In the great majority of corals we have as simple or even simpler conditions than those of Alcyonium; for as a rule the calcified partitions or septa of the coral either converge upon a central chamber (or central “columella”), or else fail to meet it and end freely. In the latter case the problem of space-partitioning does not arise; in the former, however numerous the septa be, their separate contacts with the wall of the central chamber comply with our fundamental rule according to which three lines and no more meet in a point, and from this simple and symmetrical arrangement there is little tendency to variation. But in a few cases, the septal partitions converge to meet _one another_, there being no central chamber on which they may impinge; and here the manner in which contact is effected becomes complicated, and involves problems of space-partitioning identical with those which we are now studying. In the genus Heterophyllia and in a few allied forms we have such conditions, and students of the Coelenterata have found them very puzzling. McCoy[401], their first discoverer, pronounced these corals to be “totally unlike” any other group, recent or fossil; and Professor Martin Duncan, writing a memoir on Heterophyllia and its allies[402], described them as “paradoxical in their anatomy.” [Illustration: Fig. 174. _Heterophyllia_ sp. (After Martin Duncan.)] The simplest or youngest Heterophylliae known have six septa (as in Fig. 174, _a_); in the case figured, four of these septa are conjoined two and two, thus forming the usual triple junctions together with their intermediate partition-walls: and in the {389} case of the other two we may fairly assume that their proper and original arrangement was that of our type 6_b_ (Fig. 158), though the central intermediate partition has been crowded out by partial coalescence. When with increasing age the septa become more numerous, their arrangement becomes exceedingly variable; for the simple reason that, from the mathematical point of view, the number of possible arrangements, of 10, 12 or more cellular partitions in triple contact, tends to increase with great rapidity, and there is little to choose between many of them in regard to symmetry and equilibrium. But while, mathematically speaking, each particular case among the multitude of possible cases is an orderly and definite arrangement, from the purely biological point of view on the other hand no law or order is recognisable; and so McCoy described the genus as being characterised by the possession of septa “destitute of any order of arrangement, but irregularly branching and coalescing in their passage from the solid external walls towards some indefinite point near the centre where the few main lamellae irregularly anastomose.” {390}