The origin of visual regularities

Two things about mirror symmetry are pretty clear:

• It is an abundantly present feature in both the natural world and man-made environments.
• It is a visual regularity, that is, the visual systems of humans and many other species are highly sensitive to it.

Less clear, however, is what the evolutionary relationship between these two things might be.


Evolutionary considerations

The high perceptual sensitivity to symmetry has been proposed to have evolved because symmetry is a useful cue in object recognition. An argument, sketched in the next figure, could be that it is useful because many relevant objects exhibit symmetry, which in turn, may have been caused by a symmetry preference in mating and pollinating behaviour, that is, a preference for more-symmetrical mates or flowers over less-symmetrical ones. This known symmetry preference, in turn, may have evolved because the degree of shape symmetry is a marker of an organism's quality. That is, natural growth processes tend to produce shapes that are basically symmetrical but also exhibit fluctuating amounts of asymmetry, and a higher amount of asymmetry may indicate a lesser genetic quality or a lesser ability to cope with environmental stress.




However, the foregoing argument ignores that the symmetry preference already requires a high perceptual sensitivity to symmetry. Therefore, the next figure provides an alternative argument, which does not deny the evolutionary relevance of the just-mentioned factors, but which proposes clearer causal relationships. This argument starts from growth as a fundamental natural force towards order, which forms a balance against the also-natural force towards chaos. By this, I do not mean that growth should be considered to be a "vital principle" distinct from other physical or biochemical processes (a quite common idea in the 19-th century). Instead, I simply mean that natural growth processes are responsible for virtually all symmetrical shapes in nature, be they crystals or living organisms.




Growth may also reflect the way in which visual systems build structured mental representations. That is, just as a symmetrical body grows preserving its symmetry, the mental representation of a symmetry may "grow" in a piece-wise fashion, starting from subsymmetries. This is the idea behind the holographic approach as proposed in Psychological Review 1996 (see also Symmetry perception and Symmetry processing). This suggests that the natural principle of growth may have given rise to visual systems that are pre-eminently suited to process symmetry.

The latter then may have triggered the symmetry preference in mating and pollinating behavior, which provided such visual systems with evolutionary survival value because fluctuating asymmetry indicates quality. The symmetry preference, by the way, is obviously not the only factor in mating and pollinating behavior, but the idea is that it co-evolved together with other factors. Also in other behavioural domains, this symmetry preference may have exerted selection pressure on symmetrical features, thus boosting the occurrence of symmetrical shapes -- providing such visual systems with further survival value because, thereby, symmetry reinforced itself as a useful cue in object recognition.

This second argument does not challenge the first argument, but it does suggests that the sensitivity of visual systems to symmetry is not just a passive consequence of the abundance of symmetry in the world. That is, natural growth processes are of course the primary cause of the occurrence of symmetry, but the second argument suggests that the perceptual sensitivity to symmetry is a factor that actively influences the occurrence of symmetry in the world. In fact, it might well be the cause of the following remarkable peculiarity in the distribution of multiple symmetries in nature and art.


The distribution of multiple symmetries in nature and art

A multiple symmetry is a configuration with two or more global symmetry axes. This obviously implies that multiple symmetry perception leans upon the above-mentioned perceptual sensitivity to single symmetry. At first glance, only the number of symmetry axes seems to determine how well a multiple symmetry is perceived. Furthermore, also at first glance, there seems to be no reason that 3-fold and 5-fold symmetry might have a special status in nature and art -- yet, they do have a special status.

First, in plants, the family of monocotyledons (with one seed-leaf) usually produces 2-fold and 3-fold symmetrical flowers, and the family of dicotyledons (with two seed-leafs) usually produces 4-fold and 5-fold symmetrical flowers. As depicted in the next left-hand figure, however, the distribution within each family is skewed: Approximately 80% of the monocotyledons produce 3-fold symmetrical flowers, and approximately 70% of the dicotyledons produce 5-fold symmetrical flowers. Flowers do not seem to have intrinsic properties that could evoke this skewed distribution, but notice that insect vision evolved about 400 million years ago whereas flowering plants evolved only 200-125 million years ago (see also the next right-hand figure). Hence, the preponderance of 3-fold and 5-fold symmetrical flowers might well find its origin in the insect's perceptual sensitivity to symmetry.




Second, also in human designs, 3-fold and 5-fold symmetries seem to have a special status: as a rule, they do occur in mystical art but not in decorative art. For instance, Hardonk (1999) created a database of 800 decorative bands (40 bands for each of 20 extinct and present-day cultures from all over the world). A decorative band consists of a repetition of a motif, and about 600 of these 800 bands contain motifs with 1-8 symmetry axes. As a rule, motifs with more symmetry axes occur less often, but motifs with 3 or 5 symmetry axes escape this rule and are virtually absent (see next figure). The latter motifs might be harder to produce, but this can hardly be the reason considering that they do occur in mystical art.




In mystical art, 3-fold and 5-fold symmetrical motifs -- such as the triqueta and the pentagram (see figure above) -- have often been used to symbolize supernatural powers, and nowadays, such motifs are widely used in the emblems of police forces, armies, national flags, car brands, sports clubs, and so on. The pentagram is probably the all-time favorite, with a history of over 5,000 years. In ancient Greece, its alleged supernatural status led to the definition of the golden ratio as the ratio 1.618 in which each intersection of edges in a pentagram sections these edges (this ratio was believed to have aesthetical value). Furthermore, in ancient Mesopotamia, the pentagram symbolized imperial power; among Druids and Celts, it symbolized divine power; in medieval England, it was associated with knightly virtues; and in India, it features in Tantric art. Just as flowers, human designs do not seem to have intrinsic properties that could evoke the special status of 3-fold and 5-fold symmetries, which, also this time, might well have been caused by properties intrinsic to visual systems.

In fact, as argued in Symmetry 2011, the holographic approach indeed predicts a special perceptual status for 3-fold and 5-fold symmetries, which is due to the nonorthogonal relative orientation of their symmetry axes -- see also Treder et al. (2011) and Multiple symmetry. Among other things, this special perceptual status implies that, unlike 2-fold and 4-fold symmetries, 3-fold and 5-fold symmetries can be said to contain hidden order, that is, more order than can be captured in one stable percept. It is known that such hidden order creates a perceptual tension between roughly equally strong and therefore competing percepts, which may trigger one's curiosity, interest, and aesthetical feelings. Though cautious because of differences in visual systems, I suspect that 3-fold and 5-fold symmetrical flowers have a procreation advantage over others, precisely because this perceptually hidden order attracts more pollinators. Furthermore, I suspect that, again because of this perceptually hidden order, throughout history humans have felt that 3-fold and 5-fold symmetrical motifs are more appropriate for mystical art than for decorative art.

Hence, the special status of 3-fold and 5-fold symmetries adds to the above-given evolutionary considerations. That is, it too suggests that the sensitivity of visual systems to symmetry is not so much a passive consequence of the occurrence of symmetry in the world, but rather a factor that actively influences the occurrence of symmetry in the world.

For a similar argument about the relationship between vision and the world, see Occam, von Helmholtz, and Bayes.