6 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



away from the centre O until the edges A'B' and A" B" ', which 

 correspond to an edge AB of the cube, are coincident and become 

 the common edge of two octagons (transformed squares). Tt is to 

 be noticed that as each vertex of a cube is common to three edges 

 (three members of the subject) it takes three new positions, which, 

 owing to the regularity of the cube, are the three vertices of an 

 equilateral triangle. Thus the faces of the cube have been expan- 

 ded into octagons and the vertices into triangles. 



Fig. 2a shews the e l expansion of a tetrahedron. Each face is 

 changed into a hexagon, each vertex into, a triangle. Here again 

 a vertex of the tetrahedron is common to three members of the sub- 

 ject; the result is a tT. 



Fig. '2b shows the same expansion of an octahedron. Each face 

 is changed into a hexagon; but each vertex into a square because 

 in an octahedron each vertex belongs to four edges (four members 

 of the subject) ; the result is a tO. 



From these examples it is easy to find the e v expansion of an 

 icosahedron and of a dodecahedron. 



5. This investigation leads to the determination of the e i expan- 

 sion applied to the fourclimensional polytopes. For instance in the 

 C 8 each cube is transformed (in its own space) by the e x expansion 

 and becomes a tC (Figs. \b and 2c). These transformed cubes must 

 be so adjusted that an edge which was in the C 8 common to 

 three cubes l ) is, in its new position, common to three transformed 

 cubes. Again each vertex in a C 8 is common to four edges and 

 must take four new positions which are the four vertices of a 

 regular tetrahedron. Thus the vertex of the C 8 is expanded into 

 a tetrahedron, which is said to be of vertex import. This tetra- 

 hedron might have been determined in another way; for four cubes 

 meet in a vertex of a C 8 and in each the vertex is changed into 

 a triangle; therefore a vertex of C Q is replaced by a body limited 

 by four triangles i. e. a tetrahedron. 



The two kinds of limiting body of the new polytope e 1 C 8 are 

 shewn in Fig. 2c; in Fig. 2d are shewn the limiting bodies of e x C b . 



In C 16 , where six edges meet in a vertex, the e ± expansion 

 changes each tetrahedron into a tT (Fig. 2d) and each vertex into 

 an octahedron (of vertex import) whose vertices are the six new 

 positions of a vertex of the (7 16 . 



Again in (7 24 eight edges meet in a vertex, so that the e x expan- 



x ) In order to facilitate the application of the operation of expansion it is desirable 

 to have at hand a table of incidences; this is provided on Table III. 



