“…How is it in perception that the partible appears as whole and the whole is partitioned …because, is said, there is no measure to mind; the mind being impartible divides and perceives all… Nothing is in cognition not previously in the sense[s]…”
Aristoteles De Anima (p. 407–417)
Since classical antiquity, perception of self-similarity in multidimensional space–time forms in nature and humanity has been an appealing subject in psychophysics, especially through its association with esthetics (Birkhoff’s theory ). Scale-invariant visual and aural patterns have been perpetually tempting and provoking our cognition, with eventual perceptions of beauty arising from conquering of similarity relationships through a comprehensible model of their generator law by our mind. Ostensibly stemming from the self-similar cortical and lobe anatomy of the human brain itself and the concomitant electrochemical function of neural physiology with iterative feedback patterns, science and art always shared such an original common root: Tantalizing of human mind by intelligible regular, mathematically elegant, recursively repetitive self-similar forms, reflecting both esthetic appeal as, e.g., in visual and performing arts, and simultaneously functional performance as in, e.g., natural phenomena and human activity. Harmonious beauty and optimal functionality of self-similar structures, as illustrated below, appear to be dual sides of a kernel cognitive element in the human mind’s affinity to creation.
Description and Generation
Mathematical abstraction, since the early days of Euclidean geometry of simple, continuous, smooth, finite-featured forms, has tackled complex, discrete, nondifferentiable, multiscale self-similar forms with golden ratio recursion, helical spirals, harmonic waves, Pythagorean trees, Hippocrates menisci, Apollonian packs , etc. This is continued in Byzantine architecture and Arabesque calligraphy through the middle ages into renaissance art and culminates in a number of mathematical developments and archetypes of modern times: The Weierstrass noise, Shepard tones, devil’s staircase, Cantor sets, Koch curves, Peano meanders, Hilbert scans, Minkowski sausages, Pascal triangles, Sierpinski gaskets, Menger sponges, Cayley networks, Bethe lattices, Gray code maps, Lèvy flights, Brownian trees and clouds, Barnsley ferns, Fournier constellations, Julia and Mandelbrot sets, etc., . The work of the latter coined the term fractals to unify self-similarity concepts and introduce tools such as fractal (i.e., Hausdorff) dimension, bridging between integer dimensionalities (0D, 1D, 2D, and 3D) to describe wiggledness and roughness of space-filling forms, and lacunarity, capturing transformation-invariant vacancy (trema) and fragmentation of shapes. These gave rise or were effectively implemented in a number of powerful techniques and algorithms, such as box counting, cubit/octree spatial occupancy, Voronoi diagrams, potential theory, level sets, site/bond percolation, bifurcation theory, branching models, section dimensionality, pattern subordination, spatial tiling, affine transformations, Markov processes, cellular automata, etc.
Equally importantly, this definitive work on spatial fractal forms and distributions illustrated their generative connection to the temporal dynamics of chaotic iterative systems. Strict or approximate (asymptotic) self-similar fractals are produced via infinite or finite repetitions of discrete recursion or continuous feedback-based, deterministic or stochastic chaos generator laws with certain features: (i) lack of an absolute characteristic spatial/temporal dimension (e.g., a length or time), yielding scale-invariant applicability; (ii) nonlinearity (either smooth via a frequency power law or a probability density function, or hard via a nondifferentiable function) with nonintegrability and nonunique invertibility, causing branching bifurcations of the response; (iii) unstable amplification of random variability in its initial, boundary, or functional conditions, dominating the eventual response; and (iv) domains of dynamic repellers and basins of attractors (e.g., strange attractors with nondifferentiable contours) of lower dimensionality, shaping the equilibrium orbits of the chaotic process, thus leaving a fractal structure as its fingerprint. Paradigms of such fractal-producing recursive chaos algorithms evolved from logistical parabolas, modulus functions, limit cycle frequency-doubling (Feigenbaum) fig trees to circle inversion, three-body (clover leaf) and multi-attractor games, turtle algorithms, origami folding, etc. Such laws are central in discrete/atomistic modeling techniques in nanotechnology, including Monte Carlo, molecular dynamics, density functional theory, etc., essentially mimicking natural phenomena and processes.
Throughout natural evolution from astronomical macrospace to the nanoscale and subatomic world, numerous such chaotic phenomena have produced a variety of formations exhibiting an intriguing fractal architecture across dimensional scales: In inanimate nature galaxies and galactic foams, solar systems, interstellar dust, clouds, thunderstorms, atmospheric turbulence, winds, ocean flows, sea coastlines, rivers, mountain ranges, volcanic craters, continent plaques, pyrolusite rocks, dendritic minerals, snowflake crystals, mists and colloids, air and soil particulates, animal migration and vegetation patterns, forests, wildfires, etc., offer some familiar examples. Animate organisms such as sponges, corals, trees, foliage, florals, and fungal cultures; animal systems and vasculatures including pulmonary, lymphatic, circulatory, and neural networks; and organ tissues such as alveolar, renal, intestinal epithelium, trabecular bone, and brain cortices add to this list. At the subcellular and/or submicron scale, such fractal constructs reveal an equally diverse variety of chaotic processes for matter, energy, and information transformation at both the formative and functional stages of the structure.
