Module 0 · BioGeometry
Geometry of Life
D'Arcy Thompson's transformations, physical constraints on biological form, and the scaling laws that set the stage for everything that follows.
1. D'Arcy Thompson's On Growth and Form
Published in 1917 by the Scottish zoologist and classicist D'Arcy Wentworth Thompson, On Growth and Form is the founding text of mathematical biology. Thompson's core insight was that morphological variation among species often reflects continuous geometric deformations of a common ancestral form - mathematically, diffeomorphisms of the plane.
The transformation method
Thompson overlaid a Cartesian grid on one species and deformed it smoothly to map onto a related species. The deformation \((x, y) \to (f(x, y), g(x, y))\) captures the evolutionary or developmental relationship. His celebrated example related Argyropelecus olfersi to Sternoptyx diaphana (both deep-sea hatchetfish) by a shear followed by exponential stretching along the dorsal axis.
\[ (x, y) \;\longmapsto\; \big( f(x, y),\; g(x, y) \big) \]
with \(f, g\) smooth and \(\det \partial(f, g)/\partial(x, y) > 0\) (orientation-preserving).
Modern translation: diffeomorphism metrics
Grenander and Miller's computational anatomy formalized Thompson's idea as large deformation diffeomorphic metric mapping (LDDMM): the distance between two shapes is the minimal kinetic energy of a time-dependent velocity field \(v(x, t)\) that deforms one into the other:
\[ d(M_1, M_2)^2 = \min_v \int_0^1 \|v(\cdot, t)\|_V^2 \, dt \]
This formulation underlies modern morphometrics in paleontology, neuroimaging and developmental biology.
2. Physical Constraints on Form
Not all forms are biologically attainable. Gravity, surface tension and viscosity impose hard upper and lower bounds on what bodies can look like at a given size. Thompson was particularly eloquent about this: “Everything is the way it is because it got that way.”
Gravity: the scaling paradox
For a uniformly scaled body with length L, mass grows as \(M \propto L^3\) while bone cross-section area grows as \(A \propto L^2\). Axial stress on supporting legs:
\[ \sigma = \frac{Mg}{A} \propto \frac{L^3}{L^2} = L \]
Stress grows linearly with size. Bone strength is \(\sim 100\) MPa and density\(\sim 1000\) kg/m\(^3\). Maximum geometric land animal:
\[ L_{\max} = \frac{\sigma_{\max}}{\rho g} \approx 10\,000 \; \text{m (!)} \]
This bound is far larger than real animals because other constraints (metabolism, heat dissipation, locomotion) kick in earlier. But it explains why a scaled-up spider would collapse under its own weight: exoskeleton stress = L\(^4\) (hollow tube area \(\propto L\)) so a 10-m spider needs 10000x stronger exoskeleton than a 1-cm spider.
Surface tension: small-world regime
The capillary length \(\ell_c = \sqrt{\gamma / (\rho g)}\) is ~2.7 mm for water. Below this scale, surface tension dominates gravity:
\[ \mathrm{Bo} = \frac{\rho g L^2}{\gamma}, \quad \mathrm{Bo} < 1 \text{ for } L < \ell_c \]
Water striders walk on water (Bo << 1); mosquitoes suck blood through capillary action; rainforest drops form because of \(\ell_c\). Small aquatic insects live in a world where gravity is optional.
Viscosity: Purcell's “Life at Low Reynolds Number”
The Reynolds number \(\mathrm{Re} = \rho U L / \mu\) sets whether viscous or inertial forces dominate. Purcell (1977) made the classic observation that below Re ~ 1, swimming by reciprocal motion is impossible (the scallop theorem). Bacteria must rotate helical flagella; paramecia beat cilia metachronally; spermatozoa undulate non-reciprocally.
\[ \mathrm{Re} = \frac{\rho U L}{\mu} \]
E. coli: Re ~ 10\(^{-5}\); Daphnia: Re ~ 30; Blue whale at 10 m/s: Re ~ 10\(^8\). A factor of 10\(^{13}\) that reshapes every aspect of locomotion physics.
