On the morphology of bubble clusters and polyhedral foams

Colloids and Surfaces, 21 (1986) 179-192 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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On the Morphology of Bubble Clusters and Polyhedral Foams*

SYDNEY ROSS and HARRY F. PREST

Chemistry Department, Rensselaer Polytechnic Institute, Troy, NY 12181 (U.S.A.)

(Received 13 January 1986; accepted in final form 12 May 1986)

ABSTRACT

A symmetrical cluster of thirteen equal bubbles contains a minimal pentagonal dodecahedron at its center (the Dewar cluster) and a symmetrical cluster of fifteen equal bubbles contains a minimal truncated octahedron (or tetrakaidecahedron) at its center (the Kelvin cluster). The minimal pentagonal dodecahedron does not fill space without voids, whereas the minimal truncated octahedron does; so the true mathematical model of the ideal polyhedral foam is a space- filling assembly of uniform, minimal, truncated octahedra with walls of zero net curvature between them. This structure, which has only quadrilateral and hexagonal faces, does not agree with observations of polyhedral foams. In actual foams, even when they are made from uniform bubbles, pentagonal faces predominate. The cause of the deviation is traced to the geometric inability of minimal pentagonal dodecahedra to fill space without voids, thus introducing heterogeneity of size and of form during the transition to a polyhedral foam of the original close packing of uniform spheres.

Readers of Colloids and Surfaces need look no farther than its front cover for an illustration of polyhedral foam. Dr G.D. Parfitt was one of the four original Editors who considered that the subject matter of their new journal could be typified in this way. It appears on the cover of this Memorial number to typify Geoffrey Parfitt’s life work; and so confers symbolic meaning on the dedication of this paper on polyhedral foam to his memory.

INTRODUCTION

Polyhedral aggregates of soap bubbles [ 11, plant-cell tissue [ 21, animal-cell tissue [ 31, metal crystallites [4], and cellular plastics [ 51 all carry the unmis- takable stamp of having been shaped by the force of surface tension; and so are related to the mathematical model of the equal subdivision of space with minimum partitional area. Nevertheless, the mathematical solution is mark- edly different in many respects from the observed morphology of naturally occurring polyhedral aggregates.

*Dedicated to the memory of Professor G.D. Parfitt.

0166-6622/86/$03.50 0 1986 Elsevier Science Publishers B.V.

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Assuming that the surface tension is the same at every lamellar surface, the geometric laws that govern any assemblage of foam cells are:

(I) Three, and only three, liquid surfaces meet along an edge: the three surfaces are equally inclined to one another all along the edge; hence, their trih- edral angle equals 120”. The surfaces need not be planar: if curved, the 120” - angle is formed by the tangents to the surfaces at any point on their line of contact.

(II) Four, and only four, of those edges meet at a point; the angles at which the four edges meet are equally inclined to one another in space; hence, they meet mutually at the tetrahedral angle, 109’ 28’ 16’ ’ .

The two statements given above are not independent, for one follows as a corollary of the other [6].

Those structural features are always observed in liquid lamellae in contact, as they are derived from their generating principle, itself a direct consequence of surface tension, that the shape of each foam cell is determinedby minimizing the area that encloses it, consonant with its excess pressure above atmospheric and conditions imposed by the size, number, and pressure of surrounding foam cells. The way in which the foam is generated determines the number and size distribution of its elemental cells. The purely mathematical model is of an aggregation of uniform cells, whether or not it is realizable. The physical problem is to describe what is realizable and to explain discrepancies between that and the mathematical model.

