ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 





Symbols 





Characteristic numbers. 





(vertices) 



(edges) 



(faces) 



ƒ = 



ce 2 7) = 



J 



12 



30 



20 



*l' = 



Cl 'i H D = 



tl 



00 



00 



32 



H ' = 



e 2 I) = 



RID 



60 



120 



62 



e x e 2 I = 



*j H D = 



tin 



1 20 



ISO 



62 



rc J j I = 



ce x D = 



in 



30 



60 



32 



ce 2 / = 



D = 



j) 



20 



30 



12 



ce i H I = 



<\ D = 



tn 



60 



01) 



32 



116. We prefer the isocahedron to the dodecahedron to deduce 

 the other polyhedra of the family, as its coordinates can be represented 



by a single symbol. When the length of the edge is supposed to 

 be = 4, this symbol is 



[2, 1 -f-tf, ()]: 2 



where e is written for V 5 and: 2 indicates, that all the even 

 permutations of the coordinates must be written, each of them 

 successively with positive and with negative sign (pentagonal 

 hemiedry). 



All the polyhedra of the family have the symmetry of the icosa- 

 hedron, from which they are deducible, and thence will have 

 coordinate-symbols of the form ] : 2. Only if two of the 



coordinates he equal, the indication: 2 may be omitted. 



The coordinates of the vertices of c { I , e 2 I and e { e 2 I are 

 obtained from those of / by adding to them the components of 

 the prescribed displacements along the axes of coordinates. These 

 displacements have in the case of e l for each vertex the directions 

 from the origin of coordinates to the middle-points of the adjacent 

 edges, in the case of e 2 to the centres of the adjacent faces. 



The coordinates of the vertices of a contracted form, ce, I, ce 2 I 

 or ce 1 e 2 /, are obtained from those of the uneontracted form by 

 substracting from the coordinates of each vertex those of the icosa- 

 hedronvertex from which it was deduced by expansion. 



Hence the polyhedra /, ee ] I and ce 2 I may be considered to 

 be the primitive forms of the family. Prom these the others are 

 obtained by composition, i.e. by suitable addition of the coordinates 

 of the vertices of two or three of them. So <?« 1 is obtained from 

 ce, I and I , e 2 I from ce 2 J and /. ce^ e 2 I from ce 2 I and ce x I, 

 e x e 2 f from ce 2 1, ce 1 1 and /. 



