## Compass Correlation

Posted by Random Guy on September 26, 2010 – 6:49 pm

You have two pairs of compasses. One is set to length A, and the other is set to length B. The compasses cannot be adjusted. How do you determine the ratio of length A to length B using only a straight edge and a pencil?

September 26th, 2010 at 7:43 pm

mark a line along the straight edge with the pencil, mark start point and end point of the larger compass, then take small compass and ‘walk’ it to the end of the line and count how many times it cuts the line.

September 27th, 2010 at 6:23 am

Draw a straight line. Starting at one end of the line, arc off as many lengths of A and B as needed until two marks coincide. Count lengths of A and B needed name then Ac and Bc respectively. This is your ratio then Ac:Bc

September 27th, 2010 at 4:28 pm

John24. If the ratio is irrational, you’ll never get the coincidental marks.

I don’t know how to do this one (yet!)

September 27th, 2010 at 4:57 pm

My belief is that you can construct an irrational number (but I don’t know how to do that). But you cannot construct a transcendental number. So I conclude that this problem is insoluble. But maybe my argument is flawed in that we aren’t being asked to construct a number. Besides, I hope that no-one posts an insoluble problem. Just me being noisy

September 27th, 2010 at 5:00 pm

… oops! I should have said that you can construct an algebraic number. An infinite number of the algebraic numbers are irrational. But not all irrationals are algebraic. I’ll crawl back into my hole now.

September 28th, 2010 at 2:24 am

I don’t have time now to think on it, but clearly you cannot determine the ratio as a number (exactly because of the reason Chris provided). However, it should/might be possible to derive a line segment, which length is exactly the ratio of the two radii. For the latter I don’t have a construction so far!

September 28th, 2010 at 4:56 am

ooh slavy. I think that would mean that you could construct a transcendental.

I suspect that the question is insoluble (in general). But rational A and B is easy (as John24 and dave (pending at the moment)) have suggested.

On the algebraic numbers. I’m aware that they are the roots of rational polynomials, and that a rational polynomial provides a recipe (that’s a guess) for constructing the roots. I think Galois theory is involved in that.

September 28th, 2010 at 9:44 am

The use of the compass to find a ration will involve π, which is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.

So using the compasses to draw circles, the ratio would be – (π r sq A):(π r sq B). Putting a number to it…. nope.

September 28th, 2010 at 3:18 pm

The algebraic numbers form a field, i.e., if both radii are algebraic, then so is their ratio. Not being able to square the circle is a totally different problem (at least this is what I think). For example if my radii are 1 and \pi, a can easily construct \pi – so it is not always impossible The problem is rather geometrical then algebraic, so maybe we should focus on some bisectors and perpendiculars (those are the lines we can draw with an edge and a compass).

September 28th, 2010 at 10:55 pm

The intended answer was posted by John24, but after realising Chris’s reasoning that it might be impossible, I guess this question isnt that clear…

Carry on, people.

September 29th, 2010 at 6:24 am

Thanks Random Guy. I was beginning to worry that I was getting stupid. Never mind, your problem was a good idea.

Thanks slavy. I’m not sure that I knew that the algebraic numbers formed a field; it seems pretty obvious now that you’ve mentioned it.

October 1st, 2010 at 12:43 pm

draw first circle using any compass, say A. Then using circumference of that circle draw the next circle. Join the two center and extend the joint line to meet circle A at its circumference.

Ratio between the line which joins two center and the whole line segment gives the ration between A:B

October 3rd, 2010 at 8:02 pm

Compass is round – you cannot use a straight line to measure it

October 4th, 2010 at 4:34 pm

Starting from a point on the edge, travel with each compass, one compass at a time. Do this till the end points of the compass coincide on the same point.

Now, the ratio between compass A with compass B distances would be = Number of times compass B had to travel to reach the coinciding point / Number of times compass A had to travel to reach the same point..

October 7th, 2010 at 6:30 am

You may laugh at this as being far too simplistic . You have a pencil and a straight edge and two compasses . create a straight line usign the straight edge .placing both compass at a common start point you would have a segment which reaches from one the ouside of one arc to the other . mark this distance on your straight edge as a length of “A” . repeat this measurement along the straight edge from the start point . this woudl give you ( i think ) a ratio of (BxA)/(CxA) with B beign the number of repetitions of A from the start point to the edge of the inner circle and B being the number of repetitions of A tot he edge of the outer circle

October 16th, 2010 at 5:09 pm

draw 1 line with compass A, with compass B start at one point rotating until you get to the end of the line. provided that the compass provide somewhat of a good ratio, you will either have a ratio to scale down or up ie A:B or B:A

October 26th, 2010 at 4:31 am

we can fold it !