SUBDIVISION OF SPACE WITH MINIMUM PARTITIONAL AREA

Consider a coherent foam consisting of polyhedral cells. Such cells in contact may assume many different forms, but, because they are all subject to the laws of bubble geometry, those forms are not so varied that they cannot be treated in general terms. We imagine an idealized foam or bubble cluster as consisting of uniform polyhedra of equal size capable of being extended indefinitely by the addition of more such cells. Kelvin pointed out long ago [7] that the equal subdivision of space with minimum partitional area is effected by the “minimal tetrakaidecahedron”, or truncated octahedron (one of the 13 Archimedean semi- regular solids) which has to be slightly modified from its plane-faced original in order to meet the conditions of minimum area. The modification consists of substituting tetrahedral vertices, which changes the dihedral angles from 125” 16’ for the 4-6 faces and 109”28’ for the 6-6 faces to 120” throughout. The minimal truncated octahedron of the Kelvin cluster has curved edges meeting at its vertices at the tetrahedral angle of 109’ 28’; three pairs of equal and opposite plane faces that are curvilinear squares; and four pairs of equal and opposite hexagonal faces that are nonplanar but have zero net curvature. The University of Glasgow preserves Kelvin’s wire model of the packing of minimal truncated octahedra, which shows that he demonstrated their ability

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to pack without voids by means of an actual construction (see Fig. 1). When packed together these polyhedra assume a body-centered cubic configuration.

In discussing this question Kelvin did not invoke the calculus of variations. His main recourse was an inspired interpretation of Plateau’s experiments with soap solutions and wire skeletons of a cube and of a square prism with sides in the proportions 1:l:fi. In short, he found the answer to a complex mathematical problem by observing the behavior of soap films.

Expressions for the volume and surface area of the minimal truncated octahedron are not known. Available expressions refer only to orthic polyhedra,

Fig. 1. Photograph of Kelvin’s wire model of the packing of minimal truncated octahedra, with one of them painted white for better visualization. Courtesy of Mr J.T. Lloyd, Department of Natural Philosophy, University of Glasgow. Known irreverently around the Department as Kel- vin’s “bed spring”.

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that is, those with straight edges and plane faces, such as the orthic truncated octahedron. Let V denote the volume and A the area of a given solid. Given that of all solids of equal volume, the sphere has the least area, then the ratio

36nV/A3 I1 (1)

This inequality is called the isoperimetric theorem and the quotient on the left-hand side is the isoperimetric quotient. Polya remarks that “the isoperimetric theorem, deeply rooted in our experience and intuition, so easy to con- jecture, but not so easy to prove, is an inexhaustible source of inspiration” [8]. Table 1 gives the isoperimetric quotient, 36xV/A3, and expressions for the volume and surface area for a number of such polyhedra and for the sphere, which has the largest volume for a given surface area, and so has the largest isoperimetric quotient of any solid figure.

In the course of developing his result Kelvin presented an argument, based on thought experiments with soap films, to demonstrate that the uniform subdivision of space by means of rhombic dodecahedra, which had been assumed until then to provide minimum partitional area, is not a possible configuration of soap films since they would spontaneously re-form into the minimal truncated octahedron.

The isoperimetric quotients listed in Table 1 show that a packing of pentagonal dodecahedra would have less partitional area than a packing of rhombic dodecahedra, so the question of the former packing must be considered. Reg- ular pentagonal dodecahedra cannot fill space tiithout leaving voids; but per- haps the alteration of the vertical and dihedral angles required to convert them into their minimal forms would now allow them to fill space without voids. The proof that that is not so, is as follows. A portion of a minimal pentagonal dodecahedron, in the form of a central pentagonal soap film, can be obtained by withdrawing from a soap solution a wire cage in the form of a right pentagonal

TABLE 1

Mensuration formulae of the sphere and polyhedra (R is the radius of the sphere; e is the length of the side of the polyhedron)

Isoperimetric quotient: 36n V/A3

Volume Area

Sphere Pentagonal dodecahedron Truncated octahedron Rhombic dodecahedron

1.0000 4.1888R3 12.566R’

0.7547 7.663e3 20.646e’

0.7534 11.314e3 26.784e’

0.7405 2.7108e3 10.392e’

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prism. Plateau [9] did this experiment more than a century ago. If it were possible to pack these pentagonal prisms together to fill space without voids, then their enclosed minimal pentagonal dodecahedra would necessarily also fill space without voids. This requirement reduces the question to whether uniform, regular pentagons can cover an area without anything left over (tes- selate), which is well known to be impossible. Therefore the minimal pentagonal dodecahedron, like the regular pentagonal dodecahedron, cannot pack to fill space without voids.

