with Minimum Partitional Area, 507 



7. We have now space divided into equal and similar tetra- 

 kaidecahedral cells by the soap -film ; each bounded by 



(1) Two small plane quadrilaterals parallel to one another ; 



(2) Four large plane quadrilaterals in planes perpendicular 

 to the diagonals of the small ones ; 



(3) Eight non-plane hexagons, each with two edges common 

 with the small quadrilaterals, and four edges common with 

 the large quadrilaterals. 



The films seen in the Plateau cube show one complete 

 small quadrilateral, four halves of four of the large quadri- 

 laterals, and eight halves of eight of the hexagons, belonging 

 to six contiguous cells ; all mathematically correct in every 

 part (supposing the film and the cube-frame to be infinitely 

 thin). Thus we see all the elements required for an exact 

 construction of the complete tetrakaidecahedron. By making 

 a clay model of what we actually see, we have only to 

 complete a symmetrical figure by symmetrically completing 

 each half-quadrilateral and each half-hexagon, and putting 

 the twelve properly together, with the complete small quadri- 

 lateral, and another like it as the far side of the 14-faced 

 figure. We thus have a correct solid model. 



8. Consider now a cubic portion of space containing a 

 large number of such cells, and of course a large but a 

 comparatively small number of partial cells next the boundary. 

 Wherever the boundary is cut by film, fix stiff wire ; and 

 remove all the film from outside, leaving the cubic space 

 divided stably into cells by films held out against their tension 

 by the wire network thus fixed in the faces of the cube. If 

 the cube is chosen with its six faces parallel to the three pairs 

 of quadrilateral films, it is clear that the resultant of the 

 whole pull of film on each face will be perpendicular to the 

 face, and that the resultant pulls on the two pairs of faces 

 parallel to pairs of the greater quadrilaterals are equal to one 

 another and less than the resultant pull on the pair of faces 

 parallel to the smaller quadrilaterals. Let now the last- 

 mentioned pair of faces of the cube be allowed to yield to 

 the pull inwards, while the other two pairs are dragged out- 

 wards against the pulls on them, so as to keep the enclosed 

 volume unchanged ; and let the wirework fixture on the faces 

 be properly altered, shrunk on two pairs of faces, and extended 

 on the other pair of faces, of the cube, which now becomes a 

 square cage with distance between floor and ceiling less than 

 the side of the square. Let the exact configuration of the 

 wire everywhere be always so adjusted that the cells through- 

 out the interior remain, in their altered configuration, equal 

 and similar to one another. We may thus diminish, and if we 



