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Thread: diameter-chord ratio

  1. #1
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    diameter-chord ratio

    Can I find the ratio between the number of diameters to the number of chords in a given circle?
    What can You say on the relationship between them?
    Can I say that:
    number diameter is a subset number of chords.
    Is the a this relationship is fixed number? Why? Or why not?
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  2. #2
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    Re: diameter-chord ratio

    There is no relationship between the number of chords and the number of diameters of a circle. This is because the set of all chords or diameters has a cardinality that is not a number. You can say that for any point on a circle, there is exactly one other point that will make a diameter while you can choose any other point to make a chord. So, you can say the cardinality of diameters may be represented (informally, and slightly abusing notation):

    $$\dfrac{1}{2}\dbinom{|S^1|}{1}$$

    While the cardinality of chords might be:

    $$\dbinom{|S^1|}{2}$$

    But these sets both have cardinality of the continuum, so their cardinalities are equal.
    Last edited by SlipEternal; Sep 15th 2018 at 12:58 PM.
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  3. #3
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    Re: diameter-chord ratio

    Given a single point on the circle, any line from that point to another is a chord so there are infinitely many chords through that point but only one of them is a diameter. But there are an infinite number of points on the circle so you not only have infinitely many chords you have infinitely many diameters. There is no good way to compare "infinities" except in terms of "cardinalities" as SlipEternal said.
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