Date: 01/28/2001 at 06:24:07 From: Anonymous Subject: Regions in a circle What is the maximum number of regions you can have with n chords in a circle? I've already found: With 0 chords you have 1 field With 1 chord you have 2 fields; 1 more 2 4 2 more 3 7 3 4 11 4 etc. Now I need a formula and a proof why this is so. Can you help me? Thanks.
Date: 01/28/2001 at 08:43:45 From: Doctor Anthony Subject: Re: Regions in a circle We can find a general formula for the number of regions when the interior of a circle is divided by n lines. Suppose the number of regions is given by f(n) when there are n lines drawn in the circle. Now draw one more line cutting all the other n lines. There are n points on the additional line, and so this line must traverse n+1 of the available f(n) regions, dividing each into two parts. It therefore adds n+1 more regions to those present. Thus f(n+1) = f(n) + n+1 We can write f(n+1) - f(n) = n+1 and now form a succession of equations as follows: f(1) - f(0) = 1 f(2) - f(1) = 2 f(3) - f(2) = 3 ................... ................... f(n-1) - f(n-2) = n-1 f(n) - f(n-1) = n ------------------------- adding all the equations, f(n) - f(0) = SUM( 1 to n) (note cancellation between lines) f(n) - 1 = n(n+1)/2 f(n) = n(n+1)/2 + 1 Check if this is correct n=0 gives f(0) = 1 n=1 gives f(1) = 2 n=2 gives f(2) = 4 and these are correct. With n = 4 we get 4 x 5/2 + 1 = 11 regions. And so on. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
Date: 04/19/2003 at 05:00:58 From: Sheila Subject: Regions in a circle We drew a circle and then put two points on the line and joined the points. The circle is divided into two parts. If there are 4 points (not evenly spaced) and they are each joined to all the other points, the circle is divided up into 8 regions. There is a conjecture that says that if the process is repeated, with more points along the edge of the circle, not evenly spaced, then a rule for the maximum number of regions is 2 to the power of (the number of points take one) r=2^(n-1) where ^ = to the power of. For 2,3,4,5,6 points, they follow the formula, but for 7 points, it doesn't; there are 57 regions, and if it followed the rule, there'd be 64. Am I not counting them properly? Is there a simple way of understanding how many regions there are, because there must be a constant formula, shouldn't there be?
Date: 04/19/2003 at 05:30:48 From: Doctor Anthony Subject: Re: Regions in a circle Regions of a Circle Cut by Chords to n Points ---------------------------------------------- n points are distributed round the circumference of a circle and each point is joined to every other point by a chord of the circle. Assuming that no three chords intersect at a point inside the circle we require the number of regions into which the circle is divided. With no lines the circle has just one region. Now consider any collection of lines. If you draw a new line across the circle which does not cross any existing lines, then the effect is to increase the number of regions by 1. In addition, every time a new line crosses an existing line inside the circle the number of regions is increased by 1 again. So in any such arrangement number of regions = 1 + number of lines + number of interior intersections = 1 + C(n,2) + C(n,4) Note that the number of lines is the number of ways 2 points can be chosen from n points. Also, the number of interior intersections is the number of quadrilaterals that can be formed from n points, since each quadrilateral produces just 1 intersection where the diagonals of the quadrilateral intersect. Examples: n=4 Number of regions = 1 + C(4,2) + C(4,4) = 8 n=5 Number of regions = 1 + C(5,2) + C(5,4) = 16 n=6 " " = 1 + C(6,2) + C(6,4) = 31 n=7 " " = 1 + C(7,2) + C(7,4) = 57 Note that formula 2^(n-1) starts to go wrong at n=6 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
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