Abstract
LONDON Mathematical Society, March 12.—J. W. L. Glaisher, F.R.S., President, in the chair.—Messrs, Philip Magnus and R. Lachlan were elected Members.—Mr. J. J. Walker, F.R.S., made a second communication on a method in the analysis of plane curves.—Mrs. Bryant, D.Sc., read a paper on the geometrical form of perfectly regular cell-structure. “Investigation of the properties of the rhombic dodecahedron supplies the clue to the solution of two interesting questions, which are the essential, because the pure geometrical, constituent of several questions as to actual forms in physical nature, such as the geometrical structure of compact tissues on the one hand, and the geometrical form of the honeycomb cells on the other hand. The first question is as follows:—If space were filled with spheres, and this spacefill of spheres were then crushed together symmetrically till the whole became a solid mass, what shape would each sphere ultimately assume? Since twelve is the number of spheres that can be placed round one sphere, in contact with it and with one another, it is evident that each of these ultimate solids would be dodecahedral in shape. The second question is the counterpart of the first:—If space were filled with a homogeneous solid, in which equally efficient centres of excavation were distributed uniformly, what would be the ultimate form of the cells excavated, it being supposed that when the excavators cease their work the walls of the cells are uniform in thickness? The answer to the first question is manifestly the answer to this second question also.” After a geometrical discussion the author says:—“We should expect to find this dodecahedral shape in nature wherever originally spherical cells have been uniformly pressed together in a complete manner. The condition is probably seldom fulfilled, and examples are therefore difficult to find. We may look for their fulfilment, however, in the centre of a mass of soap-bubbles,” The paper then considers the case of the honeycomb cells, with the conclusion: “The above explanation tends, however, to show that the bees need not be credited with any economical instinct to account for their work, but only with those simpler instincts, which enable them to carry out a joint work with perfect regularity and exactness, which simpler instincts, while sufficiently remarkable, are fairly within the limits of credibility.”—Mrs. Bryant illustrated her remarks with several models of the cube and the rhombic dodecahedron.—Mr. Kempe, F.R.S., and the President (who stated that he had some few years since considered the matter from another point of view) made some interesting remarks in connection with the subject.—Prof. Sylvester, F.R.S., gave an account of a paper on the constant quadratic function of the inverse co-ordinates of n + 1 points in space of n dimensions; and Prof. Cayley, F.R.S., and Prof. Hart spoke on the same subject. As the hour was late Mr, Tucker (hon. sec.) merely communicated the titles of papers by Prof. K. Pearson (on the flexure of beams); Rev. T. C. Simmons (two elementary proofs of the contact of the “N.P.” circle of a plane triangle with the inscribed and ascribed circles, together with a property of the common tangents); and by himself (two other proofs of the first part of Mr. Simmons's communication).
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Societies and Academies . Nature 31, 496–500 (1885). https://doi.org/10.1038/031496a0
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DOI: https://doi.org/10.1038/031496a0