## Abstract

Vendobiont type fossils are characterized by possessing ‘gliding symmetry’, in which quilts of the different vanes consisting the body of the Vendobionta meet along the seam with small offset. Because two, three or more vanes are joined along this seam and the quilts of different vanes can be variously offset along the seam, a variety of gliding symmetry exists among Vendobionta. Our theoretical consideration here predicts that the number of possible gliding symmetry is one, two and three for organisms having two, three and four vanes, respectively. We also predict that a special combination of morphological types appear alternatively during the growth of the Vendobionta with two vanes. Further, we were able to identify the existence of two types in gliding symmetry predicted by us for *Pteridinium* specimens with three vanes. Detailed examination of the gliding symmetry of other Vendobionta, therefore, may give important clues in solving riddles veiling the vendobiont relationships.

A considerable number of Ediacaran fossils show ‘gliding symmetry’ and are placed in Vendobionta, on the basis of the existence of this feature. The body of many vendobiont-type fossils consists of vanes that are joined along a seam. The hollow space between the two surface layers defining each vane is separated, by series of more or less parallel walls, into long and narrow quilts. Two, three or more vanes can be joined along the central seam. Quilts of different vanes meet along the seam at nearly a right angle. The quilts meet with an offset shorter than quilt width, leading to the alternation pattern along the central seam (Fig. 1).

Some researchers regard this alternating pattern as merely a variety of bilateral symmetry, whereas others, like Seilacher (1992), regard the pattern as a significant diagnostic feature demanding that the vendiobionts be given a high taxonomic rank, one example being as a class of giant rhizopods (Seilacher *et al.* 2003). Rigorous analysis of this feature, however, has seldom been carried out, although a closer look at this ‘gliding symmetry’ may bring a deeper understanding of the true nature of the organisms.

During field surveys on the Neoproterozoic in Namibia, there was the chance to study numerous specimens of *Pteridinium*. This genus has three sheets, or vanes each bearing quilts arranged more or less parallel to each other. The three vanes join along a central axis or seam, along which the quilts meet in an alternating pattern (Pflug 1970; Grazhdankin & Seilacher 2002). This detailed study of *Pteridinium*, reveals two different ways in which the quilts of three vanes were arranged in this alternating relationship. These observations, together with consideration of another Ediacaran, *Swartpuntia*, which may have had four vanes, leads to the proposal of theoretical models of the alternating quilt patterns in multi-vaned fossils. This paper reports the results of these theoretical considerations.

## Quilt arrangement in multi-vaned Vendobionta

The types of alternating quilt arrangements are examined for organisms with two, three and four vanes that join each other along a median seam. The number of vanes is expressed as *n* (*n*=2, 3, or 4). For simplicity, we assume that the width of the regularly spaced quilts has a constant value of *d*. The offset between the quilts of different vanes that meet along the seam is called ‘quilt offset’. When the vane number is two (*n*=2), the quilt offset of the two different vanes is *d*/2. When the vane number is three (*n*=3), the value of quilt offset is either *d*/3 or 2*d*/3. Therefore, it may be generalized that, for the vane number of *n*, the minimal quilt offset value is *d*/*n*. The vane is assumed to have infinite length to avoid the complication caused when taking the quilt arrangement(s) at both ends of the seam into consideration. The vanes are regarded to consist of an infinite repetition of quilts with uniform widths of *d*. Each vane is given a number from 1 to *n*, clockwise. Vanes are then arranged arbitrarily with quilt-offset values of *d*/*n* relative to the neighbouring ones along the seam. For vane number *n*, the number of all possible operations with this rule is *n*!. However, because some operations yield the same results, the number of recognizable arrangements is less than *n*. Figure 2 summarizes the theoretically possible quilt arrangements for cases with *n*=2, 3 and 4. In the following text, quilt arrangements for these three cases will be described.

