4 GEOMETRICAL DEDUCTION OF SEMIREGULAR ETC. 



for the polyhedra of this description, almost the same as those 

 given by Andreini, are as follows: 



T, C, 0, I), I indicate the five regular polyhedra, using the 

 initial letters of their ordinary names Tetrahedron, Cube, Octahe- 

 dron, Dodecahedron, Icosahedron; and if p n indicates a regular 

 polygon with n vertices we have 



tT= truncated T limited by 4p 6 and 4p t 



W = » C 'i " 6j!? 8 



tO = h O " » 8jt?g 



tl)= a D // <> 12j» 10 



tl— a I // " 20p G 



CO = C and O in equilibrium » » 6j» 4 



ID = I a I) a a » // 12/? 5 



R CO = combination of rhombic D, C and O .... » » 18jt? 4 



RID = a a a Tr l ),I a D » // 12^ 6 , 30/? 4 



WO = truncated 2 ) CO " // 6p Si 12/? 4 



tID = a ID // // 12/?i , 30^ 4 



20ft, 



2U^3, 



20^, 

 20 A . 



Moreover we want: 



P 3 , P 4 ,... for threedimensional triangular prisms, square prisms 

 (cubes), etc. 



P c , P ,... for fourdimensional prisms on a cube, an octahedron,... 

 as base, 



P (3 ; 3) or simply (3 ; 3) for a prism otope 3 ) with two groups 

 of threedimensional prisms P 3 as limiting bodies, 



P (6 ; 8) or simply (0 ; 8) for a prismotope having for limiting 

 bodies six octagonal and eight hexagonal prisms, etc. 



2. The transformation of the regular into the semiregular bodies 

 and space fillings can be carried out by means of two inverse ope- 

 rations which may be called expansion and contraction. 



In order to define these operations conveniently, the vertices, 

 edges, faces, limiting bodies,... of a regular polytope are called 



J ) By Tr we indicate tbe solid limited by 30 lozenges in planes through the edges 

 of I or D normal to the lines joining the centre to the midpoint of each edge. 



2 ) According to custom the word "truncated" is used here, though this body and the 

 next one cannot be derived from the CO and the ID by truncation. 



3 ) This body is also a „simplotope" as the describing polygons (placed here in planes 

 perfectly normal to each other) are triangles (compare Schoute's „Mehrdimensionale 

 Geometrie", vol. II, p. 128). 



In general a prismotope is generated in the following way: 



Let Sp and Sq be two spaces of p and q dimensions having only one point in com- 

 mon; let P be a polytope in Sp , Q a polytope in Sq . Now move Sp with P in it 

 parallel to itself, so as to make any vertex of P describe all the points of Q. Then 

 P generates the prismotope. Here we have to deal only with the case of two planes 

 (p = g = 2); by the symbol (6 ; 8) we will indicate the polytope limited by eight 

 hexagonal and six octagonal prisms obtained in the indicated manner if we start from 

 a hexagon and an octagon situated in two planes perfectly normal to each other. 