The pentagonal dodecahedron is the first, and the least, of what may be called the spheroidal polyhedra, which continue to progress, through the icosahedron, the truncated icosahedron, and the six-frequency icosahedron, toward more and more sphere-like shapes. Their increase in isoperimetric quotient is accompanied by reduction of their ability to fill space, from a void volume of about 3% for the best packing of pentagonal dodecahedra [lo] to about 26% for spheres. The difference between the isoperimetric quotients of the pentagonal dodecahedron and the truncated octahedron is reported in Table 1. The former has the larger isoperimetric quotient because it is a spheroidal polyhedron, sharing with the sphere an improvement in isoperimetric quotient at the expense of leaving voids on close packing. As we have seen, even in its minimal form, the pentagonal dodecahedron cannot fill space without voids, that is, it remains spheroidal; whereas the minimal truncated octahedron retains the property of filling space without voids. Although the minimal forms of these two polyhedra necessarily have larger isoperimetric quotients than their orthic counterparts, it appears that they maintain their relative ranks in the list of isoperimetric quotients. The isoperimetric quotient of the minimal truncated octahedron, the Kelvin polyhedron, therefore, is larger than 0.7534 but is less than that of the minimal pentagonal dodecahedron. The form of the minimal pentagonal dodecahedron is barely altered from its regular plane-faced form by adjusting its vertices to the tetrahedral angle; consequently its isoperimetric quotient is only slightly larger than 0.7547. We shall see what this implies about the change of morphology when uniform, close-packed spherical bubbles with twelve nearest neighbors are converted into polyhedral foam by hydro- dynamic drainage and capillary suction; but, as a useful preliminary, we discuss the geometry of clusters of equal-sized bubbles and their internal polyhedra.

VARIOUS CLUSTERS OF EQUAL-SIZED BUBBLES

Symmetrical clusters of two, three, four and seven equal bubbles are well known and frequently illustrated [ 111. Their geometry is not complicated, because the partitions that divide them are plane. Larger clusters introduce nonplanar partitions of net zero curvature, requiring an application of solutions to Plateau’s problem in order to solve for volumes and surface areas of

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the adjoining cells. The Dewar cluster of thirteen equal bubbles, with an internal minimal pentagonal dodecahedron, and the Kelvin cluster of fifteen equal bubbles, with an internal minimal truncated octahedron, are two clusters of equal bubbles with a single internal polyhedron. The Kelvin cluster can be extended to any number of cells, because minimal truncated octahedra when packed together fill space without voids; and a cluster of any number of such equal polyhedra can always be bounded by suitable quadrilateral or hexagonal truncated pyramids with convex spherical bases. The Dewar cluster cannot be extended in the same way, as its internal polyhedron does not fill space without voids. The Kelvin polyhedron, therefore, is the true mathematical model of an ideal foam; and the Dewar polyhedron, although usually taken as the model, is strictly speaking untrue, although not greatly in error for practical purposes.

Kelvin deduced that the minimal truncated octahedron would divide space with minimum partitional area when he saw the junction of six such cells sharing the edges and faces of a quadrilateral fenestra, which is demonstrated by the internal soap films that are developed inside a cubic skeleton frame on withdrawing it from total immersion in a soap solution. Even more convincing would be to see the unit foam cell demonstrated by a cluster of bubbles without the intervention of supporting wires. Half of the desired structure can be produced on a glass plate, which may be silvered so as to display the complete cluster, or on the concave side of a watch glass, as follows: The first layer of seven equal hemispherical bubbles is laid down on the glass, using a graduated syringe to produce the bubbles. They form a ring of six around a central figure of hexagonal cross section, with a spherical cap. In the next layer, the bubbles have to be made twice the volume of those in the first layer, to preserve the equality of size of the whole cells and hence a pressure difference of zero between them. Four bubbles are put into the second layer, in order to form the half- cluster of fifteen cells. These four may be added to the second layer one at a time, but the proper positioning of the last of the four is quite difficult. If placed symmetrically on top of the other three it remains there in a stable configuration as the first of an unwanted third layer, out of contact with the central bubble; if placed asymmetrically on the second layer it tends to slip down into the first layer. The four bubbles of the second layer may, however, be assem- bled separately on the convex side of a watch glass and all four transferred simultaneously to the second layer, after which the glass is gently and carefully withdrawn. When the experiment is successfully performed, the central bubble demonstrates the form of one-half of Kelvin’s minimal truncated octahedron: two hexagonal faces form the sides of a roof, with two sloping quadrilateral faces taking the place of the gables. The observation of the cluster of fifteen equal bubbles, with the central minimal truncated octahedron, has not, as far as I am aware, yet been reported; nor even the present observation of one-half

