GEOMETRICAL DEDUCTION OF SEMIHEGULAll ETC. 5 



its limits (/) and are denoted respectively by the symbols / , l x , 



^2 ' 3 ' • • 



I. The operations of expansion and contraction. 



Definition of expansion. 



3. Let O be the centre of a regular polytope and M x , M 2 , M 3 .. 

 the centres of its limits l x , l 2 , / 3 . . . The operation of expansion 

 e k consists in moving the limits l k to equal distances away from O 

 each in the direction of the line O M k which joins O to its centre, 

 the limits l k remaining parallel to their original positions, retaining 

 their original size, and being moved over such a distance that the 

 two new positions of any vertex, which was common to two adjacent 

 edges in the original polytope, shall be separated by a length equal 

 to an edge. 



The polytope determined by the new positions of the limits l k 

 will have the kind of semiregularity described above. The limits 

 4 are said to be the subject of expansion or briefly the subject-, 

 and the new polytope is denoted by the symbol of the original 

 regular polytope preceded by the symbol e k . 



A few particular cases, in 2, 3 and 4 dimensions, will now be 

 examined. 



Examples of the e x exjMnsion. 



4. Here the edges (/^ are the subject. 



It is evident that this operation applied to any regular polygon 

 changes it into a regular polygon having the same length of edge 

 and twice as many sides. In Fig. \a a square is changed into an 

 octagon by the application of the e x expansion x ). 



Fig. \b shews the e 1 expansion of a cube. The real movement 

 of any edge AB is in the direction of the line OM l but that move- 

 ment may be resolved into two. Thus instead of moving AB 

 directly to the position A l B 1 it might have been moved to A'B' 

 or to À'B" and then to A 1 B 1 . If the movements of all the edges 

 be thus resolved the result is the same as if the faces AC, AB... 

 (Fig. lc) of the original cube had been first transformed into octa- 

 gons by an e x expansion of each in its ow T n plane, and then moved 



x ) In these drawings the thick lines represent edges of regular polytopes in their 

 original or in new positions, the thin lines edges introduced by expansion. 



