The verity that nature has mathematically fashioned its building blocks from the only number systems that scale (which is a deep derivative symmetry) portends the emergence of fractal symmetry in the macroscopic domain. Fractal symmetry is the symmetry of repetitive scaling in the parts of a single object, or the shape of a sequence of objects. Fractal symmetry is often how symmetry emerges from the lower simpler realms across the non-linear effects of composition into the higher macroscopic realms. Fractal symmetry uprises strongly in the bodily organization of multicellular plants and animals.
The mathematical primitives that nature uses conform to symmetry mandates. Nature uses primitive mathematical entities that can be scaled by multiplication and division to compose low level structures—from the structure in the elementary particle internal symmetry spaces to the structure of many of the functional systems in the bodies of organisms. These primitives are members of one of the only four number systems over which multiplication and division are defined: the real numbers, imaginary numbers, quaternions, and octonions. These are the only number systems known to scale.
At the most fundamental level, nature uses real numbers to label the points of spacetime, complex numbers for electromagnetic force internal-structure, quaternions or equivalently sets of two complex numbers for weak force internal-structure and octonions or equivalently sets of three complex numbers for strong force internal-structure. In the higher realms of complex structure and behavior, the symmetry feature of these number systems, which is their intrinsic support for scaling, is readily apparent from the scaling of cosmic structures to the bodily system structures of advanced life.
Nature makes extensive use of symmetric (i.e., proportional up and down) scaling in the systems required by advanced life. When an object’s structure is composed of a sequence of successively and proportionally scaled subunits it possesses fractal symmetry. In plants, the branching of limbs, the veining in leaves and the branching of roots exemplify fractal symmetry. In animals, the circulatory, respiratory, filtration and nervous systems possess fractal symmetry.
The appearance of fractal symmetry in natural phenomena is a direct indication that—even in the face of complex, non-linear processes embedded within chaotic quantum mechanical contexts—symmetry ascends into the macroscopic realm. The reappearance of symmetry at every level of form-wise complexity is an intrinsic property of the universe. Living systems represent the most complete realization of nature’s complexity, and therefore living systems should strongly express both micro- and macro-structural symmetry, including fractal symmetry. And indeed, they do. Advanced living systems are rife with bilateral, helical, reciprocally dual, and fractal symmetry. In fact, they are dependent on it. Without leveraging architectural fractal symmetry, nature could not have produced the efficient systems essential for advanced life forms. There would be no well-functioning circulatory systems, breathing systems, neurological systems, blood filtering systems, etc.