Dividing a circle into areas
In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
Though the first five terms match the geometric progression 2n − 1, it deviates at n = 6, showing the risk of generalising from only a few observations.
The lemma establishes an important property for solving the problem.
In the figure the dark lines are connecting points 1 through 4 dividing the circle into 8 total regions (i.e., f(4) = 8).
squares, this combines to Finally, which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior).
Under this assumption of "generic intersection", the number of regions can also be determined in a non-inductive way, using the formula for the Euler characteristic of a connected planar graph (viewed here as a graph embedded in the 2-sphere S 2).
View the diagram (the circle together with all the chords) above as a planar graph.
Thus the main task in determining V is finding the number of interior vertices.
As a consequence of the lemma, any two intersecting chords will uniquely determine an interior vertex.
Further, by definition all interior vertices are formed by intersecting chords.
Therefore, each interior vertex is uniquely determined by a combination of four exterior vertices, where the number of interior vertices is given by and so The edges include the n circular arcs connecting pairs of adjacent exterior vertices, as well as the chordal line segments (described below) created inside the circle by the collection of chords.
As chords are uniquely determined by two exterior vertices, there are altogether group 3 edges.
The sum of these results divided by two gives the combined number of edges in groups 2 and 3.
one then obtains Since one of these faces is the exterior of the circle, the number of regions rG inside the circle is F − 1, or which resolves to which yields the same quartic polynomial obtained by using the inductive method The fifth column of Bernoulli's triangle (k = 4) gives the maximum number of regions in the problem of dividing a circle into areas for n + 1 points, where n ≥ 4.
Considering the force-free motion of a particle inside a circle it was shown (see D. Jaud) that for specific reflection angles along the circle boundary the associated area division sequence is given by an arithmetic series.
