{ (U6 ) 



infinite series of rectangular prisms wliicli liave a cube witli the edge 

 two as basis, and the section of this net of prisms is exactly the net 

 of rhonibohedra. That the sections which, when the intersecting space 

 has an arbitrary position, are quite irregular parallelopipeda, here 

 become rhombohedra is the result of the fact that the intersecting 

 space forms with each of the three spaces .i\ = 0, x\ =^. 0, a:^ = 



equal angles, angles with a cosine of the value |/3. Out of the 



(3 



6 

 diagram with the symbol — it is furthermore evident that the ends 



12 



BB' of the axis of this rhombohedron lie in two consecutive spaces 

 .I'j = 2^/j -\- 1 and that the distance of the parallel spaces of inter- 

 section of the intersecting space with these spaces, which spaces 

 cut the net of rhombohedra in the intersecting space into pieces, 

 must amount to 4. This tallies; for the angle between the spaces 



3''\ + ■''2 + ''"s "I" '^'4 = ^ '"^^^^^ '^'1 ^= ^ ^^^^ I '3 as cosine and there- 



fore - as sine, so that the distance of the planes must be 2 : - . 

 2 2 



From the preceding follows now likewise that the section with 

 the space 3,t'j -j- x^ -\- x^ -\-x^ = i furnishes a space-tilling consisting 

 of the parts D^, D^, D^, D^, D^, D^^; of course also this space-filling 

 consisting of three groundforms each of which appearing in two 

 opposite positions can be obtained by cutting up a net of rhombo- 

 hedra. It is also clear that by taking an intermediary position of 

 the space of intersection we are led to six quite difterent ground- 

 forms, which can be indicated by /A , Z),^, . . . . D^^t,, or in opposite 

 orientation by D^i^ , D^y • • • • ^nè • 



By cutting a block of k* cells C\ instead of a fourfold infinite net 

 (C'a) we can also deduce how one of the forms Z)^> of /(-times greater 



linear size can be built up out of the above mentioned segments 

 D„. We avoid this not to become too longwinded. 



5. Case (2, 1, 1, 0). — When treating the case (1, 1, 1, 0) we have 

 seen that the appearance of nought in the symbol causes prisms to 

 be found with the constant height 2, by which the fourdimensional 

 problem is reduced to a threedimensional one. Thus we are placed 

 before the consideration of the section (2, 1, 1) of the net of cubes 

 which in various respects for the threedimensional space forms the 

 analogon of that of the section (3, 1, 1, 1) in >Sp^. 



If we suppose that the space, in which the section (2, 1, 1) is to be 

 taken, contains the upper cube of the eightcell and the vertex P — see 

 the first of the 20 diaui'ams — is taken as origin of a rectangular 



