Graduate Research Course
BioGeometry: The Mathematical Language of Life
From the golden ratio in sunflowers to fractals in lungs, from Turing patterns in zebrafish to minimal surfaces in radiolaria — the geometric principles that shape all life.
Key Equations of BioGeometry
Golden ratio
\( \varphi = \frac{1+\sqrt{5}}{2} \approx 1.6180 \)
Vogel sunflower spiral
\( r_n = c\sqrt{n}, \; \theta_n = n \cdot 137.508^\circ \)
Turing reaction-diffusion
\( \partial_t u = D_u \nabla^2 u + f(u, v) \)
D’Arcy Thompson transform
\( (x, y) \to (f(x, y), g(x, y)) \)
Mandelbrot fractal dimension
\( D = \frac{\log N}{\log (1/\varepsilon)} \)
Kleiber’s law
\( B = B_0 M^{3/4} \)
About This Course
Life is geometric. Phyllotaxis produces sunflowers at precisely 137.5°; Kleiber's law predicts metabolism across 27 orders of magnitude; Turing's reaction-diffusion equations generate zebrafish stripes, angelfish patterns and even digit spacing in mouse paws. These geometric regularities are not ornament — they are the physical solutions to the optimization problems that life faces: pack leaves for sunlight, distribute blood through branching networks, minimize skin area for a given volume, break symmetry for development.
This graduate-level course blends differential geometry, dynamical systems and developmental biology. Each module combines rigorous derivations, SVG diagrams and Python simulations so you can feel the mathematics come alive in living forms. We draw heavily on the canonical texts: D'Arcy Thompson's On Growth and Form (1917), Mandelbrot's Fractal Geometry of Nature, Philip Ball's Self-Made Tapestry, and primary research from Turing, Kleiber, West, Brown, Enquist and Mahadevan.
Cross-linked with our Plant Biochemistry, Biophysics, and Ecological Biochemistry courses.
Nine Modules
M0
Geometry of Life
D’Arcy Thompson’s coordinate transformations, physical constraints on form, gravity and surface tension limits, length-mass-shape scaling fundamentals.
M1
Golden Ratio & Phyllotaxis
The golden ratio phi from continued fractions, Fibonacci convergence, Vogel’s sunflower model, 137.5 deg divergence angle, auxin-driven phyllotaxis.
M2
Symmetry & Chirality
Bilateral, radial and spiral symmetries, amino-acid homochirality, snail-shell chirality and nodal gene, left-right asymmetry in vertebrates.
M3
Fractals & Branching
Mandelbrot dimension, Horton-Strahler branching, West-Brown-Enquist model for lungs and trees, fractal dimensions of coastlines, neurons and vasculature.
M4
Allometric Scaling
Kleiber’s M^(3/4) law, heart-rate scaling, lifespan-mass relationships, metabolic theory of ecology, surface-area vs volume and its biological consequences.
M5
Minimal Surfaces & Soap Films
Plateau problem, mean curvature H = 0, radiolarian skeletons, cell packing in epithelia, triply periodic minimal surfaces in butterfly scales.
M6
Morphogenesis & Turing Patterns
Reaction-diffusion systems, Turing bifurcation, zebrafish stripes, digit patterning, Wolpert positional information, wavelength selection.
M7
Packing & Tessellation
Hexagonal honeycombs, Voronoi diagrams in epithelia, Kelvin and Weaire-Phelan foams, sphere packing in compound eyes.
M8
Geometry, Disease & Health
Fractal analysis of tumors, arrhythmia on geometric substrates, retinal vascular tortuosity, geometric biomarkers in radiology.
Recommended Textbooks
- [1] Thompson, D'A.W. (1917). On Growth and Form. Cambridge University Press.
- [2] Mandelbrot, B.B. (1983). The Fractal Geometry of Nature. W.H. Freeman.
- [3] Ball, P. (1999). The Self-Made Tapestry: Pattern Formation in Nature. Oxford University Press.
- [4] Murray, J.D. (2003). Mathematical Biology I & II, 3rd ed. Springer.
- [5] Meinhardt, H. (2009). The Algorithmic Beauty of Sea Shells, 4th ed. Springer.
- [6] Wolpert, L. (2015). Principles of Development, 5th ed. Oxford.
- [7] Ball, P. (2009). Nature's Patterns: Shapes, Flow, Branches (3 vols). Oxford University Press.