58 A. T. MASTERMAN [january 



type, corresponding almost exactly with the " bilateral symmetry," 

 is repetition of parts in only one dimension, so that the centre of 

 symmetry is a plane formed by the two remaining dimensional axes. 

 In this case the minimum number of secondary centres of symmetry is 

 reduced to two, and, as there is but one dimension of symmetry, this 

 corresponds to the predominating number. 



Piano-symmetry occurs in a few of the Protozoa and Coelentera, 

 and is found almost universally in the Metazoa above the Coelentera, 

 though it may be more or less disguised in certain groups {e.g. Echino- 

 derma, Tunicata), by a secondary superposed axo-symmetry. The 

 environmental conditions for the production of this form of symmetry 

 are not many. A primary physical environment fulfilling the desired 

 conditions is not common nor easy of attainment. Locomotion of the 

 organism in a definite direction, but forming some angle with the 

 perpendicular to the surface or to the bottom, will supply the requisite 

 dissimilarities of environment in two dimensions, provided that rotation 

 about the locomotive axis be prevented. In other words, the motion 

 is limited to one dimension, which, for the greatest effect, should be 

 horizontal. Nevertheless, although this is the prevalent origin of 

 piano-symmetry, a physical environment of an organism at rest is con- 

 ceivable in which the necessary conditions are fulfilled. Thus fixation 

 at one end will cause, as already shown, axo-symmetry if the axis is 

 parallel with the perpendicular to the plane of fixation, but if they 

 intersect at an angle, a factor of dissimilarity is produced, which, like 

 the last, will be more potent the greater the angle. Examples of this 

 type without locomotion- in one direction are rare, but are probably 

 exemplified in Taenia, Loxosoma, etc. In some text-books it is stated 

 that " bilateral " symmetry is confined to animals with locomotion in 

 one direction, but, as pointed out here, the necessary conditions are 

 attained in a sedentary habit. 



There are three possible sub-types of piano-symmetry, which are 

 defined according to the heterogeneity in the dimensional axes. The 

 first or tri-plano-symmetry (Fig. 4) has all three axes formed of similar 

 component radii ; and hence three planes of symmetry are present, 

 intersecting each axis respectively. The geometrical expression of this 

 type is the right rhombic prism or octahedron. It is included by 

 Haeckel in a sub-group of his Centraxonia. Di-plano-symmetry 

 (Fig. 5) has two planes of symmetry, as the radii of one axis differ. It 

 is represented by the right rhombic pyramid. Piano-symmetry (or 

 monoplano-symmetry) has only one plane of symmetry (Fig. 6) as the 

 radii of two axes differ, and hence the plane of symmetry must pass 

 through these two. It is represented by the right pyramid with a 

 rhomboidal base. Spencer distinguishes these three as single, double, 

 and triple bilateral symmetry (Figs. 4, 5, and 6). 



The true piano-symmetry is by far the commonest, though instances 

 of the others can be easily found. The typical Hexactinia, taking into 



