A uniform foam was constructed by placing bubbles made of a special soap solution, one by one, into a cylindrical dish. The volume of each bubble was controlled by setting the plunger of a Yale tuberculin syringe at 1/10 (or 1/20) of a cc., and making a bubble with the enclosed air. In this way a dish was filled with nearly 2000 (or 4000) bubbles, each of the same controlled volume. Then with a dissecting microscope it was possible to single out, by focusing, any bubble in such a mass and record with precision the number and kinds of its faces. The exact three-dimensional shape of bubbles in foam has been for many years a subject of discussion and speculation among mathematical physicists (Plateau, Kelvin), metallurgists, plant and animal histologists, and students of protoplasmic structure. Studied in this way, 400 peripheral (epidermal) bubbles, of which 200 were free and 200 in contact with the walls of the cylinder, averaged 10.99 faces, with a range from 7 to 14. More than half of the faces of the pcripheral bubbles were pentagonal, and 98 per cent were quadrilateral, pentagonal, or hexagonal. No one combination of faces among those found strongly predominated or could be considered "the type." The commonest of the 43 combinations of faces which were found had 11 faces, of which 3 were quadrilateral, 6 pentagonal, and 2 hexagonal; there was normally one hexagonal face above, in addition to 6 lateral contacts and 4 basal ones. This combination occurred 67 times and constituted 17 per cent of the total number studied. Pentagonal dodecahedra occurred 11 times among the 400 peripheral bubbles. Comparisons were made between peripheral bubbles in foam, peripheral compressed lead shot, and cells in tissues. The bubbles were more uniform than the shot or than the cells. The cells showed a much closer resemblance to the peripheral bubbles than to the peripheral shot, suggesting strongly that surface forces, which are of relatively greater importance in the bubbles than in the shot, are also distinctly significant when cell shapes are being determined. The average number of faces, 10.99, agreed with 11-faceted epidermal cells previously reported by Lewis. The average number of contacts of 600 central bubbles was 13.70, the range being from 11 to 17. More than a third (218) of the 600 foam cells had 14 faces, and more than five-sixths had 13, 14, or 15 contacts. More than two-thirds of all the faces were pentagonal, and 99.6 per cent of all the faces were quadrilateral, pentagonal, and hexagonal. The 600 central bubbles tabulated occurred in 36 different combinations of faces. As in the peripheral bubbles, there was no one combination of those occurring that was strongly preponderant and that could be considered as "the type." The most abundant combination was a 13-faceted foam cell with 1 quadrilateral, 10 pentagonal, and 2 hexagonal faces (1-10-2) which occurred 118 times among the 600 bubbles. The second and third most frequent combinations were of bubbles with 14 faces, 1-10-3 and 2-8-4, which were found 73 and 64 times, respectively. The pentagonal dodecahedron, with 12 faces all pentagons, occurred 50 times. These 4 combinations accounted for 305 of the 600 bubbles. When a large number of bubbles, all of equal volume, are placed into a cylindrical dish so that they are free to glide and adjust themselves, they do not assume the form of the orthic or of the minimal tetrakaidecahedra of Lord Kelvin. Instead, they become arranged in a mass of somewhat irregular foam cells, satisfying, in approximation at least, the equilibrium conditions of Plateau. In this system pentagonal faces predominate; consequently the foam cells, although averaging close to 14 faces as in Kelvin's tetrakaidecahedra, differ from Kelvin's figures, since the latter have all hexagonal and quadrilateral faces. In the foam pentagonal dodecahedra, in which all the faces are pentagons, are fairly common, and it is demonstrated that this figure requires but little modification to satisfy the equilibrium conditions for angles as specified by Plateau. Calculation shows further that the pentagonal dodecahedron is slightly more economical in surface per unit of volume than the orthic tetrakaidecahedron and than the rhombic dodecahedron. Comparisons were made between central bubbles and central lead shot compressed in a steel cylinder to eliminate interstices. The average number of contacts and the range in number of contacts were slightly less in the bubbles than in the shot. The number of pentagonal faces was much greater in the bubbles than in the shot. The 600 central bubbles occurred in 36 different combinations of faces, while 624 central shot of Marvin occurred in 303 combinations. The commonest combinations in the bubbles were not the most abundant ones in the shot. There were no tetrahedral angles in the bubbles, and they were frequent in the shot. While there were certain similarities between compressed shot and bubbles in foam, there were also characteristic differences, which could be ascribed, in part at least, to the surface forces in the soap film system. Detailed comparisons were also drawn between central bubbles in foam and cells in essentially undifferentiated tissues; and they were both compared with the compressed lead shot. In average number of contacts the cells were intermediate between bubbles and shot, but closer to the bubbles. In number of pentagonal, triangular, heptagonal and octagonal faces the cells again stood between bubbles and compressed lead shot. Similarly, the number of combinations of faces was least in the bubbles, and greatest in the shot, while the cells were again intermediate. Tetrahedral angles occurred abundantly in the shot, less frequently in the cells, and not at all in the soap film system. Since cells are intermediate in so many ways between the soap bubbles, in which surface forces are of controlling importance, and the compressed lead shot, in which they are not, and since the cells show a much closer similarity to bubbles in foam than to compressed shot, the conclusion is drawn that interfacial forces constitute one of the factors determining the shape of essentially undifferentiated cells at or near the plant meristems. This is not to be construed as meaning that interfacial tensions are the only or necessarily the most significant factor. The evidence here assembled indicates that surface forces are important.
Published by the Botanical Society of America continuously since 1914, the American Journal of Botany (AJB) is the Society’s flagship research journal. AJB publishes peer-reviewed, innovative, significant research of interest to a wide audience of scientists in all areas of plant biology (e.g., biodiversity, structure, function, development, genetics, evolution, reproduction, systematics), all levels of organization (molecular to ecosystem), and all plant groups and allied organisms (cyanobacteria, algae, fungi, and lichens).
Wiley is a global provider of content and content-enabled workflow solutions in areas of scientific, technical, medical, and scholarly research; professional development; and education. Our core businesses produce scientific, technical, medical, and scholarly journals, reference works, books, database services, and advertising; professional books, subscription products, certification and training services and online applications; and education content and services including integrated online teaching and learning resources for undergraduate and graduate students and lifelong learners. Founded in 1807, John Wiley & Sons, Inc. has been a valued source of information and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Wiley has published the works of more than 450 Nobel laureates in all categories: Literature, Economics, Physiology or Medicine, Physics, Chemistry, and Peace. Wiley has partnerships with many of the world’s leading societies and publishes over 1,500 peer-reviewed journals and 1,500+ new books annually in print and online, as well as databases, major reference works and laboratory protocols in STMS subjects. With a growing open access offering, Wiley is committed to the widest possible dissemination of and access to the content we publish and supports all sustainable models of access. Our online platform, Wiley Online Library (wileyonlinelibrary.com) is one of the world’s most extensive multidisciplinary collections of online resources, covering life, health, social and physical sciences, and humanities.
This item is part of a JSTOR Collection.
For terms and use, please refer to our Terms and Conditions
American Journal of Botany
© 1946 Wiley
Request Permissions