Reflections on Staggered Bilateral Symmetry in Dickinsonia and other Ediacarans
Extended Abstract – Kyoto 2006

Enrico Savazzi
Hagelgränd 8, 75646 Uppsala, Sweden.
(email: enrico5@savazzi.net)

Number of Words: 666
Number of References: 1 
Number of Figures: 0 
Number of Tables:  0

Running header text: Staggered Bilateral Symmetry 

The body-plan of Dickinsonia and other Ediacaran organisms has been interpreted as a series of fluid-filled cavities surrounded by thin membranes (Seilacher 1992). This structure is equivalent to a multiple pneu (i.e., a set of flexible membranes enclosing spaces kept under hydrostatic pressure).  An alternative way of studying the properties of multiple pneus is by analogy with foam, e.g. a set of soap bubbles in reciprocal contact.
This analysis starts with the assumption that the optimal shape for Dickinsonia is flattened, with all its pneus co-planar (which is reasonable for an elongated epifaunal organism lying on the seabed, and is in fact the normal morphology of fossil specimens). In this context, the simplest multi-pneu configuration has a uniserial row of bubbles strung along a straight axis. A flattened structure can be obtained by adding additional series of pneus on either side. Unfortunately, a uniserial row of pneus is a poor choice as the “vertebral column” of a flattened structure, because this configuration allows it to both bend and twist.
The next step toward complexity is a biserial row of bubbles. This can be implemented as two uniserial strings of bubbles joined side-by-side. A biserial row can have either a bilateral symmetry, or a staggered symmetry. In the former case, four bubbles share the same corner along the central axis of the organism. In the latter, only three bubbles share a corner.
By observing foam, it is easy to see that the case of four coplanar bubbles sharing a corner cannot occur if the bubbles are free to reposition themselves. In such a case, the final configuration will be a lowest-energy, stable one, i.e., one that minimizes the deviation of each bubble’s shape from spherical. Thus, 90º angles between bubble walls that share a corner are less stable than 120º ones. 
Two alternative events take place in this case: (1) two of the four bubbles, on opposite sides of the cluster, lose contact with each other (i.e., the configuration shifts into two separate corners, each shared by three bubbles, or (2) the four bubbles shift into a tetrahedral (i.e., non-coplanar) arrangement. This configuration keeps a central corner shared by all four bubbles, but the angles between shared faces around this corner become 120º. A further analogy can be made with the sp3 orbitals of carbon, which also have a tetrahedral symmetry because the electrical charges in each orbital repel equally those of the other orbitals, and this arrangement maximizes their reciprocal distances (i.e., it is a lowest-energy configuration). The latter configuration is not relevant to a coplanar structure, while the former can retain its coplanarity. By adding more bubbles to this configuration while keeping its character intact, one can indeed end up with a staggered, biserial and flat structure. Although the membranes of pneus and Ediacaran organisms have a finite elasticity and cannot be endlessly stretched and twisted like bubble walls (which are fluid, rather than solid), the above argument applies also to solid membranes, in that minimum-energy geometries impart less tension to all types of membranes.
This idea needs to be verified in Dickinsonia, as well as other Ediacaran organisms. This concept may prove useful, for instance, to understand the triradiate symmetry of Tribrachidium. Charnia appears to have implemented a similar constructional principle, but in the context of a branching, rather than bilateral geometry. One may also ask whether Pteridinium, with its three vanes, requires a different axial arrangement of pneus than Dickinsonia. Preliminary observations of Pteridinium do indeed show a staggered symmetry between two coplanar vanes, but it is difficult to assess how the third vane relates to these.

Reference
Seilacher, A., 1992. Vendobionta and Psammocorallia: lost constructions of Precambrian evolution. The Journal of the Geological Society of London, 149, 607-614.