Vol.43 TORREYA Tllv 194.^ 



Haphazard as a Factor in the Production of Tetrakaidecahedra* 



Frederic T. Lewis 



This paper, more fully presented than was possible in oral delivery and 

 ^^■ith added reference to sulDsequent publications, has been published in the 

 American Journal of Botany, 30: 74-81. Jan.. 1943. There it is entitled 

 "A Geometric Accounting for Diverse Shapes of 14— hedral Cells : the Transi- 

 tion from Dodecahedra to Tetrakaidecahedra."' A summary of the discourse 

 follows : 



The study of cell shapes in compact parenchyma, or in similar unspecialized 

 aggregates, has led to a series of surprises. (1 ) Such cells, instead of being- 

 rhombic dodecahedral products of surface tension, in reality have an average 

 of between 13.5 and 14 facets. — usually close to 14. (2) The cells, though 

 having an average of 14 facets, very rarely present the 14— hedral shapes 

 deduced by Lord Kelvin as dividing space into uniform bodies of minimal 

 surface. Even irregular or distorted approximations of those shapes, with 8 

 irregularly hexagonal facets and 6 quadrilaterals that are far from true squares, 

 occur in less than 1 per cent, of the cells studied. (3) Compressed solids, 

 such as shot of a given size, no longer controlled by surface tension, assume 

 the same irregular cell-like shapes with the same average of close to 14 facets 

 (Marvin). (4) Aggregations of soap bubbles of as nearly uniform size as 

 they can be made, responding to surface tension, and free to glide over one 

 another, do not assume the Kelvin shapes. With an average of 14 facets, they 

 present a variety of cell-like forms (^latzke). 



Confronted with this situation, the aggregation of geometrically perfect 

 rhombic dodecahedra was considered anew. At six corners of each rhombic 

 dodecahedron, when surrounded by others like it, six polyhedra meet at a 

 mathematical point. Let two of them deviate a hair from meeting the other four 

 at a point, and let the deviations throughout the mass occur in all directions 

 at random, and the aggregation of rhombic dodecahedra becomes an assem- 

 blage of irregular shapes with an average of 14 facets. The shapes range from 

 12- to IS-hedra, and have an abundance of pentagonal facets. A\'hen all edges 

 are more or less of the same length, these irregular poh-hedra present many of 

 the forms common to cells, bubbles in foam, and compressed shot. The average 

 of close to 14— facets in a disarranged space-filling mass of bodies of similar 

 size thus appears inevitable. The occasional occurrence of five or six cells 

 chancing to meet at a point, or of four meeting along a line, would slightly 

 reduce the average. 



* Read at the 75th Anniversary Celebration of the Torrey Botanical Club at Columt)ia 

 University. ^.londay. Tune 22. 1942. 



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