Part 2/2 can be read here.

“Mathematics is the language in which God has written the universe.”

― Galileo Galilei

Is math discovered or created? This question has troubled mathematicians and physicists for millennia. I’m certainly not an expert in this field, but I believe that math is discovered. This theory stems from the axiom that math exists independent of humans – it supersedes any man-made attempt to explain the natural world. Creating mathematical proofs, for example, could be thought of as a way to codify what already exists in the universe. The more that we uncover mathematical truths that explain the fabric of reality, the more that we can make sense of what is happening. What’s interesting to me is how we can unlock our understanding of the human system through applying knowledge of these first-principles. Physics is governed by mathematical laws, and both appear to explain the human system from micro to macro.

Geometry in Nature

When observing the natural world, two things become clear: nature leans towards efficiency and geometry explains the shape of all things. Geometry arises out of math and therefore is consistent with the first-principles of biology. Different shapes emerge from nature and give us insight into the simple rules that govern complex systems.

The most energy-efficient shape is the sphere because it encloses the largest volume with the least surface area. However, spheres do not exist in biology (1)(2). Any curve that we observe from a higher scale is actually an imperceivable amount of straight lines (1)(2). Furthermore, the efficiency of a sphere is lost due to the ‘dead-space’ that arises when we pack this shape into an enclosed area (2). Recall nature’s tendency to find efficient solutions as a means for self-preservation. Because of the problem with spheres in an enclosed space, triangular structures are actually more economic when conjoined with each other. Conveniently, triangles are the most stable shape and can resist collapse much greater than a circle or square (2). Forces applied to a triangle are distributed across the entire structure, drawing parallels to the concept of biotensegrity which will be discussed later.

In 3-dimensional space, an icosahedron is the closest approximation of a sphere constructed with triangles as its sub-units (2)(3). It is the best of both worlds: it has a large volume-to-surface-area ratio making it efficient, but it also follows the constraints of biology since its structure is dictated by straight lines. Another benefit to icosahedrons is that they are omnidirectional, meaning that they can conjoin with other icosahedrons in all directions (2)(3). Buckminster Fuller coined the term geodesic geometry which is a useful model that pulls together the above concepts. Theoretically, the human system organizes itself into polyhedral variants from cells up to larger anatomical units. If we can see through this perspective, we can start to understand the code behind our biology.

Human Anatomy and Fractals

“The language of anatomy tends to obscure the unity across scales. The fractal approach, by contrast, embraces the whole structure in terms of the branching that produces it; branching that behaves consistently from large scales to small.” – James Gleick on the branching of the bronchial tree (4)

A fractal is a scale-free pattern that is defined by its self-similarity (4)(5)(6). Complex structures emerge from the repetition of simple fractal rules. In relation to biology, fractals beginning at the molecular level could scale outwards to explain the observable shape of the human system. The elegance of nature is shown by combining simple shapes with fractal rules.

The anatomy of the lungs is a perfect example of fractal patterning (4)(5)(6). During fetal development, lung tissue starts from one source that differentiates over time. Structures are stereotyped based on inherent fractal rules, which allows for the closest packing of lung tissue in a confined space. This creates a large surface area which functions to optimize gas exchange. The bronchus, bronchioles and alveoli are “separate” subunits that have been named based on function and scale. These categories can be helpful in some circumstances, but misleading in others. When we categorize various branches, we contradict the unity that exists across scales.  

In order to bring awareness to how prevalent fractals are, here are some examples seen in the human body and nature:

Nature

• Bifurcating fractals: tree branches and root systems, river systems, lightning bolts

• Spirals (based on the Fibonacci sequence/golden ratio): flower heads, cacti, pinecones, Romanesco broccoli, nautilus shells, barrel waves, hurricanes/typhoons, galaxies

• Fractal patterning: coastlines, clouds, honeycomb, frost patterns, snowflakes

The Human Body      

• Bifurcating fractals: bronchial tree, neural networks, capillary networks, urinary collection system, biliary duct in the liver, His-Purkinje network in the heart

• Spirals (formed around the Fibonacci sequence/golden ratio): cochlea in ear, DNA double helix, electrical spiral of the heart, PNF patterns, spiral lines and functional lines as per Thomas Myers, helical motion and left precession of viscera as per Bill Hartman

Biotensegrity - Bridging the Gap between Geometry and Fractals

Biotensegrity is a model that better illustrates the bodies’ viscoelastic nature held together by an equilibrium of compression and tension (3)(7)(8). It is based on the laws of physics which are fundamental to our understanding of the universe. Biotensegrity is also consistent with polyhedral and fractal modelling. From this perspective, we are interconnected from head-to-toe – our bones are suspended in a sea of myofascia. In reality, all of our tissues act in a viscoelastic manner and even bones have the ability to bend and torque in order to accommodate loads (9)(10). Thinking through the lens of biotensegrity allows us to appreciate the body as an integrated system made for movement, rather than a reductionist view of muscles and joints. Nothing acts in isolation. Much like the physics of a triangle, tensegrity structures – and the human body – distribute loads across the entire structure.

