Prof. Jeffrey Grossman of MIT is investigating 3D shapes for solar cells that optimize the performance of _non-tracking_ photovoltaic solar arrays. He is assuming that the cost of the PV material is no longer the limiting factor. He is using so-called "genetic algorithms" to optimize initially random patterns: http://dmse.scripts.mit.edu/news/?p=1753 His recent talk in Palo Alto (available somewhere on the web) leads me to pose the following question: Consider a 2D case. The sun rotates around a horizontal plane(actually line). We wish to come up with a 2D solar array (made up of plane(actually line) elements that maximizes the power generated _averaged over the whole day_. In our simplified 2D case, the sun only shines for 12 hours out of the 24 hour day, and during the daylight, the sun moves at a constant rate in a semicircle above the plane(actually line). We assume that the performance of small flat plane(line) element is proportional to length*cos(theta), where theta is the sun's angle with the element. We also assume that every element is large relative to the wavelength of the light, so diffractive effects can be ignored. We hold constant the area(actually length) of the _footprint_ (vertical projection) of the array, but allow any shape for the array itself. What would an optimal shape look like? (We also assume that any part in a shadow produces zero power.) Ditto, but now we also allow a combination of PV elements and _mirror_ elements -- i.e., an element can be either a PV element or a mirror element, but not both. However, an element could be a PV element on one side and a mirror element on the other. We also assume that the area outside the footprint of the array is non-reflective, so that any sun energy hitting outside won't bounce into the array. (A flat passive mirror outside the footprint could be part c of this question.) I presume that the optimum shape would be some kind of fractal, but I haven't been able to figure out how to visualize the problem yet. It would appear that the answer to part c might be a simple circle at some distance above the plane/line, because the mirror effect of the plane/line would make the solar array think that there were _two_ suns at the same time -- at angles +-phi above/below the horizon.