Best way to estimate number of hexagonal facets on an insect eye if assume a sphere?

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/imgoct12/dwDSC00039.jpg
Image of part of surface of facets under my microscope: http://www.microscopy-uk.org.uk/mag/imgoct12/dwIMG_0930.jpg

Re: Best way to estimate number of hexagonal facets on an insect eye if assume a sphe

Vlasev said:

I like your method of calculating facet area and dividing the total area by the facet area. It's simple and you don't have to count too much. The question is, how accurate do you want this measurement of the number of facets to be?

Maybe you can measure the area of several different facets, take the average, and divide the sphere area by this area. You should be careful to include half the width of the facet wall. Say if the thickness of a wall is t and the distance between two walls is d , the area of the facet is \sqrt{3}d^2/2 . The correction is A = \sqrt{3}(d+t)^2/2 , and this is the effective area of the facet in your measurement.

This is all I can say on this for now.

Click to expand...

Hello Vlasev

Many thanks for your very useful reply. Given the likely variations between insect specimens for a species, accuracy is not vital, but I became intrigued with the best maths approach to this sort of problem.

Good to hear the total sphere area divided by an averaged facet area seems sound as wasn't certain if it was as yet to find any naturalist articles describing this method.
Thanks for the wall thickness correction which I hadn't considered, which I can measure from my facet images. Given the facets are very small, over 6000 present, the wall thickness will have a significant effect.

Perhaps I can throw some loose ends into the forum as some aspects still puzzle me of the insect eye geometry.

- a sphere can't be made from flat hexagons and there seems a very low density of pentagons on insect eye to give wrapping (ie like a soccer ball of hexagons and pentagons)

- if an insect makes the hexagons curved does this from a geometry view give a sphere ie can a sphere surface be perfectly inscribed with hexagons to give perfect packing? As an aside are there many regular curved shapes that can perfectly pack when drawn on a curved sphere, pentagons?

Thanks for any insight.

regards
David