Biotensegrity challenges the common view of classical mechanics where the body acts as a machine with pillars, arches, levers and pulleys that oppose the downward pull of gravity (1)(7). This is an architectural perspective that assumes linear mechanics; bones are thought to be stacked on top of one another like a building (2). This representation is based on inanimate objects and does not accurately portray the non-linear dynamics of living tissues (3)(7). One of the contributors to this view is the interpretations of anatomy based on cadaver dissections. Cadavers are dried-out versions of living tissue and therefore do not accurately represent how we move. We are 2/3rds water by weight, which means we must behave like water through shape change and pressure management (11). During dissection, anatomist must also cut away the fascial system. This is problematic because fascia is continuous along muscles and their subunits; our “proprioceptive suit” connects all tissue and transmits tension in relation to human movement.

Biotensegrity seems to be a better representation of what is actually happening in the human body and has several advantages over classical mechanics. Tensegrity structures allow for fluid-like movement without sacrificing stiffness or stability (1)(7)(8). They can also self-organize by automatically orienting themselves towards positions of equilibrium, thus preserving energy (1)(8).

Application to Clinical Practice

“Everything should be made as simple as possible, but no simpler.”

― Albert Einstein

Practically, why do these concepts matter? When dealing with the human system, it’s easy to get lost in complexity and minutia. Fractals allow us to create simple rules that are seen across scale and body region. They provides us with a deeper level of analysis on external anatomy and shape change. If we look for self-similarity in the body instead of differences, we realize that many structures appear and behave in a consistent way. In terms of geometry, we can identify triangular and polyhedral patterns in our anatomy which can provide insight into our internal and external helical angles (11). Lastly, biotensegrity is a framework for regional interdependence. It explains how we can intervene at a distant area to affect the patient’s primary complaint.

Here are some examples of self-similarity in our external anatomy. We can justify our interventions based on the relationships identified. This is just the tip of the iceberg, but it should be sufficient to get the point across.

• Iterations of the infrasternal angle, infrapubic angle and sphenoid (11)

• Synchrony of thoracic diaphragm and pelvic diaphragm during respiration

• Mirroring of scapular and innominate mechanics during respiration (11)

• Sacrum and lumbar spine interrelated with the structure/function of upper dorsal-rostral and cervical spine (11)

• Thorax expansion/compression patterns in relation to upper/lower extremity variability (11)

• Inguinal ligament in the pelvis and clavicle are embryologically similar (11)

• Homologies within the brachial and lumbo-sacral plexus (11)

• Joint-by-joint approach as per Gray Cook and Mike Boyle – self-similarity of structure/function of hip and shoulder, knee and elbow, foot and hand etc.

• PNF – pisiform and lateral calcaneal contacts to drive external rotation; thenar eminence and medial calcaneal contacts to drive internal rotation

• PNF – D1/D2 patterns for upper and lower extremity

Hopefully I’ve made a compelling argument for viewing the body in this interconnected way. In my opinion, this approach better resembles reality since it’s rooted in math and physics; however, I will continue to evolve my thought process and am open to new ideas.

I still have plenty of questions about the utility of these principles outside of the field of rehabilitation and human performance. Some of these ideas are definitely fun to explore. If fractals really are scale-free, could these principles repeat inward and outward into infinity? Could fractals provide a solution to the Theory of Everything? If this is true, are we just iterations of the universe? Is everything, everything?! Needless to say, these questions are well above my pay grade.

References:

1. Scarr G. Helical tensegrity as a structural mechanism in human anatomy. International Journal of Osteopathic Medicine. 2011;14(1):24-32.

2. Scarr G. Simple geometry in complex organisms. Journal of Bodywork and Movement Therapies. 2010;14(4):424-444.

3. Levin S. Tensegrity: the new biomechanics. Oxford Textbook of Musculoskeletal Medicine. 2015;:150-162.

4. Gleick J. Chaos. London: The Folio Society; 2015.

5. Fractal Foundation Online Course. Presentation presented online; 2020.

6. Sapolsky R. Part 1: Chaos and Reductionism ; Part 2: Emergence and Complexity. Presentation presented online; 2010; Stanford.

7. Scarr G. BIOTENSEGRITY: A DIFFERENT WAY OF THINKING. Fascia: scientific advances. 2018;:167-180.

8. Swanson R. Biotensegrity: A Unifying Theory of Biological Architecture With Applications to Osteopathic Practice, Education, and Research—A Review and Analysis. The Journal of the American Osteopathic Association. 2013;113(1):34.

9. Wu Z, Ovaert T, Niebur G. Viscoelastic properties of human cortical bone tissue depend on gender and elastic modulus. Journal of Orthopaedic Research. 2011;30(5):693-699.

10. Martens M, van Audekercke R, de Meester P, Mulier J. Mechanical behaviour of femoral bones in bending loading. Journal of Biomechanics. 1986;19(6):443-454.

11. Hartman B. The Intensive IX. Presentation presented at IFAST; 2019.