10 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



centre 0; for it goes without saying that a space central to the 

 central eightcell of the block is in general excentric with respect 

 to the other eightcells of it. 



If the projection / of the excentric intersecting space is inclined 

 to the edge PQ, the section is either a whole rhonibohedron or a 

 part of it. The first case occurs when the two endpoints of the 

 segment of / lying within the rectangle PQQ'P' are points of the 

 edges PQ, P'Q'; in the other case the section is a truncated rhoni- 

 bohedron , truncated at one end or at both ends according to 

 whether one or neither of these endpoints lie on these edges. If it 

 is truncated at one end it may happen that we have to deal with 

 a rhonibohedron of which only a triangular pyramid is cut off, or 

 left; or that the endplane bears a hexagon, and the section is 

 greater than, equal to, or less than half the rhombohedron. If it 

 is truncated at both ends the two endplanes can be situated on 

 different sides of the middle plane containing a régulai' hexagon 

 equal to // & , or on the same side; so we may even obtain a trun- 

 cated triangular pyramid. 



9. The method of investigating the sections of an eightcell by a 

 space rotating about a central plane t by means of the projection 

 in the plane %' in the centre totally normal to it has been brought 

 most to the front in the preceding pages, as indeed it is our 

 opinion that this method is preferable for its generality to any 

 other 1 ), which opinion is strengthened by its applicability to a block 

 of eightcells. But still we must acknowledge that, with regard to 

 the construction of the side-faces of the rhombohedrical sections, 

 a more direct method, more suitable to the wants and needs of 

 the maker of cardboard models, may be obtained by cutting one 



') If we suppose in space S n of n dimensions a measure-polytope P 2n to be given 

 and we have to consider the sections of this polytope by a pencil of central spaces 

 S n _ l normal to the plane n' passing through two opposite edges PQ, P'Q', the same 

 method can be used. In this case the projection of the given polytope P 2 „ on to the 

 plane i is a rectangle divided into n — 1 equal rectangles by n — 2 lines parallel to 

 PQ, P' Q'- The section is either a measure-polytope -P2(n-1) of n — 1-dimensional space, 

 stretched out in the direction of one of the diagonals figuring as axis with the period 

 n — I, which may be called a rhombotope, or a mutilated rhombotope truncated at 

 both ends. With regard to the section of P 2n with the central space S n _ 2 to which 

 the plane x' is entirely normal, which n — 2-dimensional solid is common to all the 

 n — 1-dimensional sections, we can only remark here, that its vertices are vertices of 

 P, n rr midpoints of edges of P. 2n according to n being even or odd. We add that a 

 new method of dealing with this n — 2-dimensional section, which is indeed a generali- 

 sation of the fact that in fig. 2 the hexagon li e is the part of the plane * common to 

 the two equilateral triangles PQR, P'Q'R', will appear shortly in the Proceedings 

 of this Academy {Verslagen, Dec. 1907, p. 467, Proceedings, Jan. 1908, p. 485). 



