9712

Overlapping Circles

In this Demonstration, the gray circle is a unit circle centered at the origin, and unit circles are placed evenly on its circumference. The area contained in regions with overlapping circles is calculated and compared to the area of the bounding circle (the dotted circle) that has radius 2.

THINGS TO TRY

SNAPSHOTS

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DETAILS

As and is held constant, the colored area will fill the bounding circle. If the integer is chosen to be proportional to , so , then the colored area approaches as . Since the range of valid is , the ratio can be tuned so that the colored area approaches anywhere between none or all of the bounding circle's area.
It is interesting to note that for , the area with at least overlaps is exactly half the bounding (dotted) circle. The first three such cases are snapshot 3, 4 and 5 respectively.
Also, whenever , the intersection point lies exactly on the central unit circle.
There is an infinite number of cases where , and such that the intersection lies on the unit circle and the intersecting regions take up exactly half the bounding circle.
The first such case () is when , the intersection points lie on the central unit circle and the regions have exactly half the area of the bounding circle. The symbol is known as the "seed of life", which is a basic component in the "flower of life". This is shown in the fourth snapshot.
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