Overlapping Circles

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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.
Contributed by: Simon Tyler (December 2011)
Open content licensed under CC BY-NC-SA
Snapshots
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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