Suppose one has come into ownership of a company made from the fusion of N equally important little companies. Let us not worry how this strange fait accompli of plutocratic shenanigan may have come to pass – doubtless there was intrigue involved!
Anyhow, it is important that none of the employees or managers of any of the N little companies feel slighted as we are given the task of designing a logo that represents each of the N logos “fairly.”
By this, we require (for sake of keeping our designers from going mad) that
a) All the little logos can be inscribed in a circle of area pi/N
b) The new logo will be circular and subtend area pi.
c) Yet instead of slicing the circle into N tiny slices (none of which is very suitable for displaying a logo inside), we wish each of the N slices, to be as close to circular as possible, but polygonal, so as not to waste space.
By close to a circle, we mean that the proportion of overlap between the polygon in which each logo is inscribed and a circle should be maximal. (minimizing the proportion of non-overlap)
First, before you think of simply dividing a circle into pie slices, take a look at http://www.mathteacherctk.com/blog/2008/02/dividing-circular-area-into-equal-parts/ . Like pie slices, though, these shapes don’t work well as shrink-wraps for logos!
Packing circles within circles has been studied extensively (see Ron Graham’s work on the subject at http://www.math.ucsd.edu/~ronspubs/98_01_circles.pdf or more modern inquiry, for example, at: http://hydra.nat.uni-magdeburg.de/packing/cci/d2.html _ Hence, we might consider importing the centers of the N circles found from an optimal packing at letting those become the centers of a Voronoi diagram. The problem though, is that the resulting polygons will not have equal area and for certain values of N (like 20, 23, 29, or 32) the asymmetry induced will result in large area and/or shape distortion near the edges of the circle. (packings tend to be more honeycomb like toward the center, with our corncobs more likely to miss teeth at the edges)
Sometimes, like when N=31, the packing is hexagonally tight in the center.
So what ideas might have you folks? Clearly the concept of packing the logos so they subtend equal areas, become more complicated as their individual geometries diverge from circular and from one another.
I suspect that for N=4, four circular sectors (see http://mathworld.wolfram.com/CircularSegment.html ) would be more similar to the circle than would be a solution involving a single circle placed atop three triangular sectors.
For five, however, the Zia sun sign would seem to be a more effective solution (for maximizing areas of the five logos) than five circular sectors which become awfully triangular (hence cropping out too much of each of the individual logos).
For nine, the solution at http://math.stackexchange.com/questions/220161/divide-circle-into-9-pieces-of-equal-area appears quite workable.
So what would be a best design for 5, 6 or 7?
Note that we may rephrase the above with, instead of circles, arbitrary polygons. It then starts to look a bit like tiling polyominoes with congruent polyominoes (which you may google, if so inclined).