738 A NOTE ON POLYHEDRA [ch. 



This applies at once to the tetrahedron, the cube and the regular 

 dodecahedron, and at once excludes the possibihty of the closed 

 hexagonal network. 



We found in Dorataspis a closed shell consisting only of hexagons 

 and pentagons; without counting these latter we know, by our 

 formula, that they must be twelve in number, neither more nor 

 less. Lord Kelvin's tetrakaidekahedron consists only of squares 

 and hex:agons; the squares are, and must be, six in number. 



In a typical Peridinian, such as Goniodoma, there are twelve 

 plates, all meeting by three-way nodes or corners; we know^ and 

 we have no difficulty in verifying the fact, that the twelve plates 

 are all pentagonal. 



T. 345. Goniodoma, from above and below. 



Without going beyond the elements of our subject we may want 

 to extend our last formula, and remove the restriction to three-way 

 corners under which it lay. We know that a tetrahedron has four 

 triangles and four trihedral corners ; that a cube has six squares and 

 eight trihedral corners ; an octahedron eight triangles and six four- way 

 corners ; an icosahedron twenty triangles and twelve five-way corners. 

 By inspection of these numbers we are led to the following rule, and 

 may estabhsh it as a deduction from Euler's Law : 



ifs + c,) = 8 + 0if, + c,) + {f, + c,) + 2if, + c,)+etc (5). 



This important formula further illustrates the limitations to which 

 all polyhedra are subject; for it shews us, among other things, that 

 (if we neglect the exceptional case of dihedral facets or 'Uiths") 

 every polyhedron must possess either triangular faces or trihedral 

 corners, and that these taken together are never less than eight in 

 number. 