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of such a structure. For Kelvin’s recognition of its potential existence and for convenience of reference, it is appropriately named the Kelvin cluster.

On the way toward the forming of the Kelvin half-cluster of fifteen equal- sized bubbles we pass a smaller half-cluster of equal bubbles. Placing three bubbles on the second layer gives a central cell that is half of a dodecahedron. The results compiled in Table 1 lead one to anticipate its being a pentagonal rather than a rhombic or quadrilateral dodecahedron, and that is indeed the result. The dodecahedral foam cell cannot be an orthic pentagonal dodecahedron, however, because it has vertex angles of 109” 28’ instead of 108”) and dihedral angles of 120” instead of 116”23’; the slight curvature of the edges and faces that must be there is not discernible by eye. These modifications of the regular polyhedron, since they were introduced by liquid lamellae under tension, imply that a dodecahedron of minimal area is created. The full cluster of thirteen equal-sized bubbles, with a central pentagonal dodecahedron, was actually observed by Dewar [ 121, who obtained it once by accident, reported it without comment, and did not (or could not) investigate it further. It may for convenience be named the Dewar cluster.

In both the Dewar and the Kelvin clusters of equal-sized bubbles the liquid lamellae between the cells have zero net curvature, because no pressure differ- ences exist across them. In the Dewar cluster, the twelve identical cells that surround the central dodecahedron have the form of frustra of pentagonal pyramids terminated by spherical bases. In the Kelvin cluster, the surrounding cells consist of six frustra of square pyramids and eight frustra of hexagonal pyramids, both kinds of pyramid being terminated by spherical bases. The gas inside the cells of any cluster is at a lower pressure than when it was in the spherical bubble from which it came, so that it has undergone an expansion on fusing into the cluster; simultaneously the liquid lamellae containing the gas have undergone a contraction on fusing into the cluster. Both these changes are spontaneous, and so account for the spontaneous formation of bubble clusters from single bubbles; whereas the reverse effect, namely, the separation of a cluster into its component single cells, does not occur spontaneously. The expansion of the gas (dV) and the contraction of the liquid surface (AA) are related [ 131 by the equation:

3PdV+BodA=O (2)

where P is the pressure of the external atmosphere, and c is the surface tension of the liquid. Equation (2) tells us that the twelve cells at the boundaries of the Dewar cluster of equal bubbles have the same volume and the same surface area as the central pentagonal dodecahedron. The proof is as follows: The final volume of each of the thirteen cells is the same as each contains the same quantity of gas at the same pressure and temperature; therefore each was formed from its original sphere with the same expansion of volume; hence by Eqn (2) each has also undergone the same contraction of area. The minimal pentagonal

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dodecahedron, therefore, has the same volume, the same area, and the same isoperimetric quotient as the pentagonal pyramidal frustra that surround it. By the same reasoning, each of the six quadrilateral pyramidal frustra and each of the eight hexagonal pyramidal frustra that surround the central minimal truncated octahedron in the Kelvin cluster has the same volume, the same area, and the same isoperimetric quotient as the central cell. The same conclu- sion applies to the two different forms of cell in the four-bubble cluster, and to the two different forms of cell in the seven-bubble cluster.

THE SPHERE OF EQUIVALENT CURVATURE

For any foam cell we can conceive a corresponding free bubble, a spherical liquid lamella of the same surface tension as the lamellae of the foam cell, and of such a radius that the gas it contains is at the same pressure as the gas contained in the foam cell. This conceptual bubble is the sphere of equivalent curvature - not to be confused with the real bubbles with which Dewar and Kelvin clusters can be built. This conceptual bubble simplifies calculations of foam geometry, because no matter what the shape of the foam cell being considered, and the variety of their forms is legion, some of its significant geo- metrical relations can be deduced from the geometry of a sphere.

