( 68) 



the vertices and the triplets of digits of each following tO. It goes 

 without saving that here too the centre is found by drawing tO 

 twice over and putting these tO in such a manuer upon each other 

 as to have — with a diffei-ence in orientation of 72° — a hexagon 

 in common. In the case of the two tO deduced from (2345) and 

 (3451), which T of fig. 1« have the face (345) in common, this 

 common hexagon is characterized by this that the six vertices bear 

 the digits 3, 4, 5 in all possible permutations. 



The digit 1 stands foremost at the triplets of twelve vertices, the 

 vertices of a tT of vertex import ; b}- omitting from these triplets 

 the 1 we get not only in position but also in notation the ^ Tof fig. 4. 

 So the five tT of vertex import are accounted foi'. Moreover, as to 

 the ten P, of edge import we can refer to the development given 

 under ^,. 



Circumpolygon a semiregular decagon with sides alternately s and 

 d. Six wreaths of ten vertices, all of them semiregular decagons. 



e,e, aS'(5) — (60, 150, 120, 30) — \^UT, 10 P„ lOP,, f>CO). 



In this case - — for the result compare fig. 13 — the T of fig. 2 

 is transformed by the ^.-operation into the tT of fig. 4, after which 

 this tT is moved out as a wdiole; as by this process each vertex of 

 é?i^S(5) assumes three different positions we must follow once more 

 the notation of the triplets of digits, which can be done here by 

 placing after each pair of digits of fig. 4 the digit 1 not occurring 

 at the vertices of the tetrahedron (2345) moved out. If these triplets 

 have been inscribed in fig. 13, rotation about the centre and augmen- 

 tation of the digits by unity gives all that is wanted, as soon as the 

 centre has been constructed. We arrive as soon as possible at the 

 construction of this centre by determining the prisms of face import 

 first. In the case e^ they were the prisms P^ represented by fig. 8 

 and 9 ; by applying to the T the ^^-operation, the triangles of the 

 T pass into hexagons, which includes that the P^ are transformed 

 into Pg, which can be drawn immediately. By applying to the end- 

 planes (13, 23, 53), (14, 24, 54) of the upper prism P, of fig. 10 

 the éj-operation we obtain the upper prism P, of fig. 1 3 represented 

 separately by fig. 14. Consideration of this prism Pg shows that 

 the limiting polygon is a semiregular decagon, the sides of which 

 are alternately .!> and s-{-d; from this the centre can be deduced. 

 In the same way the prism P, of fig. 10 with the endplanes 

 (12, 32, 42), (15, 35, 45) passes into the P, represented by fig. 15. 

 Farthermore the two P, with the pairs of endplanes (341, 342, 345), 



