16 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



II. Application to space fillings. 

 Expansion applied to the nets. 



26. A space filling or net in any space £> n may be considered 

 as a polytope with an infinite number of limiting spaces of n dimen- 

 sions in a space S n+i of one dimension higher. 1 ) According to this 

 view the operations of expansion and contraction and their com- 

 binations may be applied to nets; but the fact that the net is a 

 particular case of a polytope modifies the manner in which the 

 operation is to be applied. 



Expansion has been defined as a movement of any set of limits 

 away from the centre of a polytope. This movement in general 

 separates the members of the subject. 



In a polytope in S n with an infinite number of n — 1 -dimensional 

 limits (a net) the centre is at an infinite distance in a direction 

 normal to the space S n of the net and no movement away from 

 the centre can separate the limits forming the subject, in other 

 words can expand the net. Now it has been shewn that the real 

 movement taking place in an expansion may be resolved into two, 

 one of which transforms the limits each in its own space and the 

 other adjusts those transformed limits. In this way the operation 

 can be applied to the special case under consideration. Thus if the 

 e { expansion be applied to a net of squares (Fig. 12) they are 

 transformed into overlapping octagons and then the octagons must 

 be moved apart until an edge which was common to two squares 

 becomes common to two octagons. 



This adjustment leaves a gap A x A 2 A B A k (vertex gap) between 

 the octagons corresponding to the vertex A common to four squares. 

 Thus the transformed net of squares is composed of two constituents, 

 octagons corresponding to the squares, and squares corresponding to 

 vertices of the original net. 



27. In threedimensional space there is only one regular space filling 

 i. e. the net NC of cubes. The net N(0,T) of octahedra and 

 tetrahedra is semiregular. 



If the e l expansion be applied to a net of cubes each cube is trans- 

 formed into a tC. These will overlap and must be moved apart 

 until an edge which was common to four cubes becomes common 

 to four tC (Fig. 13) By this adjustment octahedral gaps (vertex gaps) 

 are left at the vertices. So the net e x NC is formed of tC and 0. 



In order to determine the octahedra it is necessary to observe 

 that as a vertex of the original net belongs to six edges, i. e. 



*) See the quoted paper of Andreini , art. 47. 