For *n*=2, the number of possible quilt arrangement is 2!=2. Each vane is given a number 1 and 2 clockwise around the axis. Here a symbolic notation for arrangement types will be introduced as follows: a *d*/2 quilt offset of vane 1 ahead of vane 2 will be described as [12], whereas a 1/2*d* quilt offset of vane 2 ahead of the vane 1 will be described as [21]. However, both types can be perfectly overlapped, if one type is shifted *d*/2 along the axis relative to the other. Therefore there is only one quilt arrangement type for *n*=1.

For *n*=3, the number of possible quilt arrangements is 3!=6. However, [231] and [312] can be overlapped to [123] by shifting operations along the axis. The same can be said for [213] and [321], which can be overlapped to [132]. As a result, there are two quilt arrangement types (Fig. 2). These two cannot be overlapped by shifting and rotational operations along the axis, they are mirror images to each other.

A theoretical description of the shifting operations yielding the above results is as follows. For any numbers *a*, *b*, and *c* (*a*≠*b*≠*c*) from {1, 2, 3}, the obtained vane constellation, (for example [*a b c*]) can be overlapped to [*b c a*] by shifting *d*/3 along the axis and to [*c a b*] by shifting 2*d*/3 along the axis.

For *n*=4, the number of possible quilt arrangements is 4!=24. However, after excluding the identical ones that can be overlapped by shifting operations along the axis, the number of arrangements is reduced to six: [1234], [1243], [1324], [1342], [1423] and [1432] (Fig. 2). [1243] can be overlapped to [1324] when the former is rotated by 90° anticlockwise around the axis. In the same way, [1243] and [1423] are equivalent to [1342] after rotation around the seam by 180°, and after rotation by 270°, respectively. Consequently there are only three different arrangement types [1234], [1432] and [1243] (Fig. 2) left for the case *n*=4. [1234] and [1432] are mirror images to each other. [1243] is the mirror image of itself.

Although the above results are for vanes with infinite length, correlation between them and patterns exhibited by real Vendobionta is easy to carry out, as long as the fossils preserve at least a fragment containing more than one quilt structure on each vane along the seam.

## Quilt patterns and mode of growth

Vendobionts have finite lengths defined by the two ends of the seam. The existence of tips introduces more constraints. Here, such constraints are considered for the case with *n*=2. Quilt arrangement between two tips can be classified into four patterns. If the total number of quilts in both vanes is odd, and the difference in quilt numbers is minimal between the two vanes, the number of quilts in one vane is by one quilt larger than that of the other vane. According to the condition that odd-numbered quilts are on the left or right vane, there are two types (A and B in Fig. 3). In both cases, the head end of longer quilt sequence is by *d*/2 ahead of the shorter sequence and the tail by *d*/2 behind the even one. If the total number of quilts in the two vanes is even and the difference in quilt number between the two vanes is minimal, both vanes must have the same number of quilts. In this configuration, there are two possible variations: either the quilts on the left or right hand vanes are offset ahead by *d*/2 relative to those on the opposite vanes (C and D in Fig. 3).

This recognition is very important for inferring the mode of growth of those organisms. As an example, we consider the case that the organism grows by adding quilts alternately at one tip and grows in such a manner that the difference between the quilt numbers stays minimal (i.e. one or zero). If the organism of type A is taken as the starting point, the new quilt is added at the right tip, changing the organisms to type D. In this organism of type D, then, the new quilt is added at the left tip, bringing it back to type A. Consequently, in a fossil assemblage with this manner of growth, we can find only the combination of type A and D. Therefore, the distinction of these types in fossils may provide us with the mode of growth of organisms with quilted vanes (see Fig. 3 for other possible growth modes and accompanying occurrence of tip types). This type of analysis may be usefully applied to investigating growth modes of *Yorgia* or *Spriggina* (Fedonkin 2003; Seilacher *et al.* 2003).

## Pteridinium specimens from Aar Farm

*Pteridinium* is a well-known Ediacaran fossil, which has three quilted vanes meeting along a seam. Along the seam, quilts meet each other in an alternating fashion. Previous studies have described in some detail morphological characters of *Pteridinium*. Pflug (1970) reconstructed *Pteridinium* as a complex of vanes connected to form a canoe-shaped body. Grazhdankin & Seilacher (2002) studied taphonomical and morphological features and described the general morphology of *Pteridinium* as follows. Two sides of the boat (i.e. two lateral vanes) meet along a median seam. Along this seam, the mould can be broken to reveal the impression of the third, median vane. All vanes are patterned by parallel and evenly spaced furrows of quilts, which meet the seam at nearly a right angle.

