BI A SPACE KOTATING ABOUT A PLANE. 5 



of intersection of C 8 with a space parallel to these bearing the upper 

 and lower cubes. So any point U within the rectangle PP l Q 1 Q 

 (or P 2 P'Q'Q 2 ) is the projection of an equilateral triangle, which is 

 the intersection of C Q with the plane, in U entirely normal to the 

 plane of the diagram. Likewise any point V within the rectangle 

 P l P 2 Q 2 Q i is the projection of an equiangular hexagon with alter- 

 nately equal sides, which becomes a regular hexagon for the points 

 V situated like O at equal distances from P 1 Q 1 and P 2 Q 2 . Of 

 these triangles and hexagons the projections £7 and V form the centres. 

 An intersecting space /S s (tt) through x projects itself on the 

 plane of the diagram as a line / through 0; therefore the section 

 itself is represented in projection by the segment of / situated 

 within the rectangle PP'Q'Q. According to the position of that 

 line-segment we distinguish three different cases of intersection; if 

 (P designates the absolute value of the acute angle between / and 

 its position 1 Q in which it is parallel to PP' , which angle tt we 

 call the angle of rotation, the three cases are: 



1 st . 0°<(p<30 o , 2 nd . 30°<(|><60 o , 3 rd . 60°<^<90°. 



In the first case the endpoints of the segment lie in the edges 

 PQ, P'Q' and the space /S s (t) has no points in common with the 

 upper and lower cubes: the section is limited by three pairs of 

 parallel planes, whilst in the two other cases it is included by 

 four pairs of parallel planes. In the second case the segment still con- 

 tains points lying outside P l P 2 Q 2 Q l : by some of the planes normal 

 to / the section is cut in semiregular hexagons, by others in 

 equilateral triangles. In the third case no point of the segment lies 

 without P 1 P 2 Q 2 Q 1 : the sections by planes normal to / are exclu- 

 sively hexagons. We will consider each of these cases separately. 

 But first we wish to make a general remark. 



If we consider the line-segment that forms the projection of the 

 solid of intersection as the locus of the points it contains, the 

 section itself appears as built up of an infinite number of infinitely 

 thin slices. Now in the space S 3 (r), bearing the section, the pro- 

 jection / normal to tt in O cuts the plane of any of these slices 

 at right angles in its centre. As the slices of the two different 

 kinds (equilateral triangles and equiangular hexagons) equally admit 

 of a rotation of + 120° about their centres, in their planes, we find: 



"By a rotation of + 120° about its line of projection / the 

 section is transformed into itself; in other words / is an axis of the 

 section, of period three". 



4. The case ty <C 30°. In this case the six side-cubes are cut 



