20 GEOMETRICAL DEDUCTION OF SEMIREGULAlt ETC. 



gap are therefore 16 tetrahedra and 8 CO (Fig. 19/3). The regu- 

 lar net of 6^4 has thus been transformed into one of three constituents : 



(1) e 2 C* (limited by ECO, P 3 , CO), Fig. 19tt 



(2) P Ti Fig. 19/3 



(3) ce^Cs (limited by 8(70, 16P), Fig. 19a. 



In this net two polytopes (tt) have an 11 CO, a (tt) and a (j3) 

 have a P 3 , a (tt) and an (a) have a CO, and an [a) and a (/3) 

 have a T in common. 



77/e e d expansion applied to a block of cubes. 



32. The figure 20 shews the result e 3 NC clearly. It has 

 already been remarked that this expansion leads to a block of cubes 

 of different kinds, some having face import {a), some edge import 

 (b), and some vertex import (c). 



In figure 21 is shewn the result of the operation e 1 e 3 NC; the 

 cubes corresponding to those of the original net are changed into 

 tC; the cubes of edge import (subject of the second operation 

 e^) remain cubes: those of face and vertex import are changed 

 respectively into _P 8 and ECO (A. 22). 



The e 3 expansion applied to a net of C m . 



33. Each C i6 is expanded according to the rule and produces 

 a polytope limited by T, P 3 , P 4 , C (Fig. 22t). 



When these are adjusted, so that tetrahedra which were common 

 to two C i6 are common to two e 3 C i6i there are face, edge, and 

 vertex gaps; these are defined respectively by three parallel positions 

 of a face, 12 parallel positions of an edge, and 96 positions of a 

 vertex; since in the NC i6 a face is common to three, an edge to 1 2, 

 and a vertex to 96 tetrahedra (members of the subject). It remains 

 only to determine the limiting bodies surrounding these gaps. 



34. In order to find those of the face gap the three new parallel 

 positions of the face ABC are represented by the triangles A { B y C { , 

 J,B 2 C 2 , J Z B Z C Z (Fig. 23). 



It follows from the definition of expansion that the lines A i A 2f 



A2A3, A s A i are normal to the face ABC and equal to an 



edge. Thus the face gap is surrounded by two groups of three 

 P 3 ; one group consists of the P 3 : A i B l C { A 2 B 2 C 2 , A 2 B 2 C 2 A 3 B S (7 3 , 

 A i B â C 3 A 1 B i C\ of face import and the other of A l A 2 A ó B 1 B. 1 B 3 , 

 B i B 2 B s C\ C 2 6 3 , C i C 2 C B A ± A 2 A d of edge import. 



