Unequipolar stauraxonia are the pyramids, a fundamental form that plays an important part in the configuration of organic bodies. They were formerly described as regular or fundamental forms. Such are the regular blooms of flowering plants, the regular echinoderms, medusæ, corals, etc. We may distinguish several groups of them according to the number of the horizontal transverse axes that cut the vertical main axis in the middle.
Two totally different divisions of the pyramidal types are the regular and the amphithecta pyramids. In the regular pyramids the transverse axes are equal, and the ground-surface (or base) is a regular polygon, as in the three-rayed blooms of the iris and crocus, the four-rayed medusæ (A-f, 16, 28, 47, 48, etc.), the five-rayed "regular echinoderms," most of the star-fish, sea-urchins, etc. (A-f, 10, 40, 60), and the six-rayed "regular corals" (A-f, 9, 69).
The amphithecta (or two-edged) pyramids, a special group of pyramidal types, are characterized by having as their basis a rhombus instead of a regular polygon. We may, therefore, draw two imaginary transverse axes, vertical to each other, through the ground-surface, both equipolar, but of unequal length. One of the two may be called the sagittal axis (with dorsal and ventral pole), and the other the transverse axis (with right and left pole); but the distinction is arbitrary, as the two are equipolar. In this lies the chief difference from the centroplane and dorsiventral forms, in which only the lateral axis is equipolar, the sagittal axis being unequipolar. We find the bisected pyramid in a very perfect form in the class of the ctenophora (or comb-medusæ, A-f, 27), where it is quite general. The striking typical form of these pelagic cnidaria is sometimes called biradial, sometimes four-rayed and bilateral, and sometimes eight-rayed-symmetrical. Closer study shows it to be a rhombus-pyramid. The originally four-rayed type, which it inherited from craspedote medusæ, has become bilateral by the development of different organs to the right and left from those before and behind.
Similar rhombo-pyramidal forms to those of the ctenophora are also found in some of the medusæ and siphonophora, many of the corals and other cnidaria, and many flowers. The name "two-edged" which is given to this special type is taken from the ancient two-edged sword. Its chief axis is unequipolar, the handle being at the basic pole and the point at the verticle pole; but the two edges left and right are equal (poles of the lateral axis), and also the two broad surfaces (dorsal and ventral, joined by the sagittal axis).
III. Centroplane Types.—The natural middle of the body is a plane, the median or chief plane (planum medianum or sagittale); it divides the bilateral body into two symmetrical halves, the right and the left. With this is associated the characteristic antithesis of back (dorsum) and belly (venter); hence, in botany this type (found, for instance, in most green leaves) is called the dorsiventral, and in zoology the bilateral in the narrower sense. One characteristic of this important and wide-spread type is the relation of three different axes, vertical to each other; of these three straight axes (enthyni) two are unequipolar and the third equipolar. Hence, the centroplanes may also be called tri-axial (triaxonia). In most of the higher animals (as in our own frame) the longest of the three axes is the principal one (axon principalis); its fore pole is the oral or mouth pole, and its hinder pole is the aboral or caudal (tail) pole. The shortest of the three enthyni is, in our body, the sagittal (arrow) or dorsiventral axis; its upper pole is at the back and its lower pole at the belly. The third axis—the transverse or lateral axis—is equipolar, one pole being called the right and the other the left. The various parts which make up the two halves of the body have relatively the same disposition in each half; but absolutely speaking (namely, in relation to the middle plane) they are oppositely arranged.
Further, the centroplane or bilateral forms are also characterized by three vertical axes which may be drawn through each of the normal axes. The first of these normal planes is the median plane; it is defined by the chief axis and the sagittal axis, and divides the body into two symmetrical halves, the right and left. The second normal plane is the frontal plane; this passes through the chief axis and the transverse axis (which is parallel to the frontal surface in our body), and divides the dorsal half from the ventral half. The third normal plane is the cingular (waist) plane: this is defined by the sagittal and transverse axes. It divides the head half (or the vertical part) from the tail half (or the basal part).
The name "bilateral symmetry," which is especially applied to the centroplane and dorsiventral types, is ambiguous, as I pointed out in 1866 in an exhaustive analysis and criticism of these fundamental forms in the fourth book of the General Morphology. It is used in five different senses. For our present general purpose it suffices to distinguish two orders of centroplane types, the bilateral-radial and the bilateral-symmetrical; in the former the radial (pyramidal) form is combined with the bilateral, but not in the latter.
The bilateral-radial type comprises those forms in which the radial structure is combined in a very characteristic fashion with the bilateral. We have striking examples in the three-rayed flowers of the orchids (A-f, 74), the five-rayed blooms of the labiate and papilionaceous flowers, etc., in the plant world; and in the five-rayed "irregular" echinoderms, the bilateral sea-urchins (spatangida, clypeastrida, A-f, 30) in the animal world. In these cases the bilateral symmetry is recognizable at the first glance, as is also the radial structure, or the composition from three to five or more raylike parts (paramera), which are arranged bilaterally round a common central plane.
The bilateral-symmetrical type is general among the higher animals which move about freely. The body consists of two antithetic parts (antimera), and has no trace of radial structure. In the free moving, creeping, or swimming animals (vertebrates, articulates, mollusks, annelids, etc.) the ventral side is underneath, against the ground, and the dorsal side upward. This form is clearly the most useful and practical of all conceivable types for the movement of the body in a definite direction and position. The burden is equally distributed between the two sides (right and left); the head (with the sense organs, the brain, and the mouth) faces frontward and the tail behind. For thousands of years all artificial vehicles (carts on land and ships in water) have been built on this type. Selection has recognized it to be the best and preserved it, while it has discarded the rest. There are, however, other causes that have produced the predominance of this type in green leaves—the relation to the supporting stalk, to the sunlight that falls from above, etc.
Special notice must be taken of those bilateral forms which were originally symmetrical (by heredity), but have subsequently become asymmetrical (or of unequal halves), by adaptation to special conditions of life. The most familiar example among the vertebrates are the flat-fishes (pleuronectides), soles, flounders, turbots, etc. These high and narrow and flattened boney-fishes have a perfect bilateral symmetry when young, like ordinary fishes. Afterwards they form the habit of laying on one side (right or left) at the bottom of the sea; and in consequence the upper side, exposed to the light, is dark colored, and often marked with a design (sometimes very like the stony floor of the ocean—a protective coloring), while the side the flat-fish lies on remains without color. But, what is more curious, the eye from the under side travels to the upper side, and the two eyes lie together on one side (the right or left); while the bones of the skull and the softer parts of each side of the head grow quite crooked. Naturally, this ontogenetic process, in which a striking lack of symmetry succeeds to the early complete symmetry of each individual, can only be explained by our biogenetic law; it is a rapid and brief recapitulation (determined by heredity) of the long and slow phyletic process which the flat-fish has undergone for thousands of years in its ancestral history to bring about its gradual modification. At the same time, this interesting metamorphosis of the pleuronectides gives us an excellent instance of the inheritance of acquired characteristics, as a consequence of constant œcological habit. It is quite impossible to explain it on Weismann's theory of the germ-plasm.