608 On Reciprocal Figures and Diagrams of For 



res. 



On Reciprocal Figures in three dimensions. 



Definition. — Figures in three dimensions are reciprocal when 

 they can be so placed that every line in the one figure is perpen- 

 dicular to a plane face of the other, and every point of concourse 

 of lines in the one figure is represented by a closed polyhedron 

 with plane faces. 



The simplest case is that of five points in space with their ten 

 connecting lines, forming ten triangular faces enclosing five 

 tetrahedrous. By joining the five points w T hich are the centres 

 of the spheres circumscribing these five tetrahedrons, we have a 

 reciprocal figure of the kind described by Professor Rankine in 

 the Philosophical Magazine, February 1864; and forces propor- 

 tional to the areas of the triangles of one figure, if applied along 

 the corresponding lines of connexion of the other figure, will 

 keep its points in equilibrium. 



In order to have perfect reciprocity between two figures, each 

 figure must be made up of a number of closed polyhedra having 

 plane faces of separation, and such that each face belongs to two 

 and only two polyhedra, corresponding to the extremities of the 

 reciprocal line in the other figure. Every line in the figure is 

 the intersection of three or more plane faces, because the plane 

 face in the reciprocal figure is bounded by three or more straight 

 lines. 



Let s be the number of points or summits, e the number of 

 lines or edges, / the number of faces, and c the number of poly- 

 hedra or cells. Then if about one of the summits in which 

 polyhedra meet, and a edges and 77 faces, we describe a polyhe- 

 dral cell, it will have (/> faces and a summits and 77 edges, and 

 we shall have 



7] = cf) + cr — 2; 



s, the number of summits, will be decreased by one and in- 

 creased by cr ; 

 c, the number of cells, will be increased by one ; 

 /, the number of faces, will be increased by cf> ; 

 e, the number of edges, will be increased by 77 ; 



so that e + c — (*+/) will be increased by 77 + 1 — (cr-f<£— \\ 

 w r hich is zero, or this quantity is constant. Now in the figure 

 of five points already discussed, e=10, c = 5, 5 = 5, /=10; so 

 that generally 



e + c = s +f s 



in figures made up of cells in the way described. 



The condition of a reciprocal figure being indeterminate, de- 