3. Length-Mass-Shape Fundamentals
Three scaling relations organize biological form across 20 orders of magnitude of mass:
- Isometric scaling: \(L \propto M^{1/3}\) (geometric similarity)
- Elastic similarity (McMahon 1975): limb length \(L \propto M^{1/4}\); diameter \(d \propto M^{3/8}\)
- Stress similarity: \(L \propto M^{1/3}\) but bones thicken super-linearly
Surface-to-volume ratio \(S/V \propto L^{-1}\) - the origin of scaling laws in heat loss, respiration and water balance.
Why a 10-meter spider cannot exist
A tarantula (5 cm body, 8 g) has leg segments 1 mm thick. Scaling isometrically to 10 m: leg thickness would be 20 cm, body mass \(8 \mathrm{g} \times 200^3 = 64\,000\) kg. Bending stress on cylindrical exoskeleton tube scales as \(ML/(r^3 t)\) with tube wall thickness t fixed fraction of r: \(\sigma \propto L^{4}/L^4 = L\). But chitin yield stress is only 80 MPa vs bone's 100 MPa, so the giant spider would shatter in its first breath. This is why arthropods plateau at ~1 m in air (compared to the 3 m of the giant crab Macrocheira - which lives in water where gravity effectively halves due to buoyancy).
4. SVG: D'Arcy Thompson Transformation
5. Simulation: D'Arcy Thompson Grid Transformation
Three panels: reference grid with pufferfish, sheared grid (Argyropelecus-like), and a radial inflation (sunfish-like). Continuous diffeomorphisms relate the three forms.
Click Run to execute the Python code
Code will be executed with Python 3 on the server
6. Simulation: Physical Scaling Limits
Four panels: leg stress vs body mass with bone-strength cap; Reynolds number across size scales; Bond number showing surface-tension dominance; length-mass-shape scaling with isometric, elastic and allometric lines.
Click Run to execute the Python code
Code will be executed with Python 3 on the server
7. Simulation: Dimensionless Groups Across Taxa
Reynolds, Froude, Peclet numbers for 10 organisms spanning bacteria to blue whales - showing the enormous range of dimensionless physics that life must navigate.
Click Run to execute the Python code
Code will be executed with Python 3 on the server
7b. Case Study: The Blue Whale as Scaling Limit
The blue whale (Balaenoptera musculus, 150 tonnes, 30 m) is the largest animal to have ever existed. Its existence is enabled by water: buoyancy removes the gravitational scaling constraint, and viscous drag at Re ~ 10\(^8\) is minor compared to pressure drag.
Skeletal mass fraction in terrestrial mammals scales as \(M_{skel}/M_{body} \propto M^{0.09}\)(Schmidt-Nielsen 1984): elephants are 27% bone by mass while mice are only 5%. This drives why elephants cannot jump. Blue whales break this trend: they are only 12% bone because seawater does the structural work.
Heat transfer scaling
Metabolic heat generation \(\propto M^{3/4}\) (Kleiber, M4), while heat loss via the skin\(\propto M^{2/3}\). The difference drives thermal regulation:
\[ \frac{\dot{Q}_{gen}}{\dot{Q}_{loss}} \propto \frac{M^{3/4}}{M^{2/3}} = M^{1/12} \]
Large animals overheat easily. Desert elephants dissipate heat through large ears (Jumbo's ears add 20% to surface area); whales shunt blood to flukes and fins; small mammals face the opposite problem and cannot afford to be diurnal in cold climates.
8. Lessons for the Rest of the Course
- Biological form is constrained but not determined by physics
- Geometric relations among species reflect continuous deformations (M0)
- Packing problems favor spiral arrangements (M1 Phyllotaxis)
- Symmetry and chirality emerge from development (M2)
- Branching networks solve transport optimization (M3, M4)
- Surface tension minimizes area and sculpts soft tissues (M5)
- Reaction-diffusion mathematics generates patterns (M6)
- Cell packing shapes epithelia and compound eyes (M7)
- Deviations from expected geometry diagnose disease (M8)
Throughout the course, we will keep returning to the interplay of physics, geometry and biology that D'Arcy Thompson opened 100 years ago.
