman's understanding, I was using right-angles as my starting point and was never going to get from there to the 4' arc that is essential to finishing the job.

Does this mean that a 360-gon is actually constructable then?

 

[Number Nine]

I believe disregard that is right. A n-gon is constructible iff n is a product of a power of 2 and 2 mersene primes. Since 360 is not, a 360-gon is not constructible.

 

[micromass]

Does this mean that a 360-gon is actually constructable then?

Depends on the instruments you use. With a compass and a straightedge: no. With a compass and a marked straightedge: yes.

 

[mintparasol]

I believe disregard that is right. A n-gon is constructible iff n is a product of a power of 2 and 2 mersene primes. Since 360 is not, a 360-gon is not constructible.

Would it not be constructible by Alephzero's method, above?

 

[micromass]

Would it not be constructible by Alephzero's method, above?

AlephZero stated that he worked with a marked straightedge. So with one of those, it's constructible.

If you don't have a marked straightedge then it's not constructible.

 

[mintparasol]

Depends on the instruments you use. With a compass and a straightedge: no. With a compass and a marked straightedge: yes.

Well, by 'ruler', I meant a marked straight edge.

edit:-

And by 'Euclidean', I meant basic high school geometry. This little experiment came to me the other night after reading of base 12 and base 60 number systems and I wondered exactly how the ancients might have divided a circle into 360'.

It's only since I've followed the links posted in this thread that I've seen that the use of a compass and unmarked straightedge was a restriction imposed by the Greeks on many plane geometry problems! So perhaps the thread title was misleading..

Thanks for the replies and the help,
ad