Answer: (m, 45k) where k is an integer such that
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Reason:
Any polar point is of the form (r, theta)
r = distance from the pole to the terminal point
theta = angle formed by the positive x axis and the segment of length r.
The center of the octagon is at the pole, which is the origin.
Then we can go m units either left or right to arrive at a vertex point.
Going m units to the right and we arrive at (r,theta) = (m,0).
Going m units to the left and we arrive at (r,theta) = (m,180).
We could also go m units up or down to arrive at two more vertices.
Going m units up and we arrive at (r,theta) = (m,90).
Going m units down and we arrive at (r,theta) = (m,270).
So far, we've split the polar plane into 4 quadrants. We can split each quadrant in half along a diagonal cut. That gives 4*2 = 8 different regions. This of course fits perfectly with the 8 vertices needed for the octagon.
This means 360/8 = 45 degrees is the angle of separation from one polar point to its next door neighbor.
For instance, one such pair of neighbors is (m,0) and (m,45).
In general, we have the eight vertices of the general form (m, 45k) where k is an integer such that
If you wanted, you can choose to expand things out like this
(m, 45k) = (m, 45*0) = (m, 0)
(m, 45k) = (m, 45*1) = (m, 45)
(m, 45k) = (m, 45*2) = (m, 90)
(m, 45k) = (m, 45*3) = (m, 135)
(m, 45k) = (m, 45*4) = (m, 180)
(m, 45k) = (m, 45*5) = (m, 225)
(m, 45k) = (m, 45*6) = (m, 270)
(m, 45k) = (m, 45*7) = (m, 315)
which cleans up to this list here
(m, 0)
(m, 45)
(m, 90)
(m, 135)
(m, 180)
(m, 225)
(m, 270)
(m, 315)
Though I think there's more elegance in being able to express all 8 solutions as one expression. So I think writing (m, 45k) is the better path.
Side note: if you need to convert to radians, then multiply each degree measure by pi/180. You should find that 45 degrees = pi/4 radians.