8b. The Buckingham Pi Theorem in Biology
Behind every scaling law is Buckingham's pi theorem: a physical relation among n variables with k independent dimensions reduces to a relation among n - k dimensionless groups. For swimming, the relevant variables are speed U, body length L, fluid density ρ, viscosity μ, and gravity g. Five variables, three dimensions (MLT): two dimensionless groups - Reynolds and Froude.
\[ \mathrm{Re} = \frac{\rho U L}{\mu}, \qquad \mathrm{Fr} = \frac{U}{\sqrt{g L}} \]
Any drag-coefficient function must be \(C_D = f(\mathrm{Re}, \mathrm{Fr})\) alone. This is why a fish 1000x smaller than a whale but at Re = 10\(^5\) can use exactly the same hydrodynamic principles.
Example: Alexander's dimensionless walking
Alexander (1989) showed that walking-to-running transition occurs at \(\mathrm{Fr} \approx 0.5\)across mammals - from mice to elephants. This is a direct consequence of pendulum dynamics in the limb (\(T \sim \sqrt{L/g}\)) combined with Froude scaling, and has been used by paleontologists to estimate dinosaur walking speeds from footprint geometry.
9. Interdisciplinary Bridges
Physics connections
Dimensional analysis, fluid mechanics, elasticity, statistical mechanics. Every organism lives at a particular intersection of these physical regimes.
Mathematics connections
Differential geometry, topology of surfaces, dynamical systems, bifurcation theory, reaction-diffusion PDEs, group theory (symmetry).
Biology connections
Developmental biology (morphogenesis), evolutionary developmental biology (evo-devo), paleontology (morphometric variation), ecology (species-area relationships).
Engineering applications
Computational anatomy, medical imaging, morphometric diagnosis, bioinspired design, computer animation of organisms.
9c. A Note on Units and Scaling Practice
Mature biomechanics practice uses dimensionless groups wherever possible: they are the actual controls that nature tunes. When you read a paper claiming a new bio-inspired design, ask: what dimensionless group is being exploited? Low Reynolds, high Peclet, small Bond? Each regime has a different physics.
9b. Morphospace
“Morphospace” is the multi-dimensional space of all possible forms. Real species occupy a tiny fraction. Raup (1966) studied shell morphospace with just three axes (W, D, T); only 0.1% of the space contains extant species. McGhee (2006) extended this analysis to Paleozoic corals, showing that macroevolution explores limited corridors of morphospace, constrained by developmental genetics and function.
This course will return repeatedly to the question: given all forms that physics allows, why does biology occupy only a narrow slice? The answer combines genetic programming, developmental inertia, and the fitness landscape.
References
- Thompson, D'A.W. (1917). On Growth and Form. Cambridge University Press.
- Purcell, E.M. (1977). Life at low Reynolds number. American Journal of Physics, 45, 3-11.
- McMahon, T.A. (1975). Using body size to understand the structural design of animals: quadrupedal locomotion. Journal of Applied Physiology, 39, 619-627.
- McMahon, T.A. & Bonner, J.T. (1983). On Size and Life. Scientific American Library.
- Grenander, U. & Miller, M.I. (1998). Computational anatomy: An emerging discipline. Quart. Appl. Math., 56, 617-694.
- Vogel, S. (2003). Comparative Biomechanics: Life's Physical World. Princeton University Press.
- Alexander, R.McN. (1996). Optima for Animals. Princeton University Press.
- Schmidt-Nielsen, K. (1984). Scaling: Why is Animal Size So Important?. Cambridge University Press.
- de Gennes, P.G., Brochard-Wyart, F., Quere, D. (2004). Capillarity and Wetting Phenomena. Springer.
- Ball, P. (2009). Nature's Patterns: Shapes. Oxford University Press.