Let R be the radius of the sphere of equivalent curvature of a foam cell, then:

R=4a/AP (3)

where AP is the additional pressure inside the foam cell above that of the external pressure P.

Comparing the sphere of equivalent curvature with any one of the equal bubbles in a Dewar or Kelvin cluster, the equation of state of a foam [ 131 gives:

PV, =qRT- (2/3)oA, for the sphere (4)

and

PV,=n,RT- (2/3)aA2 for the foam cell (5)

also, since the pressure inside the sphere is the same as that inside the foam cell,

n,lV, =n21V2 (6)

Combining the above equations gives:

Al/V, =A,/V,=6/R

or

R=GV,/A, (7)

The ratio 6V,/A, defines a length equal to the diameter of the in-sphere of

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any convex polyhedron that has a well-defined in-sphere (touching all faces), such as a regular polyhedron or the rhombic dodecahedron [ 141. The sphere of equivalent curvature of the polyhedral foam cell has a radius twice that of the inscribed sphere of the polyhedron.

The volumes and areas of the minimal Dewar and Kelvin polyhedra are not known with precision; but the mensuration formulae listed in Table 1 for their orthic counterparts serve as approximations. These give:

R = 2.227e for the Dewar cluster

and

R = 2.534e for the Kelvin cluster

where e is the distance between adjoining vertices.

THE TRANSITION TO POLYHEDRAL FOAM

During the transition from a close-packed array of uniform spherical bubbles to polyhedral foam cells, the surface area of each cell increases, its volume decreases, and its capillary pressure increases, as described by Eqn (2). These changes are brought about by the withdrawal of liquid from between the bubbles by gravitational drainage and capillary suction and the consequent com- pressive force as the bubbles are squeezed together. A fundamental difference exists between the formation, as already described, of bubble clusters from bubbles with double-sided walls, and the transition of sphere-type foam, that is, bubbles with single-sided walls, to polyhedral-type foam. The former is a process of degradation in which the system loses free energy by spontaneously decreasing both the area of the liquid surface and the pressure of the encapsulated gas. The transition of the foam from spherical type to polyhedral type is driven mainly by the capillary suction, and is accompanied by an increase in both the area of the liquid surface and the pressure of the encapsulated gas. Equation (2) applies to both processes: in the former, dA is negative and dV is positive (expansion); in the latter, dA is positive and dV is negative (compression.)

Pentagonal faces have been found to predominate in the polyhedral cells that are produced; that is, pentagonal faces do not disappear in favor of quadrilateral and hexagonal faces, which means that close-packed spherical bubbles do not rearrange smoothly into the regular body-centered-cubic packing of truncated octahedra. Indeed, they could not be expected to do so; for the very condition for the packing of uniform spheres, namely, twelve nearest neighbors, switches the course of events into a different direction. The close-packed uniform spheres would, if they could, transform smoothly into packed minimal dodecahedra; but geometry does not allow it. Rhombic dodecahedra fill space

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without voids, but the facial and interfacial angles are too far from Plateau’s requirements to be accommodated; and minimal pentagonal dodecahedra are unable to be packed without voids. Some of the cells must accommodate to their surroundings to ensure space filling. These accommodations introduce heterogeneity of size and of form. The disturbance of the structure occurs not just in its immediate vicinity but possibly throughout the foam mass. Another disturbance of the structure occurs at the walls of the container. Polydispersity is, therefore, a consequence of a polyhedral foam derived from a monodisperse spherical foam. A foam consisting of uniform truncated octahedra can indeed be created, as we have in effect demonstrated. The condition to be observed is meticulous care to ensure that each bubble have fourteen nearest neighbors. Such a foam cannot be constructed by the simple route of letting uniform spheres assemble into a close-packed structure: a polydisperse assembly of polyhedra is the inevitable result. In brief, wherever the minimizing action of

Fig. 2. Skeleton model of a minimal pentagonal dodecahedron, made from “Framework Molecular Models”.

