Bring it on!
ken ----------------------
Okay. There are several pieces to this idea and the relationships will not be obvious at first. Try and bare with.
There is a biological problem trying to explain how plants and animals have taken on their generalized shapes. By shape I mean a crude symmetrical form. For example, we have a bilateral symmetry and starfish have a radial symmetry. Although the shapes make sense in a functional way, and for animals this is usually explained in terms of mobility, there doesn't seem to be any biological principle involved. In other words, in biology it is any form that works. But this just kicks the question out the door. What determines what works?
While the detail varies in the extreme, the general symmetries do not. Flies and bears are symmetrical in the same way. If you cut them in half along their longitudinal axis you get two sides that mirror each other. Most higher plants on the other hand have arranged themselves around a central axis. It is this last observation that provides a clue. The central axis colines with a gravitational vector or line of force. The apparent functionality of this arrangement is that, trees for example don't fall over--most of the time.
In terms of energy, a point or colined axial symmetry, is a position of minimum potential energy or maximum stability in a field such as gravity. But the same physical principle appears in electro-magnetic fields. There are a variety of protein aggregates that form symmetric configurations which are the result of both the rigidity of the particular protein molecule and their mutual attractions. In these cases the field or attraction is based on electro-magnetic properties, and again the minimization of potential energy or maximum stability seems to be the principle involved. Some examples of this phenomenon are the formation of aggregate protein coats for viruses which create symmetric shapes. Here the symmetries are spherical or point symmetries, and helical symmetries, that is radial turns about a linear axis.
So how is it that these phenomenon have taken on what appears to be the classical shapes from elementary geometry? Some are platonic solids no less. There are also many examples from chemistry (molecular steric number), in particular crystal formations that generate the same abstract symmetries. In fact the whole field of crystallography is devoted to examining these shapes. And, the way we know what various proteins look like is to crystallize them and then make x-ray photographs of their structures.
What is going on here?
The organizing principle is the attractive point, axis, or field line itself that determines the possible arrangements of form in space. The point or field line acts as an invariant under the possible rigid transformations in that particular configuration space.
It turns out that these symmetries are the result of the number of unique point or axial permutations possible in a particular finite set of spatial dimensions. [This is the group theory part. I am describing the polyhedral groups which compose a subgroup of the full symmetric group. The common representation of S(n) is to write out all the possible permutations on (n) elements. For example, given four elements the possible permutations there 4! = 1 x 2 x 3 x 4 = 24. The polyhedral group representation takes the form a tetrahedron in three dimensions where the invariant is the center point or null element. In two dimensions the representation of four points is a square and the null is the center or the axis of rotation and reflection. With the loss of one dimension of rotation, the resultant rigid motions or transformations are reduced to 8 or D(4), the dihedral group order four.]
The connection to platonic solids requires some explanation. These three dimensional shapes and their two dimensional analogues are created following the rule that you must divide a sphere or circle into equal parts with a rule and compass. The figures and shapes that are created when the division points are connected yield the platonic figures. This geometric idea of an equipartition of a sphere or circle is equivalent to the idea of stability or minimum energy in the sense that all forces present are equalized to cancel each other out. In other words, there is zero relative motion. A similar equalization of force principle exists in both the atomic and molecular worlds that result in the same symmetric arrangements.
To return to the gravitational world, then a colinear point or axis is the position of maximum stability and determines the maximum number of unique finite permutations about that axis. In the case of continuous and infinite point to point transformations in three dimensions the result is a sphere. However, if a lateral motion (or transformation) perpendicular to a radial axis is introduced, then the result is a loss of some rotational degree of symmetry. This means that motion across the grain of a gravitational field produces forms of bilateral symmetry as a consequence. An ellipse is a geometric example of a circle that has been dilated or under gone a transformation in the direction of a diameter to produce a major axis. So an ellipse has a bilateral symmetry or a reflection through its major axis.
Now it depends on how much further you want to go with how a gravitational field has determined the evolutionary parameters of biology. I just read Ian Murray's current post to the effect that gravitational forces are nil for microorganisms. Well, most of those that move around under their own steam are in fact ellipsoids. When you rotate an ellipse through its major axis you get a ellipsoid. With a random choice of ends, pretty soon a head and tail end appear and before you know it, we are talking salmon like the one I just caught, cleaned and ate last night.
The more important point about gravity is to remember that as a vector it has two components, one of magnitude and the other of direction. While the argument that small floating organisms can barely be effected directly by the force or magnitude of gravity, they are always subject to its directional component. And while relative to an external medium like water, perhaps the surface forces are negible. However, the internal components are still subject to the same field forces. For example, if you are flying in an aircraft and don't jump out, you still walk down the isle instead of float.
But in more concrete terms there are a whole variety of internal cell components that respond to unidirectional forces like gravitation or omnidirctional forces like pressure. Actin and tublin are examples. These are the molecular equivalent to muscle and bone and provide the structural and/or motility components of most cells.
In even more primative contexts, you also have to remember that gravitation partitions matter by density, that is creates environmental sedimenations, which are thought to contribute to various molecular arrangements in a primodial soup.
Chuck Grimes