By Hermann Weyl
Defines symmetry via a dialogue of its many makes use of in a wide selection of fields either educational and natural.
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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 FIG. FIG. 54 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.
One way of including improper rotations is simply more precisely by adding to by adjoining the proper rotations S of r all the improper rotations of the form
Yet again repetition in FIG 35 dimensional plane with a center 0 . Magnificent examples of such central plane symmetry are provided by the rose windows of Gothic cathedrals with their brilliant-colored glasswork. The richest I remember is the rosette of St. Pierre in Troyes, France, which is based on the number 3 throughout. Flowers, nature's gentlest children, are also conspicuous for their colors and their cyclic symmetry. Here (Fig. 35) is a picture of an iris with its triple pole. The symmetry of 5 is most frequent among flowers.