Why the average cell has six sides
Cells in a sheet — an epithelium, a soap froth, a leaf surface, the cells of a seed head — average six sides. The standard explanation is that hexagons are the most efficient way to tile a plane, which is a true statement about a different question.
The mean is six because a tiling cannot average anything else.
The counting argument
Euler’s formula for a connected planar graph says V − E + F = 2. In a tiling where three cells meet at each vertex — which is what soap films, and most epithelia, and a generic Voronoi diagram all do — the counting is short.
Each vertex has three edges meeting it, and each edge has two ends, so 3V = 2E. Each face has some number of sides, and each edge is shared by two faces, so the sum of all the side counts is 2E.
Substituting into Euler’s formula and letting the tiling get large, the mean number of sides per face goes to exactly six.
Nothing in that argument mentions area, perimeter, efficiency or preference. It is a counting identity, and any large tiling with three-way vertices satisfies it whatever the cells are doing.
What this rules out
The efficiency argument says that of all the ways to divide a plane into equal-area cells, hexagons have the least total perimeter. That is true — it is the honeycomb theorem, proved by Hales in 1999 — and it is a statement about the optimal tiling.
The counting argument is a statement about every tiling. A sheet of cells that were not optimising anything at all, arranged at random, still averages six sides.
So the observation “epithelial cells average six sides” does not distinguish the two stories. It is consistent with cells minimising perimeter and equally consistent with cells doing nothing of the kind, because the mean is forced either way.
That is the kind of observation this site is most interested in: one that appears to support an explanation and is actually compatible with its negation.
What would distinguish them
The spread, not the mean.
A tiling of cells that really were minimising perimeter under equal-area constraints would be nearly all hexagons, with a narrow distribution. A tiling produced by an indifferent process has a characteristic width — a substantial fraction of fives and sevens, some fours and eights.
Measured on a golden-angle head, six-sided cells are a clear majority and there is a real tail either side. The distribution’s width is the measurement that carries information, and it is the one that usually goes unreported in favour of the mean that cannot vary.
There is also a well-known empirical regularity in real tissue — Lewis’s law, that a cell’s area rises roughly linearly with its number of sides — which is a statement about the joint distribution and says considerably more than either the mean or the spread alone.
Why a patch cannot be all hexagons
There is a sharper version of the same counting, and it explains the exceptions.
On a sphere — a closed surface rather than a plane — the same argument with Euler’s characteristic 2 instead of 0 forces a deficit: a closed three-way tiling must contain twelve pentagons, however many hexagons it has. That is why a football has twelve black patches whatever its size, why a fullerene has twelve pentagons, and why a spherical virus capsid does too.
On a flat patch the boundary plays a similar role, and the interior cannot be uniformly hexagonal while the boundary closes. The non-hexagons are not errors in the tiling; they are what the topology requires.
The general shape
An observation that seems to confirm an efficiency story turns out to be forced by counting, and the thing that would actually test the story — the spread — is the thing nobody reports.
The same pattern appears throughout this subject. The spiral counts are quoted as a property of a flower when they are a property of an annulus. The golden angle is quoted as optimal packing when the packing measurement does not single it out. In each case a number that carries no information has been mistaken for evidence, and the informative quantity beside it has gone unmeasured.