m A SPACE ROTATING ABOUT A PL\NE. 25 



represent the horizont.il middle sections of the 19 (P\ Moreover 

 the 4 -J- 12 -\- 24 -j- 12 + 4 line-segments equal to unity, forming 

 together the sides of the 19 squares, represent the horizontal edges 

 of the tetrahedra 2 1{r> . Any of these 56 segments belongs to two 

 tetrahedra if there are two segments lying in two adjacent planes 

 crossing it at right angles and having their centres in the vertical 

 through the centre of the chosen segment. If there is only one 

 such segment the chosen segment belongs to only one tetrahedron. 

 So we rind in the layers between the planes successively 

 4+12-1-12 + 4. = 32 T*\ 16 right-handed ones and 16 left- 

 handed ones. 



Case V. Here we find B = A + l 2 C + 2 B ^- 2 B. Of these 

 different pieces A and B occur in the middle layer only, C and D 

 in the extreme layers only. The forms A are all white , the forms 

 B all black. The forms C and D show this particularity that not 

 even the equal forms belonging to the same extreme layer corres- 

 pond in shade. So the upper layer contains 13 6, six white ones 

 (tV' 4") an( l seven black ones (| -, y 7 ^), etc. 



Case VI. Here B = A -f 2 C + 2 E + 2 B -j- 2 I). Moreover 

 all A, E, B are white, all (', B black. 



Case VII. This case leads us back to the well-known plane- 

 filling by regular hexagons and equialateral triangles. 



Liscard I X1 , -. nn „ 

 . November, 1907. 



Groningen ) 



