Hello

I'm a naturalist and unclear what maths approach would best estimate the number of regular hexagonal facets on an insect compound eye when the facets are small and assume eye is part of a sphere?

The facet count on different insect eyes are widely stated but not clear how they were calculated. Some insects have many thousands on an eye so impractical to count them.

Microscopy with calibrated optics can give:

The hexagonal facet side length / width

The diameter of the hemisphere of eye.

Less accurately the number of hexagonal facets along a great circle of hemisphere.

Using any one or more of these parameters what would be the best modern approximation for estimating total facets?

- an early microscopist (Leeuwenhoek 1702) counted the number of facets along a quadrant of a great circle and used this figure alone and using the method ‘after Metius’ (quadrature?). He counted 35 facets along quadrant and calculated 6236 on a sphere (ie number in both eyes of cultivated silk moth).

A 19th century worker gave geometric arguments why this approach needed a correction to 7213 facets (quadrature assumes squares not hexagons). His paper is here if of interest.

Original Communications: Remarks on the Cornea of the Eye in Insects, with reference to certain sources of fallacy in the ordinary mode of computing the Microscopic hexagonal Facets of this membrane: with an Appendix, containing a brief notice of a n

He suggested punching a circle of facets from eye then count facets in circle and work from there (or now could do this virtually on a digital image), but need to account for cut hexagons along circle perimeter; he describes counting facets in a rhomb which gives better packing.

- modern optical microscopy can directly measure facet size and hemisphere size in absolute units, eg microns. So could divide the calculated area of one facet into the total area of the eye assuming part of a sphere. Is this a better method than the above? So from my measurements, a facet 25.6 microns wide and sphere 1400 microns diameter gives 10850 facets on sphere (both eyes).

- I tried the old method counting facets along quadrant, this is difficult, but averaged 45 facets along a great circle quadrant to give 10309 by Leeuwenhoek’s method (uncorrected).

There's many assumptions from the nature viewpoint, not least variation between specimen of given species, but interested in best approach from the maths. I gather a regular polyhedra of flat hexagons doesn’t exist, although was unclear if a spherical surface could be perfectly inscribed with 'curved' hexagons. I notice the insect eye studied had occasional pentagons and other irregularities, not sure if this reflects inability to perfectly pack hexagons.

Thanks for persevering with these ramblings and any insight appreciated!

regards

David

Image of a whole cultivated silk moth eye I cleared to show facets: http://www.microscopy-uk.org.uk/mag/...dwDSC00039.jpg

Image of part of surface of facets under my microscope: http://www.microscopy-uk.org.uk/mag/...dwIMG_0930.jpg