GEOMETRICAL DEDUCTION OF SEM1REGULAR ETC. 1 7 



six members of the subject, each vertex takes six new positions, 

 forming the six vertices of an octahedron (see rule, art. 6) whose 

 eight faces are supplied by the expanded vertices of the eight cubes 

 meeting in a vertex of the original net. 



28. The application to fourdimensional space is simple. 



For instance if the e Y expansion be applied to a net of C 8 each 

 C 8 is changed into an e ± C 8 (Fig. 2c), two adjacent ones having a 

 tO in common. As a vertex in the net C s belongs to eight edges 

 (eight members of the subject) each vertex takes eight new posi- 

 tions which are the eight vertices of a 6 7 16 . 



The limiting bodies of this C m may be identified as follows. 

 In the net C 8 each vertex is surrounded by 16 members. Each 

 vertex of a C 8 is changed by expansion into a tetrahedron, so that 

 the vertex gap in the net is surrounded by 1 6 tetrahedra, the 

 limiting bodies of a C i6 . Thus by the e x expansion a net of C 8 

 has been converted into a net e i NC s of two constituents, e i C 8 and 

 C i6 , in which two adjacent e 1 C 8 have a tC in common, while an 

 e i C 8 and a C iQ have a tetrahedron in common. 



29. Again the e 2 expansion may be applied to a plane net. 

 In this case the constituents of the net are moved apart until an 

 edge assumes two positions, the opposite sides of a square, and 

 the vertex gap is a polygon with as many vertices as there were 

 constituents meeting in a point in the original net; figure 14 (a and b) 

 shews this with regard to a net of triangles. 



If the e. z operation be applied to a net of squares, it moves 

 apart the squares and the result is again a net of squares; but 

 they are not all of the same kind, some being the squares of the 

 original net, some of edge import, others of vertex import 

 (Fig. 15). From this simple example it may be seen that the e n 

 expansion applied to a net of measure polytopes in ^-dimensional 

 space produces again a net of measure polytopes; but the latter is 

 composed of constituents with different imports, and the subject 

 of any further expansion must be suitably chosen. For instance if 

 the e i e 2 expansion be applied to a net of squares the subject of 

 the e { expansion comprises only those squares of edge import intro- 

 duced by the e 2 expansion in a net of squares (Fig. 153). The 

 result is that the squares of the subject remain unchanged except 

 in position. Those of vertex import and those corresponding to 

 the squares of the original net are changed into octagons of 

 different imports. The corresponding double expansion of the net 

 of triangles is shewn in figure 14c. 



30. If the e 2 expansion be applied to a net of cubes each cube 



Verhand. Kon. Acad. v. Wetensch. (I e Sectie) Dl. XI. A 2 



