Moser Circle Regions and Concurrence
The familiar doubling pattern breaks at six circle points, and one hidden concurrence changes the count again.
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Moser Circle Regions and Concurrence
The problem of finding the maximum number of regions created by connecting n points on a circle with straight chords is a classic example in mathematics illustrating the danger of early extrapolation.
Theory
For the first few values of n, the number of regions follows a doubling pattern: 1, 2, 4, 8, 16. It is tempting to predict that for n=6, there will be 32 regions. However, the actual maximum number of regions for n=6 in general position is 31.
The formula for the maximum number of regions in general position (where no three chords intersect at a single interior point) is given by:
R = 1 + C(n, 2) + C(n, 4)
When chords concur (three or more intersecting at a single point), the number of regions decreases. Specifically, if k chords intersect at a point, they add k-1 to the region count instead of the C(k, 2) intersections that would occur if they were slightly perturbed.
Worked Example
Let us calculate the maximum number of regions for n=6 in general position using the formula.
- The base region count is 1.
- The number of chords is C(6, 2) = 15, which corresponds to 15 edges added.
- The number of distinct interior crossings is C(6, 4) = 15 (assuming no concurrences).
- Adding these together: 1 + 15 + 15 = 31.
Thus, the expected maximum number of regions is 31, breaking the doubling pattern sequence of 32.
Experiment
You can interact with a live model to observe these patterns and the effects of concurrences:
Experiment with the following steps:
- Set Circle points to 5 and verify that the actual regions count is 16.
- Set Circle points to 6 and Configuration to "Generic" to see 31 regions.
- Change the Configuration to "Central symmetry". Notice that three chords now intersect at a single central point.
- Observe that the distinct interior crossings drop to 13 and the actual regions drop to 30, creating a concurrence deficit of 1.
Limitations
The algebraic model uses precise rational points to avoid floating-point inaccuracies when determining exactly coincident intersections. However, finding points that are both on a circle and perfectly symmetric computationally can be non-trivial. The provided symmetric configuration uses a projective representation to guarantee exact algebraic intersections.
References
For more information on the history and mathematics behind Moser's circle regions problem, see the following resource:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Moser Circle Regions and Concurrence module.
Try Moser Circle Regions and Concurrence on Riano →