A couple of quick thoughts:
The arts & crafts, or Ruskin approach:
Let it be approximate. Your projects are beautiful. Do they have to approximate laser-cut, computer-controlled machined wood?
The geometric constructivist approach (spelled out in words, this appears far more complex than it is to do):
1. Set your canvas... 480 x 480... manually draw from corner to corner, establishing the center.
2.A. draw a working circle using the intersection of these lines as the origin.
2.B Strike a new working circle greater than 240pix radius, using the intersection of the working circle in 2.A. and one of the 45 degree lines from 1.
2.C. Clone this circle and move its origin to the corresponding origin opposite, right.
2.D. Clone this circle, move to corresponding origin bottom right.
2.E. Clone this circle, move to opposite origin, bottom left.
2.F. Draw lines from the intersections of these circles (fat ellipses in diagram below).
3. Continue bisecting areas as in step 2.
The attached XAR file has layers you can turn on and off to see the steps more clearly.
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