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surface tension creates pentagonal dodecahedra out of spheres, dislocations in the packing occur elsewhere, as the regularity of the packing cannot be main- tained. The occurrence of dislocations introduces a random element into the morphology, which accounts for so little difference being observed in the morphology of foams created from uniform and those created from non-uniform bubbles [ 11.

By assuming that the average cell in a foam composed of random polyhedra is a regular polyhedron with planar faces, Schwarz [ 151 calculated its number of faces, vertices and edges, given the only further condition that the angles between the edges are tetrahedral. The two suppositions are incompatible, but the assumption of planar faces is not likely to introduce much error. His result is an average polyhedron with 5.10 edges per face, 13.39 faces, and 22.79 vertices. For comparison, the truncated octahedron has 5.143 edges per face, 14

Fig. 3. Skeleton model of a minimal truncated octahedron, made from “Framework Molecular Models”.

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IDEALIZED POLYHEDRAL FOAMS

The mathematically ideal polyhedral foam consists of uniform minimal truncated octahedra. The boundaries of the foam are not usually included in the conception but must be considered at least to establish the fact, made evi- dent by their spherical facets, that the gas inside them, and hence inside each foam cell, is at a pressure greater than that of the external atmosphere. Not everyone who has considered an ideal foam, consisting of uniform, minimal polyhedra with walls between them of zero net curvature, has comprehended this fact. Even as reputable an authority as Bikerman wrote [ 161:

“Since [in such a foam] there is no capillary pressure difference across a plane boundary [ or one of zero net curvature], the pressure in all bubbles would be identical, and equal to the pressure outside.” (Italics added.)

Fig. 4. Skeleton model of a minimal P-tetrakaidecahedron, Models”.

made from “Framework Molecular

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One use of an ideal polyhedral foam is for computer simulation of the kinet- ics of foam drainage and foam decay, required to evaluate the separation effi- ciency of surface-active solutes by foam fractionation [17]. The structure of foam also affects mechanical properties of cellular plastics [El].

The problem remains, however, that the distribution of faces observed in natural packings of aggregates of soap bubbles, plant cells, and metal crystallites, showing a preponderance of pentagons, does not correspond to the distribution in Kelvin’s minimal truncated octahedron, which has no pentagonal faces. We have seen how this might have come about in the transition from close-packed spheres to the incomplete space-filling of pentagonal dodecahedra. Williams [ 181 has pointed out that topological transformations of vertices and sides permit a number of other space-filling polyhedra to be derived from Kelvin’s polyhedron. Such derived polyhedra have fewer elements of symmetry than their parent polyhedron, but one of them can be singled out that closely matches naturally occurring distributions of faces. Such a tetrakaidecahedron, designated beta by Williams, retains the same average number of edges per face (5.143), faces (14), vertices (24), and edges (36) as the Kelvin polyhedron; but it has two quadrilateral, eight pentagonal, and four hexagonal faces, which reproduces the predominance of pentagonal faces in the distribution that is reported in all observations of polyhedral aggregates and that is so signally lacking in Kelvin’s polyhedron.

Williams’ /I-tetrakaidecahedra pack together as a body-centered tetragonal lattice, which, because it is not isometric, would, if it were metastably composed of soap films, rearrange spontaneously to an assembly of Kelvin’s polyhedra. This would occur because the /3-tetrakaidecahedron has a smaller isoperimetric quotient than Kelvin’s. Williams himself confirms this conclu- sion by reporting that his preliminary calculations show the P-tetrakaidecahedron to require approximately 4% more surface area to enclose the same volume as the truncated octahedron. This feature makes it impossible to produce with soap films a Williams cluster of 15 equal-sized bubbles having an enclosed /3-tetrakaidecahedron.

Skeleton models of foam polyhedra can be made, however, from the “Frame- work Molecular Models” [ 191, in which the metal vertices are fixed at tetrahedral angles and flexible plastic tubes, serving as edges, readily conform to the required curvatures to close the figures. Photographs of the minimal pentagonal dodecahedron (Dewar’s), the minimal truncated octahedron (Kel- vin’s), and the minimal P-tetrakaidecahedron (Williams’,), made in this way, are shown in Figs 2-4.

