On the sections of a block of eightcells by a space 

 rotating about a plane 



BY 



Mrs. A. BOOLE STOTT and Dr. P. H. SCHOUTE. 



INTRODUCTION. 



1 . If a space of three dimensions S 3 rotates in /S" 4 about a given 

 fixed plane tt the general case is that /S. d and ir have only a line 

 / in common. We will restrict ourselves here to the special case 

 where S 3 passes through tt. 



If we start from a fourfold infinite block of eightcells and cut 

 it by a space the polyhedra of intersection form a threedimensional 

 space-filling. When the position of the intersecting space is an 

 arbitrary one, the number of the polyhedra of different shape is 

 infinite. Here we will restrict ourselves once more to the special 

 cases where the number of the polyhedra of different shape is 

 finite. These comme mur able cases are characterized by the property 

 that any space parallel to the considered position of the rotating 

 space of intersection and passing through a vertex of one of the 

 eightcells cuts the four edges through the opposite vertex in points 

 the distances of which from that last vertex are commensurable 

 with the length of the edge. 



Finally we restrict ourselves to the case of a finite block con- 

 sisting of 3 4 = 81 eightcells, forming together an eightcell of three 

 times the size, and we suppose that the fixed plane 7r passes through 

 the centre of this block and is totally normal to a plane ir' con- 

 taining two opposite edges of it. 



G 1* 



