Symmetry by Hermann Weyl

Author:Hermann Weyl [Weyl, Hermann]
Language: eng
Format: epub, pdf
Tags: Ensayo, Ciencias exactas, Divulgación
Publisher: ePubLibre
Published: 1952-01-01T00:00:00+00:00

FIG. 48.

FIG. 49.

According to the laws of capillarity a soap film spanned into a given contour made of thin wire assumes the shape of a minimal surface, i.e. it has smaller area than any other surface with the same contour. A soap bubble into which a quantum of air is blown assumes spherical form because the sphere encompasses the given volume with a minimum of surface. Thus is it not astonishing that a froth of two-dimensional bubbles of equal area will arrange itself in the hexagonal pattern because among all divisions of the plane into parts of equal area that is the one for which the net of contours has minimum length. We here suppose that the problem has been reduced to two dimensions by dealing with a horizontal layer of bubbles, say, between two horizontal glass plates. If the froth of vesicles has a boundary (an epidermal layer, as the biologist would say), we observe that it consists of circular arcs each forming an angle of 120º with the adjacent cell wall and the next arc, as is required by the law of minimal length. After this explanation one will not be surprised to find the hexagonal pattern realized in such different structures as for instance the parenchyma of maize (Fig. 50), the retinal pigment of our eyes, the surface of many diatoms, of which I show here (Fig. 51) a beautiful specimen, and finally the honeycomb. As the bees, which are all of nearly equal size, build their cells gyrating around in them, the cells will form a densest packing of parallel circular cylinders which in cross section appear as our hexagonal pattern of circles. As long as the bees are at work the wax is in a semi-fluid state, and thus the forces of capillarity probably more than the pressures exerted from within by the bees’ bodies transform the circles into circumscribed hexagons (whose corners however still show some remains of the circular form). With the parenchym of maize you may compare this artificial cellular tissue (Fig. 52) formed by the diffusion in gelatin of drops of a solution of potassium ferrocyanide. The regularity leaves something to be desired; there are even places where a pentagon is smuggled in instead of a hexagon. Here (Fig. 53 and 54) are two other artificial tissues of hexagonal pattern taken at random from a recent issue of Vogue (February 1951). The siliceous skeleton of one of Haeckel’s Radiolarians which he called Aulonia hexagona (Fig. 55) seems to exhibit a fairly regular hexagonal pattern spread out not in a plane but over a sphere. But a hexagonal net covering the sphere is impossible owing to a fundamental formula of topology. This formula refers to an arbitrary partition of the sphere into countries that border on each other along certain edges. It tells that the number A of countries, the number E of edges and the number C of corners (where at least three countries come together) satisfy the relation A + C − E = 2.

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