While the formation of an hexagonal pattern on the basis of ready-formed and symmetrically arranged material units is a very common, and indeed the general way, it does not follow that there are not others by which such a pattern can be obtained. For instance, if we take a little triangular dish of mercury and set it vibrating (either by help of a tuning-fork, or by simply tapping on the sides) we shall have a series of little waves or ripples starting inwards from each of the three faces; and the intercrossing, or interference of these three sets of waves produces crests and hollows, and intermediate points of no disturbance, _whose loci are seen_ as a beautiful pattern of minute hexagons. It is possible that the very minute and astonishingly regular pattern of hexagons which we see, for instance, on the surface of many diatoms, may be a phenomenon of this order[363]. The same may be the case also in Arcella, where an apparently hexagonal pattern is found not to consist of simple hexagons, but of “straight lines in three sets of parallels, the lines of each set making an angle of sixty degrees with those of the other two sets[364].” We must also bear in mind, in the case of the minuter forms, the large possibilities of optical illusion. For instance, in one of Abbe’s “diffraction-plates,” a pattern of dots, set at equal interspaces, is reproduced on a very minute scale by photography; but under certain conditions of microscopic illumination and focussing, these isolated dots appear as a pattern of hexagons. ―――――――――― A symmetrical arrangement of hexagons, such as we have just been studying, suggests various simple geometrical corollaries, of which the following may perhaps be a useful one. We may sometimes desire to estimate the number of hexagonal areas or facets in some structure where these are numerous, such for instance as the {324} cornea of an insect’s eye, or in the minute pattern of hexagons on many diatoms. An approximate enumeration is easily made as follows. For the area of a hexagon (if we call δ the short diameter, that namely which bisects two of the opposite sides) is δ^2 × (√3)/2, the area of a circle being _d_^2 ⋅ π/4. Then, if the diameter (_d_) of a circular area include _n_ hexagons, the area of that circle equals (_n_ ⋅ δ)^2 × π/4. And, dividing this number by the area of a single hexagon, we obtain for the number of areas in the circle, each equal to a hexagonal facet, the expression _n_^2 × π/4 × 2/(√3) = 0·907_n_^2, or (9/10) ⋅ _n_^2, nearly.