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Re: Circles puzzle
Posted:
Feb 5, 2012 6:39 AM
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> OK.
>
> Imagine a circle A. Regularly spaced on this
> circle's circumference
> are the centres of a number of additional circles who
> just touch
> (inside circle A) but don't overlap each other.
>
> If you have two such circles, spaced 180 degrees
> apart, they'll touch
> at the centre of circle A, so their radii will be
> equal to that of
> circle A, and their total area will be twice that of
> circle A.
>
> If you have 1 million such circles, they'll be just
> very tiny, looking
> like a thicker line on the circumference of circle A,
> and their total
> summed area will be less than that of circle A.
>
> What is the lowest integer number of circles whose
> total area is equal
> to or less than that of circle A?
> What are two positive real numbers of circles whose
> total area is
> equal to that of circle A? Can these numbers be
> expressed as anything
> other than arbitrary decimals?
>
> Does the puzzle extend to higher dimensions?
>
> Eric
Hi Eric,
let A be a unit circle, then the area Sn of n circles placed on A as described is:
Sn = n*pi*(sin(pi/n))^2 .
The area of A is:
So = pi .
For n = 10, Sn < So .
The puzzle can not be extended to higher dimensions. For example in 3D there are only 5 regular polyhedra, in contrast to 2D where there are regular polygons of any number of sides.
Best regards,
Avni
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