IX] OF EULER'S LAW 737 



further and treat any cluster of cells, such as a segmenting ovum, 

 as a species of polyhedron and study it from the point of view 

 of Euler's Law and its associated theorems. We should have to 

 include, as the geometer seldom does, the case of two-sided facets — 

 facets with two corners and two curved sides or edges — hke the 

 " hths " * of a peeled orange ; but the general formula would include 

 these as a matter of course. On the other hand, we need very 

 seldom consider any other than trihedral or three-way corners. 



When we limit ourselves to polyhedra with trihedral corners the 

 following formula applies : 



4A+ 3/3+ 2/4+/6± 0./,-/, - 2/3- ... = 12 (4). 



That this formula apphes to the tetrahedron with its four triangles, 

 the cube with its six squares and the dodecahedron with its twelve 

 pentagons, is at once obvious. The now famihar case of our four- 

 celled egg with its polar furrows (Fig. 486, B, etc.) appears in two 

 forms, according as the polar furrows run criss-cross or parallel. 

 In the one case we have a curvilinear tetrahedron, in the other a 

 figure with two two-sided and two four-sided facets; in either case 

 the formula is obviously satisfied. 



But the main lesson for us to learn is the broad, general principle 

 that we cannot group as we please any number and sort of polygons 

 into a polyhedron, but that the number and kind of facets in the 

 latter is strictly Umited to a narrow range of possibiHties. For 

 example, the case of Aulonia has already taught us that a poly- 

 hedron composed entirely of hexagons is a mathematical impossi- 

 bihtyt; and the zero-coefficient which defines the number of 

 Jiexagons in the above formula (4) is the mathematical statement 

 of the fact J. We can state it still more simply by the following 

 corollary, hkewise hmited to the case of three-way corners : 



* Lith, a useful Scottish word for a joint X)t segment. Cromwell, according to 

 Carlyle, "gar'd kings ken they had a lith in their necks." 



t That hexagons cannot enclose space, or form a "three-way graph," has been 

 recognised as a significant fact in organic chemistry : where, for instance, it limits, 

 somewhat unexpectedly, the ways in which a closed cyclol, or space-enclosing 

 protein molecule, can be imagined to be built up. Cf. Dorothy Wrinch, in 

 Proc. R. S. (A), No. 907, p. 510, 1937. 



J Euler shewed at the same time the singular fact that no polyhedron can exist 

 with seven edges. 