To test whether or not our simple model can be applied to the morphology of this real Vendobiont, we studied specimens from Aar Farm in Southern Namibia (Fig. 4). Preserving their three-dimensional morphology, they are suitable for this analysis. Normally, quilts within each vane are arched in the same direction (Fig. 4), indicating the polarity of the organism. Here we define the convex side of the arch as anterior facing. Specimen A in Figure 4 is a fragment preserving a lateral vane and a median vane. Figure 4A1 shows the anterior and Figure 4A2 the left side of the median vane of the specimen (see viewing directions for A1 and A2 in the upper left corner of Fig. 4). Figure 4A3 is a line drawing of A2. Specimens B and C are counterparts. Figure 4B2 and Figure 4C2 are views from the top for the two specimens (viewing directions in upper left corner). Line drawings for both specimens are also shown (Fig. 4B2 and Fig. 4C2). In about 80 examined specimens, including the three illustrated ones, quilts of the two different vanes meet along the seam in alternating fashion, indicating that in *Pteridinium* the quilts of all three vanes alternate along the seam. Further, the shift between two vanes is about 1/3 of the quilt width. In specimen B of Figure 4, quilts of the right vane shift forward by 1/3 quilt width relative to those on the left side. In the specimen Figure 4C, the quilts of the left vane shift forward by 1/3 quilt width relative to the right vane. The existence of two types of quilt arrangements are predicted for the case where the vane number is three. In fact, the quilt arrangement of the specimen B in Figure 4 corresponds to the [123] type, and the quilt arrangement of the specimen C in Figure 4 to the [132] type. Almost half of the 80 examined specimens shows [123] type and the other half [132] type.

## Discussion

Our theoretical consideration predicts that there is a finite number of quilt arrangements along the seam in multi-vane vendobionts, and that the number of possible arrangements increases with the number of vanes. For the case of *n*=3, we predicted two types ([123] and [132]). The examination on three *Pteridinium* from Aar Farm confirms this prediction.

The two quilt arrangement types observed in the *Pteridinium* are mirror images to each other. In nature there is analogous phenomenon. For example, gastropod shells can coil in sinistral or dextral direction. In gastropods, the chirality (sinistral or dextral of different chirality) is genetically determined. Further, the difficulty of copulation between different chiral individuals of the same species, one of the two would eventually eliminate the other. Thus, the study of the chirality of certain organism, or groups of organisms, can tell demonstrate how genetics and environmental factors influence the evolution of organisms concerned. Further investigation on quilted Vendobionta, by analogy, may provide us with information about the genetic controls on some of the vendobionts.

We do not apply the results of our consideration about the growth of the quilted patterns on all vendobionts. However, at least one tip is preserved as large ‘head region’ in *Yorgia* and *Spriggina*. Therefore, a statistical study of these forms, might provide clues about their mode of growth, which in turn may shed light on the taxonomic status of vendobionts.

## Conclusion

There appears a good match between theoretically predicted quilt arrangements of different vanes meeting together along one seam and a those occurring in the vendobiont *Pteridinium* from Aar Farm, southern Namibia. Nearly half of the 80 specimens examined possessed [123] type and the remainder [132] type. Theoretical considerations predicted that the ‘handedness’ in quilt arrangement between two tips is determined by the mode of growth, and or what Vendobiont is. Better understanding of the possible genetic control of such growth patterns may in the future assist in determining the true systematic position of vendiobionts.

## Acknowledgments

We express our cordial thanks A. Seilacher and R. Jenkins for their useful comments on our study, and G. Schneider (Director of the Geological Survey of Namibia) for her kind permission and support for our research activities in Namibia. We also express our great thanks to Barbara and Bruno Boehm (owners of Aar Farm) for their gracious permission to carry out our research on their land.

- © The Geological Society of London 2007