

42 Prof. Tait on Listing's Topologie. 



alternately of two of these groups (such as a/3 a/3 &c.) can be 

 found, usually in many ways, so as to exhaust both groups 

 and pass once through each of the angles. That is, in another 

 form, eyerj such polyhedron may be projected in a figure 

 of the type shown in fig. 9, where the dotted lines are sup- 

 posed to lie below the full lines. But, in the words of the 

 extraordinary mathematician Kirkman, whom I consulted on 

 the subject, " the theorem .... has this provoking interest, 

 that it mocks alike at doubt and proof"*. Probably the 

 proof of this curious proposition has (§11) hitherto escaped 

 detection from its sheer simplicity. Habitual stargazers are 

 apt to miss the beauties of the more humble terrestrial objects. 

 (17) Kirkman himself was the first to show, so long ago as 

 1858, that a " clear circle of edges " of a unique type passes 

 through all the summits of a pentagonal dodecahedron. Then 

 Hamilton pounced on the result and made it the foundation 

 of his Icosian Game, and also of a new calculus of a very 

 singular kind. See figures 9, 10, 11, which are all equivalent 

 projections of a pentagonal dodecahedron. 



At every trivium you must go either to right or to left. Denote 

 these operations by r and I respectively. In the pentagonal 

 dodecahedron, start where you will, either r b or l b brings you 

 back to whence you started. Thus, in this case, r and I are 

 to be regarded as operational symbols — each (in a sense) a 

 fifth root of +1. In this notation Turkman's Theorem is 

 formulated by the expression 



rlrlrrrlllrlrlrrrlll = 1 ; 



or, as we may write it more compactly, 



[(W)W] 2 = 1, or [(Zr)W] 2 = l. 



It may be put in a great many apparently different, but really 

 equivalent, forms; for, so long as the order of the operations is 

 unchanged, we may begin the cycle where we please. Also 

 we may, of course, interchange r and I throughout, in conse- 

 quence of the symmetry of the figure. 



It is curious to study, in such a case as this, where it can 

 easily be done, the essential nature of the various kinds of 

 necessarily abortive attempts to get out of such a labyrinth. 

 Thus if we go according to such routes as (?'Z) 2 /r/ 3 , or r s lr s 

 (sequences which do not occur in the general cycle), the next 

 step, whatever it be, brings us to a point already passed through. 

 We thus obtain other relations between the symbols r and I. 

 We can make special partial circuits of this kind, including 

 any number of operations from 7 up to 19. 



* ' Reprint of Math. Papers from the Ed. Times/ 1881, p. 113. 



