( 392 ) 



direction of the axis with period four, whilst sections normal to tha 



axis are characterized by yearrings and medullary rajs. 



If we wish to treat in an analogous manner the second case — 



i.e. that of the second and the fourth figure of column four we 



can start from a net of cubes with edge 2[ y '2. If we transform this 



net by assuming inside these cubes concentric and nomothetic cubes 



3 

 with edge I 2 we find, bv omitting the boundaries of the original 



a 



cubes and producing the boundaries of the new ones, a mixed space- 

 tilling bv cubes and rectangular parallelopipeda characterized by the 



Ï 

 triplets of edges (1,1,1), (3,3,3) and (1,1,3), (1,3,3) with - | 2 



as unit. By splitting up each of the cubes of the two sizes into six 

 equal pyramids and joining these pyramids to the adjacent parallelo- 

 pipeda we get the required space-filling. Here for clearness' sake 

 the figures i>" and 9 6 show the sections corresponding to those of 

 8" and S b . 



In general the space-tilling presenting itself here consists of two 

 different polyhedra occurring in three different orientations; in two 

 particular cases one finds however only one polyhedron occurring 

 in three different positions. 



Space-fillings normal to OK^. - Here the problem of the deter- 

 mination of the section of the net C 8 loses one dimension only ; SO 

 the consideration of the section of a threedimensional net of cubes 

 by a plane normal to a diagonal shows that three sections always 



go together whose characteristic fractions differ by 4> from one 



another. Moreover we have still to bear in mind two things. First we have 



to observe that the three sections of ( ' s ' corresponding to the frac- 



12 . 



tions a a 4- . a 4- do not always give three sections ot a 

 ' r 3 3 



bodily inscribed sixteencell. If we assume for simplicity a to be 

 situated between the limits and , we shall find three sections oi 



o 



13 

 Cic 2) if <i lies between and } . i.e. in half the possible cases. 



i _ i — 



hi the second place Ave have to remember that each of two or three 

 sections of sixteencells occurs in two different orientations, being one 

 another*s mirror-image with respect to the middle plane of the prism, 



section of the bodily circumscribed Cg . If ,lie intersecting space is 

 normal to the line connecting the centre O of the chosen eightcell 

 with the midpoint h\ of its edge PQ, then we have only to con- 



