I then decided to experiment with other numbers that are easier to plot to find a pattern.
So, realizing that it's much simpler to divide everything into halves and plotting the points that way, I went on to use 2, 4, 8, and then 16 equally spaced points, drew them on a piece of paper using a Starbucks cup and labelled each point clockwise (as shown in photo).
I noticed that for each diagram I have drawn, the pairs of diametrical opposite points look like this:
Number of Points Opposite Pairs
Two points point 1 - point 2
Four points point 1 - point 3
point 2 - point 4
Eight points: point 1 - point 5
point 2 - point 6
point 3 - point 7
point 4 - point 8
Sixteen points point 1 - point 9
point 2 - point 10
point 3 - point 11
point 4 - point 12
point 5 - point 13
point 6 - point 14
point 7 - point 15
point 8 - point 16
Consequently, I realized that there is a pattern and formula to find the opposite point for each particular point on the circle. Suppose that there are n equally spaced points, then:
Opposite point = Base point + n/2, when base point < n/2; and
Opposite point = Base point - n/2, when base point > n/2
Using this formula, the opposite point for point #7 on a 30-point circle is 7 + 30/2 = 22, so point #22.
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