( 391 ) 



image with respect to the limiting space common to the two eightcells, 

 the sections of both the sixteencells with the intersecting space normal 

 to ihal limiting space arc one another's mirror-image with respect to 

 the plane of intersection of intersecting space and limiting space, i.e. 

 with respect to one of the vertical faces of the rectangular square 



prism that forms the section of ( V . As the axis MX of the figures 

 of column four forms an angle of 4.V with each of these four faces, 

 it will be normal to its mirror-image. 



In the first of the three cases — . thai of the first and the third 

 figure of the third column only one polyhedron appears, the 



square double-pyramid, the base of which is a square with side 

 2 i 2, the height of which also is 2 I 2. In the following way it is 

 easily proved that this not entirely regular octahedron can form a 

 space-filling by itself Let us consider a net ot cubes with edge 

 2 1 2 and divide each of- these cubes into six equal square pyramids 

 admitting as base one of the faces of that cube and as common 

 vertex the centre of the cube; then the required net is obtained 

 it* we join together to a double-pyramid each pair of pyramids 

 standing on the same base; according to the directions of the 

 axes with period four of these pyramids this net consists of three 

 equally strongly developed groups of polyhedra. We remark that 

 the regular octahedron cannot till space, but that we obtain a poly- 

 hedron that (.Iocs (ill space, as has just been proved, by compressing 

 the regular octahedron in such a manner that the distances of the 

 points of the surface from a plane through four of the six vertices 

 are diminished to h I 2 times the original value. 



In the third of the three cases i.e. in that of the middle figure 



of the five sections of column four - we have again to deal with 

 onlv one polyhedron, vizi a cube with edge I 2 bearing on two 

 opposite faces a square pyramid with height }, I 2. We show easily 

 that this body has the space-tilling property as follows. Let us -tart 

 from a net of cubes with edge I 2 and suppose the centre of one 

 of these cubes to be the origin of a rectangular system of coordi- 

 nates the axes of which are parallel to tic edges of the Cube. Then 

 let us divide into six equal square pyramids «'ach cube the centre 

 of which has for coordinates either only even or only odd multiples 

 of l 2, and join each of these pyramids to the adjacent cube; then 

 the required space-tilling is obtained. Of these the two figures 8° 

 and s show the sections with planes u=2k\ 2 and u={2/,-\-\ | 2, 

 where u stands for any of the three coordinates ; here have heen 

 indicated the fibres of the combination (10,20.12 running in the 



26 



Proceedings Royal Acad. Amsterdam. Vol. XI. 



