GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 13 



tually coincide. Moreover the direction of the real movement cannot 

 be represented; but valid conclusions may be drawn from diagrams 

 such as these, if the mind always distinguishes between the actual 

 and the apparent relation of parts. 



These two examples suffice to show how the result of the com- 

 bination of operations may be applied to the fourdimensional cells. 

 There are seven expansions of each: 



■ e l> e 2> e 3> e \ e 2> e i e 3 » e 2 e S ' e i e 2 e 3 ' 



but owing to the reciprocity of some of the figures these are not all 

 different. 



Thus it appears that in any expansion a set of limits , which 

 define the body and which is such that each member is in contact 

 with other members of the same set, may be made the subject of 

 expansion. 



Definition of contraction. 



19. In each of the expansions e l9 e 2 , e d . . . the resulting semi- 

 regular polytope may be reduced to the regular one from which 

 it was derived, by an inverse operation which may be called con- 

 traction. 



Here the limits which formed the subject of the expansion are 

 moved towards the centre and brought back to their original positions. 



The direct operation separates the members of the subject; the 

 inverse operation brings them again into contact, annihilating the 

 edges introduced by expansion. In both positions they define the 

 polytope of which they are the limits. 



The conditions necessary to the inverse operation are: 1 st , the 

 limits forming the subject must define the polytope; 2 nd , no two 

 members of the subject can be in contact before contraction. 



The polytopes of vertex import always satisfy these conditions 

 and can be made the subject of contraction. The symbol c is 

 used to denote contraction. The import of the limits forming the 

 subject is shown by means of subscripts, as in expansion. 



Examples of contraction. 



20. The inverse operation will be made clear by one or two 

 examples. 



In figure 10 the square A B C D has been expanded by the 

 e x operation; the edges of vertex import in the resulting octagon 

 have been made the subject of the inverse operation, that is, they 

 have been moved nearer to the centre so far that the edges of 

 the original square are annihilated, and the final result is the square 



