to tissues and beyond, in the morphogenetic process. That is, symmetry is hierarchical and scale dependent. Changes from juvenile to adult stages as well as differences between sexual and asexual reproduction will exhibit symmetry characteristics at each stage of development. Matching symmetry states to morphogenetic stages may inform the developmental process.
Symmetry is, then, a compilation of perception or observation as well as the arrangement of structural features (internally or externally) defining dimension (3D) and geometry (contours, surfaces, and boundaries) in terms of “balance” and “likeness”, and this compilation can be evaluated at any step in the morphogenetic and developmental processes of an organism at the individual, population, or higher taxonomic grouping level.
2.1.1 Recognition and Symmetry
Object recognition is the term applied to our understanding of how to perceive or observe a three-dimensional (3D) object and interpret it from its two-dimensional (2D) projection. Perception studies with Attneave’s cat [2.9] or Biederman’s cup [2.13] as iconic images to discern the process of recognition utilize the geometric aspect of objects as points connected with straight edges or partial contours, respectively (cf. visual tracking of eye movements [2.4]). These studies showed that recognition is initiated from perception of outlines rather than just points on an object. Information acquisition is accomplished via this perception or observation. One perspective is the preference for the simplest interpretation over others to infer a structured whole, and this forms the basis of structural information theory [2.80]. Another perspective involves preference for a minimum (not necessarily simple) description forming the basis of algorithmic information theory [2.21]. In any case, information theory is at the heart of object recognition. Perceived or observed information may occur as an object that is recognized. Initial recognition may occur of a whole scene as an object. In contrast, individual objects in the scene and their boundary and surface attributes such as shape, transparency, pattern, location, size, or texture may be perceived, and perception violations of individual objects in a scene may be recognized (Figure 2.1, after Mezzanotte’s scenes, in [2.14]). Symmetry as a mode of recognition can apply to the scene as an object or the parts of this object in relation to the boundary and surface attributes that comprise the scene. Information about symmetry is acquired with respect to an external or anatomical viewpoint.
Body shape is important in symmetry determination of organisms. Shape deformation relies on a point-based approach with regard to object recognition requiring identifiable equivalences on multiple shapes. The shape decomposition approach represents an implicit reference to contours in recognition as the conduit to assessing symmetry [2.127] (Figures 2.2a–c). Shape decomposition relies on the delamination of surface boundary height via contours with respect to object recognition and implicitly influences shape in symmetry assessment [2.72] (Figures 2.2d–f).
For reflective (also called bilateral, mirror, mirror-image, line, plane, or left/right) symmetry [2.159], shape is considered to be the most ubiquitous among organisms and is especially noticeable in animals and plants. Colored symmetry [2.132] can also occur, as in the left/right asymmetry of otherwise bilaterally symmetric organisms [2.41, 2.144]. However, symmetry is present as more than rotational or reflective in organisms (Figure 2.3). In 2D, dihedral (rotational plus reflective) [Weisstein 2002], translational, and glide (reflective plus translational) [2.159] symmetries are evident. The 2D and 3D quality of whole organisms or their parts lends itself to the related symmetries of knots (including chirality) [2.154], helices/spirals (including handedness) [2.159], and compound or multiple spirals (a variant of knot symmetry), [2.159] as well as scale (fractal or dilation) [2.159] and related conformal symmetry [2.159] with associated Mobius transformations [2.159]. Symmetries associated with loop networks [2.36, 2.88, 2.89, 2.159, 2.164] are present at various scales in organisms as well. See Figure 2.3 for examples of all symmetries on the diatom valve surface. Note that 1) rotational and reflective symmetries apply to the whole valve; 2) dihedral, translational, and glide symmetries apply to pores on the valve face; 3) knot, helical, compound spiral, and conformal symmetries apply to valve central areas; 4) scale symmetry applies to the replication of five-fold symmetry within a Triceratium valve face; 5) loop networks are seen in the pores on the valve surface of the Trigonium and Triceratium valves depicted.
Figure 2.1 A drawing composed by JLP that is reminiscent of one of Mezzanotte’s scenes [2.15]. Although the scene is the whole perceived object, individual objects diatom (fish, jelly fish, boat, oars, waves, and shore) and their attributes such as shape, transparency, location, size, and texture and perspective of the scene may be viewed as perception violations.
Figure 2.2 Shape perception of Arachnoidiscus ehrenbergii from image ProvBay5_12lx450. (a-c) Shape deformation via an increasing number of points defining valve features. (d-f) Shape decomposition via an increasing number of contours defining valve features.
Symmetry has implications for evolutionary processes in animals and plants [2.54]. Symmetry may be adaptive so that active organisms or those in rapidly flowing water [2.45] will have reflective symmetry while sedentary organisms in low flow shear regimes will have rotational or dihedral symmetry. Organisms with large surface to volume ratios or repetitive parts tend to have translational symmetry (Figure 2.4). Organisms that exhibit energy efficiency have scale symmetry [2.158]. In general, symmetry is a dynamical property of organisms.
Figure 2.3 Diatom valve face symmetries. (a) Arachnoidiscus ehrenbergii-rotational; (b) Biddulphia sp.-reflective; (c) Coscinodiscus sp. microstructure-translational; (d) Asterolampra marylandica-dihedral; (e) Coscinodiscus sp. microstructure-glide; (f) Trigonium americanum-helical/spiral central area; (g) abnormal Cyclotella meneghiniana-multiple helices/spirals central area; (h) Coscinodiscus sp.-helical/spiral; (i) Actinoptychus splendens-knot (interlooping valve structure); (j) Arachnoidiscus ornatus-knot (interlocking rings under valve ribs); (k) Triceratium pentacrinus fo. quadrata-scale; (l) initial valve of Biddulphia sp. 3D valve surface-conformal symmetry or Mobius transformation; (m) Trigonium dubium-loop network; (n) Triceratium bicorne-loop network. (All SEMs by Mary Ann Tiffany). See “Recognition and symmetry” section in the text.
Figure 2.4 Double translational symmetry at two scales exhibited in a Paralia sulcata chain colony of repetitive cells. Light micrograph by Mary Ann Tiffany.
2.1.2 Symmetry and Growth
Morphogenesis has been characterized as a system of detached movement of ensembles of cells in animals in contrast to lack of detached movement via ensembles of cells in plants. That is, locomotion is one way to associate a specific kind of symmetry to motile organisms in contrast to sessile organisms which may exhibit another kind of symmetry [2.59]. Modes of growth contribute to the recognition of specific types of symmetry. Branching growth exhibits continuous symmetry decrease and increase in size during growth [2.47]. Stepwise growth and size reduction are modes of discrete growth where size increases or decreases over time. Growth changes are concomitant with symmetry changes as potential indicators of stress associated with development, morphogenesis, cytogenesis, epigenesis [2.154], and embryogenesis [2.43, 2.149], depending on evolutionary or ecological conditions. Levels of environmental or teratological stress are said to influence fitness and adaptability