101 



Ith Minimum Partitional Area. 



509 



composing its edges are not in this plane, but in planes making, 

 as we shall see (§ 12), angles of 54° 44' with it. For three 

 edges of each hexagon, the planes of the arcs bisect the angle 

 of 109° 28' between the planes of the six corners of contiguous 

 hexagons : and for the other three edges are inclined on the 

 outside of its plane of corners, at angles equal to the supple- 

 ments of the angles of 125° 16' between its plane of corners 

 and the planes of contiguous quadrilaterals. 



12. The planes of corners of the eight hexagons consti- 

 tute the faces of an octahedron which we see, by symmetry, 

 must be a regular octahedron (eight equilateral triangles in 

 planes inclined 109° 28' at every common edge) . Hence these 

 planes, and the planes of the six quadrilaterals, constitute a 

 plane-faced tetrakaidecahedron obtained by truncating the six 

 corners * of a regular octahedron each to such a depth as to 

 reduce its eight original (equilateral triangular) faces to equi- 

 lateral equiangular hexagons. An orthogonal projection of this 

 figure is shown in fig. 3. It is to be remarked that space 

 can be filled with such figures. For brevity we shall call it 

 a plane-faced isotropic tetrakaidecahedron. 



1 3. Given a model of the plane-faced isotropic tetrakaideca- 

 hedron, it is easy to construct approximately a model of the 

 minimal tetrakaidecahedron, thus : — Place on each of the six 

 square faces a thin plane disk having the proper curved arcs of 

 19° 28' for its edges. Draw the three long diagonals of each 

 hexagonal face. Fill up by little pieces of wood, properly cut, 

 the three sectors of 60° from the centre to the overhanging 



x_y 



Fig. 3. Fig. 4. 



edges of the adjacent quadrilaterals. Hollow out symmetrically 

 the other three sectors, and the thing is done. . The result is 

 shown in orthogonal projection, so far as the edges are con- 

 cerned, in fig. 4 ; and as the orthogonal projections are equal 

 and similar on three planes at right angles" to one another, 



* This figure (but with probably indefinite extents of the truncation) 

 is given in books on mineralogy as representing a natural crystal of red 

 oxide of copper. 



