Quantitative results of packing 12,000 telescopic heliostats along a phyllotaxis spiral

12,000 circles packed by phyllotaxis using Bumgardner factor = 0.86. Land utilization is 0.701.

12,000 ellipses (36:1 aspect ratio) packed on a phyllotaxis spiral with Baumgardner factor = 0.86. The view fill factor is 0.639; the blocking factor, 0.008, is less than one percent. (Multiple overlaps, as occur in the center, are neglected in calculating the blocking.)

Here are some results from generating high-resolution off-screen images in Processing and using a histogram analysis sketch to measure land fill, view fill, and blocking factors when telescopic heliostats are arranged along a phyllotaxis spiral.



PRIMARIES
Bumgardner factor            Land fill factor             Blocking factor

0.92                                  79.5%                           1.6%
0.89                                  75.4%                            0.6%
0.87                                  72.4%                            0.2%
0.86                                  70.1%                              0%              



SECONDARIES
Bumgardner factor             View fill factor              Blocking factor

0.86                                    63.9%                           0.8%

Bumgardner's "fudge" factor is the ratio of the radius of a heliostat to the radius of the imaginary circle encompassing the heliostats share of the field area. With zero blocking, as there is for circles when fudge = 0.86, the land fill factor should be fudge * fudge = 0.74. The difference between 0.74 and the measured 0.70 is the extra field area added beyond the center of the farthest heliostat so that the whole heliostat lies within the field. The importance of this extra margin diminishes as the number of heliostats increases. In a very numerous array, we may take 0.74 as the land fill factor.

It may look like there are a great many heliostats in the images above, but in a GW-scale heliostat field, this many heliostats and more would be displaced by the central optics! Even placing this many heliostats along a phyllotaxis spiral requires a value for the golden angle more accurate than Processing's single-precision functions can calculate.