496 THE FORMS OF TISSUES [ch. 



space, and three equal sides make the simplest of triangles. But 

 it by no means follows that equilateral, or any other, triangles 

 combine to form the simplest of polygonal associations or patterns. 

 On the other hand, three straight lines meeting in a point are the 

 least number by which we can subdivide or partition two-dimensional 

 space ; the simplest case of all is when the three partitions meet at 

 co-equal angles, and a pattern of hexagons, so produced, is, geo- 

 metrically speaking, the siiT^plest of all ways in which a surface can 

 be subdivided — the simplest of all two-dimensional '"space-fiUing" 

 patterns. So it comes to pass that we meet with a pattern of 

 hexagons here and there and again and again, in all sorts of plane 

 symmetrical configurations, from a soapy froth to the retinal 

 pigment, from the cells of the honeycomb to the basaltic columns 

 of Staffa and the Giant's Causeway. 



We pass to solid geometry, and arrive by similar steps at an 

 analogous result. Four plane sides are now the least number which 

 enclose space, and (next to the sphere itself) the regular tetrahedron 

 is the first and simplest of solids; but its simplicity is that of a 

 solitary or isolated figure, and tetrahedra do not combine to fill space 

 at all. But as the partitioning of an equilateral triangle was the 

 first step towards the symmetrical partitioning of two-dimensional 

 space, so we draw from the regular tetrahedron a first lesson in the 

 partitioning of space of three dimensions ; and as three lines meeting 

 in a point w«ere needed to partition two-dimensional space, so here, 

 for three-dimensional space, we need four. The simplest case is, as 

 before, when these meet at co-equal angles, but we do not see quite 

 so easily what those four co-equal angles are. 



For as the centreof symmetry of our equilateral triangle was defined 

 by three lines bisecting its three angles and meeting one another in 

 a point at co-equal angles of 120°, so in our regular tetrahedron four 

 straight fines, running symmetrically inwards from the four corners, 

 meet in a point at co-equal angles, and again define the centre of 

 symmetry. If we make (as Plateau made) a wire tetrahedron, and 

 dip it into soap-solution, we find that a film has attached itself to 

 each of the six wires which constitute the httle tetrahedral cage; 

 that these six films meet, three by three, in four edges; and that 

 these four edges meet at- co-equal angles in a point, which is the 

 centi'oid, or centre of symmetry, or centre of gravity, of the system. 



