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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?

This post is under “Tom” and has 17 respond so far.
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  1. 1. dave Said:

    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.

  2. 2. John24 Said:

    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

  3. 3. Chris Said:

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

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

  4. 4. Chris Said:

    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 :)

  5. 5. Chris Said:

    … 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.

  6. 6. slavy Said:

    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!

  7. 7. Chris Said:

    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.

  8. 8. Karl Sharman Said:

    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.

  9. 9. slavy Said:

    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).

  10. 10. Random Guy Said:

    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.

  11. 11. Chris Said:

    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.

  12. 12. vedsar Said:

    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

  13. 13. Irfaan Said:

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

  14. 14. Indian Said:

    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..

  15. 15. Turtle Said:

    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

  16. 16. sean Said:

    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

  17. 17. lavanya Said:

    we can fold it !

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