>> I question the idea that bees could build cells 17% smaller than
>> they are. How would a bee fit inside a cell 4.3 mm wide to build it?
>> The thorax of the European honey bee is about 4 mm.

> I wonder the same thing.  In normal hives, there are bees of varying
> size, but one would winder why they would make cells that tight.  
> I seem to recall 4.3 cells having been the smallest cells reported in 
> EHB natural comb.  Natural comb can sag, however, and cells change 
> shape, so I don't know.

It's a straightforward issue of geometry.  Circles have larger diameters
than hexagons of equal perimeter.

Bees do not build hexagonal ("toth structure") cells.  They build cylinders,
which then deform under pressure into the "toth" crystalline structure we
know as a bee cell, just as adjacent soap bubbles blown between two sheets
of glass will deform.  (see below)

So, take a 4.9mm diameter circle as the size of the opening to the cylinder
made by the bees.
Its radius would be half the diameter = 2.45
Its circumference would be (pi * diameter) = 15.393804
Is surface area would be (pi * radius^2) = 18.8574099

When it deforms to a hexagon, the circumference (now a "perimeter") would
remain the same, 15.393804, but it would now be broken into six equal flat
sides.

So, each side would be the perimeter / 6 = 2.566
The diagonal (between corners) is twice the side length = 5.132
The distance between the flat sides would be the diagonal divided by 1.155 =
4.44

So, "4.3" measured across the flats of the cells is well within the margin
of measurement error for a 4.44 distance across the flats hexagon that
started out as a 4.9mm circle.

The area of the circle ( pi*r^2 ) = 18.8574099 is also larger than the area
of the hexagon of the same perimeter
(3 / 2 *a^2 * sqrt(3)) = 17.102


One of the surprising basic things that the small-cell acolytes are not
taught is how comb actually gets built.  This is pretty jaw-dropping given
the minutiae about comb to which great attention is given, such as the
properties ascribed to the (actually randomly-formed) cell base edge
patterns termed "Housel Positioning".  It is the inevitable result of the
self-described philosophy of "no scientists, just really good beekeepers"
(formerly "no scientists, just good beekeepers").  This misinformation has
sadly made its way into at least one recently-published but obscure book,
stating "bees...use their mandibles (mouth parts) to shape the wax into
perfect hexagons". (See
http://www.dheaf.plus.com/warrebeekeeping/idiot.pdf )

On the surface of water, close-packed bubbles will deform into hexagons,
which may be an easier test set up for a demo for the doubting.

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