At the nanoscale material accumulation, for example, occurs via diffusion-limited aggregation, electrolytic, and colloidal deposition from solvents, cloud precipitation, gaseous condensation, liquid solidification, crystallization, dendritic nucleation and growth, resin curing, adhesion, directional self-assembly, powder agglomeration and bonding, colloidal clustering, bacterial agglomeration (Eden model), etc. Matter ablation takes place through dielectric breakdown, electrochemical corrosion, mechanical erosion and fracture, cavitation, melting and evaporation, solutionization, porosification, and leaching among similar others. Material transformation is witnessed by innumerable reactions and phase transitions, polymerization, pulverization, atomization, Brownian mixing, turbulent eddy flows, viscous fingering (Hele-Shaw flows, Fig. 1), diffusive alloying, defect formation and motion, formation of long-order quasi-crystals, Ising spin glass distributions, etc. Such matter transformation is governed by nonlinear dynamics of logistic parabola-controlled population growth, evolutionary chain reaction of inflatory theory, Baker’s fold and mixing transforms, flow instabilities, bifurcation and percolation theory, frequency locking and synchronization, among countless other chaotic phenomena.
The structure–function connection of fractal forms, such as cloudlike globular agglomerates in an Apollonian pack and Brownian-tree branching networks (Fig. 2), with their respective transport performance during both formation and action, can be exemplified by an electrical discharge of a thunderbolt producing a familiar Lichtenberg figure  (Fig. 3). It appears that nature dissipates or harvests electrical charge distributed over a higher-dimensional space (3D atmospheric volume), through permeation of an intermediate-dimensional interface (2D cloud surface), via linear transport over a lower-dimensional network of conduits (1D thunderbolt branches), to a lowest-dimensional point sink (0D lightning pin) or from appoint source. In such percolation across domains of multiple integer dimensionalities, a high fractional Hausdorff dimension of the Apollonian cloud and Brownian thunderbolt fractals controls percolation properties (electrical conductance) for given effort distribution (potential field) to optimize charge flow (electrical current), so as to maximize transport of generalized energy (electrical power). In such general transport of mass, energy, and/or information, nature creates and/or operates under quantitative preservation laws (e.g., continuity, first law of thermodynamics, and quantum conservation), but also promoting qualitative loss of transformability (e.g., diffusion laws, second law of thermodynamics, and Shannon entropy), eventually down to complete irreversibility (third law of thermodynamics). To this end, it appears that nonlinear, scale-invariant, chaotic transport phenomena generate fractal domains of optimal dimension and lacunarity in space–time, as already explained above.
Such optimal transport is of course of keen interest to modern engineering and biomedicine beyond the natural science and mathematical descriptive context above, which has already catalyzed significant developments in mass and heat transfer, crystallography, colloidal chemistry, geology, life science taxonomy, medical anatomy, etc. The focus here shifts onto a functional performance (operative system design) and a constructive processing (fabrication and manufacture) framework . Design and manufacturing of electromagnetic antennae, photovoltaic electrodes, batteries and supercapacitors, membranes for fuel cells, water desalination, catalytic oxidation, pollutant filters, CO2 capture and storage, mining pathways, highway networks, aviation logistics, and construction of 3D cities are just but a few examples of potential engineering impacts of fractals. Biomedical physiology and technology call for biocompatible implant materials, composites and foams, vascularized scaffolds for tissue engineering, brain–machine interface transceivers, neural inference for vision and thought, computer tomography via inverse radon transforms, cardiac resuscitation by heartbeat fractal analysis, spread of contagious epidemics, among others. Finally, chaos-generated fractals, through their intertwining with behavioral psychology, linguistics, sociology, architecture, sculpture (Fig. 4), and music, present an original, holistic educational paradigm for the sciences, technology, engineering, arts, and mathematics.
Such an embarking onto fractal engineering and biomedical research and education hinges on three hypotheses:
The chaotic process–fractal structure–transport function relationships delineated above can be reverse-engineered to optimality for specific criteria of each application.
Inverted function–structure–process relations above lead to similar chaotic generator mechanisms for optimal construction and operation of fractals, enabling their simultaneous formation and action as in live structures (dynamic biomimetics).
Functional optimality in bioengineering dually mirrors esthetic appeal in the arts through mathematical modelability in cognitive sciences.
Rather than an exhaustive mapping of prior scientific and technical literature on the subject (intended for upcoming review papers), the previous introduction to fractal engineering and biomedicine was therefore aimed at establishing the underpinning principles of a future research and educational framework. Toward this direction, this topic hosts a number and diversity of papers highlighting underlying issues, methodological approaches, and application scope and potential in this area. Throughout this topic, the reader is challenged and encouraged to establish fertile connections and extrapolations to his/her own investigation area and teaching endeavors. It is hoped that such seminal work would chaotically stimulate and inspire future contributions, ideally propagating in fractal patterns, in the specific and broader areas of knowledge, along with fruitful benefit to the society and humanity.
The author wishes to acknowledge Dr. Quan Wang, Associate Editor and Professor at the Khalifa University and Dr. Boris Khusid, Editor of the ASME Journal of Nanotechnology in Engineering and Medicine, for their encouragement and help in publication of this special topic.