We now have three candidates from which to construct an idealized polyhedral foam, each of them, in one particular or another, failing to be entirely satisfactory. The minimal pentagonal dodecahedron, with its familiar geometry and nearest-neighbor correspondence to the packing of uniform spheres, nevertheless does not fill space; Kelvin’s minimal truncated octahedron, with

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its mathematically correct conditions of space-filling and minimum partitional area, yet fails to match naturally occurring morphologies; and Williams’ minimal /3-tetrakaidecahedron, with its good agreement with the distribution of faces found in natural packings, and its conformance with the space-filling requirement, yet does not meet the condition of minimum surface area that the existence of surface tension imposes. No single polyhedral cell can be abstracted from foam that meets all these requirements.

Our last point is philosophical. The morphology of foam displays a certain degree of order, but that does not imply that it is orderly through and through; indeed there is not so much simplicity and order in nature as people think. As Poincare points out, we analyze our data just so far as to obtain simplicity and no further. “11 faut bien s’arrbter quelque part, et pour que la science soit possible, il faut s’arrQter quand on a trouve la simplicite.”

REFERENCES

1 2 3

8 9

10

11 12

13

14 15 16 17 18 19

E.B. Matzke, Proc. Nat. Acad. Sci. USA, 31 (1945) 281; Am. J. Bot., 33 (1946) 58. F.T. Lewis, Proc. Am. Acad. Arts Sci., 58 (1923) 537. 0. Biitschli, Investigations on Protoplasm and Microscopic Foams (Trans. by E.A. Minchin), A. and C. Black, London, 1894; W.E. Seifriz, Protoplasm, McGraw-Hill, New York, 1936. For similar considerations in the context of colloid science see the papers of F. Sebba, Colloid Polym. Sci., 257 (1979) 392; J. Theor. Biol., 78 (1979) 375. C.H. Desch, J. Inst. Met., 22 (1919) 241. F.A. Shutov, Adv. Polym. Sci., 151 (1983) 55. F.J. Almgren, Jr. and J.E. Taylor, Sci. Am., 235 (1976) 82. W. Thomson (later Lord Kelvin), Phil. Mag., [s5], 24 (1887) 503; Mathematical and Physical Papers, Vol. 5, Cambridge Univ. Press, 1911, p. 297. G. Polya, Mathematics and Plausible Reasoning, Vol. 1, Princeton Univ. Press, 1954, p. 188. J.A.F. Plateau, Statique Expdrimentale et Theorique des Liquides soumis aux seules Forces moleculaires, Vol. 1, Gauthier-Villars, Paris, 1873, Fig. 78. T. PBschl, 2. Metallkd., 35 (1943) 25; E. Manegold, Schaum, Strassenbau, Heidelberg, 1953, p. 83. C. Isenberg, The Science of Soap Films and Soap Bubbles, Tieto Ltd, Clevedon, 1978, p. 96. J. Dewar, Proc. R. Inst. G. B., 22 (1918) 359; CollectedPapers, Vol. 2, Cambridge Univ. Press, 1927, p. 1218. S. Ross, Ind. Eng. Chem., 61[ lo] (1969) 48; I.D. Morrison and S. Ross, J. Colloid Interface Sci., 95 (1983) 97. H.S.M. Coxeter, Introduction to Geometry, 2nd edn, Wiley, New York, 1969, Ch. 22. H.W. Schwarz, Reel. Trav. Chim. Pays-Bas, 84 (1965) 771. J.J. Bikerman, Foams, Springer-Verlag, New York, 1973, p. 194. G. Narsimhan and E. Ruckenstein, Langmuir, 2 (1986) 230. R.E. Williams, Science, 161 (1968) 276. G.C. Brumlick, U.S. Patent 3,333,349; August 1, 1967. Framework Molecular Models are available from Prentice-Hall, Englewood Cliffs, NJ.

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On the morphology of bubble clusters and polyhedral foams