Module 5 · Biomimetics

Materials & Self-Assembly

How life organizes atoms into hierarchical materials - from DNA origami and nacre to self-healing polymers and spider silk expressed in microbial factories.

1. Self-Assembly: Principles & Nelson's Octet

Self-assembly is the autonomous organization of components into patterns or structures without external direction. Life exploits it at every scale: lipid bilayers, protein quaternary structure, viral capsids, phyllotaxis, and even entire organs arise from local interactions minimizing free energy. Nelson's octet of self-assembly lists eight physical forces that organize soft matter: van der Waals, hydrogen bonding, electrostatic, hydrophobic, steric, entropic (depletion), magnetic, and capillary.

Free energy of assembly

A system assembles when the free-energy change is negative, \(\Delta G = \Delta H - T\Delta S < 0\). Assembly is generally enthalpy-driven (binding) but pays an entropy cost per ordered component (\(\Delta S < 0\)). Equilibrium self-assembly minimizes:

\[ G(\{r_i\}) = \sum_{i<j} U(r_i - r_j) - k_B T \ln \Omega(\{r_i\}) \]

Summed pair potential \(U\) and configurational entropy \(\Omega\).

Derivation: minimum-energy folding of a polymer

Consider a linear chain with \(N\) segments, each with folding probability \(p_i\). The free energy is \(F = \sum_i \left[ \varepsilon_i p_i + k_B T (p_i \ln p_i + (1-p_i)\ln(1-p_i)) \right]\). Setting \(\partial F / \partial p_i = 0\) yields the Boltzmann distribution for each contact:

\[ p_i^\star = \frac{1}{1 + e^{\varepsilon_i / k_B T}} \]

For DNA origami, \(\varepsilon_i\) is the hybridization energy of each staple (typically \(-30\) to \(-50\) kcal/mol). Cooperative folding emerges from the seam nucleation barrier.

2. DNA Origami

Introduced by Rothemund (Nature 2006), DNA origami folds a long single-stranded scaffold (commonly M13mp18 phage, ~7249 nt) into arbitrary 2D or 3D shapes via hundreds of short “staple” strands. The folding free energy is dominated by Watson-Crick base-pairing:

\[ \Delta G_{\text{hyb}} = \Delta H - T \Delta S \approx -(0.2 - 0.5) \, n_{bp} \; \text{kcal/mol} \]

A 32-nt staple binding the scaffold contributes \(\sim -60\) kcal/mol of binding enthalpy. With 200 staples, the total folding enthalpy is \(\sim -12{,}000\) kcal/mol - overwhelming entropic penalties once nucleation is complete.

Folding pathway: Douglas et al. (2009)

Heating to 95 °C denatures scaffold and staples
Slow cooling (1 °C/min) through 65-25 °C allows thermodynamic folding
Seam formation at \(T \approx 55\) °C nucleates the structure
Remaining staples bind cooperatively; yields typically 70-95%

SVG: Folding pathway

3. Biomineralization

Biomineralization is the templated precipitation of inorganic crystals guided by organic matrices. The hierarchy spans nanometers (mineral platelets) to millimeters (tooth, shell, spine) and yields composite materials orders of magnitude tougher than their geologic counterparts.

Sea urchin spines: single crystal with a twist

A sea urchin spine (Heterocentrotus mammillatus) diffracts X-rays as a single calcite crystal yet fractures conchoidally like glass because 0.1 wt% incorporated proteins deflect cracks. Fracture toughness: \(K_{Ic} \approx 1.5\) MPa m\(^{1/2}\) vs 0.2 for geologic calcite.

Tooth enamel: 95% hydroxyapatite rods

Human enamel is the hardest biological material: 95 vol% hydroxyapatite Ca\(_{10}\)(PO\(_4\))\(_6\)(OH)\(_2\), arranged in 5-µm rods separated by <5 nm of protein. Hardness ~5 GPa;\(K_{Ic} \sim 0.9\) MPa m\(^{1/2}\).

Diatom silica shells (frustules)

Diatoms precipitate amorphous SiO\(_2\)·nH\(_2\)O in silica deposition vesicles (SDVs). Silaffin peptides (H1: polylysine bearing polyamines) catalyze silica polymerization at pH 5.5 - the same chemistry exploits in room-temperature synthesis of nanostructured ceramics.

Derivation: classical nucleation rate

For a spherical nucleus of radius \(r\), the free energy combines volumetric driving force and surface penalty:

\[ \Delta G(r) = -\frac{4}{3}\pi r^3 \, \rho k_B T \, \sigma + 4\pi r^2 \gamma \]

where \(\sigma = \ln(\text{IAP}/K_{sp})\) is supersaturation and \(\gamma\) the interfacial tension. Setting \(d\Delta G/dr = 0\) gives the critical radius\(r^\star = 2\gamma/(\rho k_B T \sigma)\) and barrier height:

\[ \Delta G^\star = \frac{16 \pi \gamma^3}{3 (\rho k_B T \sigma)^2} \]

Nucleation rate: \(J = A \exp(-\Delta G^\star / k_B T)\). Organic matrices lower\(\gamma\), dramatically enhancing nucleation at low supersaturation - this is the thermodynamic trick of biomineralization.

4. Nacre Hierarchy (SVG)

Red abalone nacre exhibits seven hierarchical levels spanning six orders of magnitude: from nanoscale aragonite platelets to macroscale shell geometry.

5. Self-Healing Polymers

Biological tissues repair themselves through inflammation, proliferation and remodeling. Synthetic self-healing materials borrow this playbook with two broad strategies: extrinsic (encapsulated healing agents) and intrinsic(reversible bonds in the polymer network).

Extrinsic: White et al. (2001)

White's landmark system embedded microcapsules of dicyclopentadiene (DCPD) and Grubbs' ruthenium catalyst in an epoxy matrix. On cracking, capsules rupture, DCPD flows into the crack, contacts the catalyst, and undergoes ring-opening metathesis polymerization (ROMP) - recovering 75% of virgin fracture toughness.

Intrinsic: vitrimers

Leibler's vitrimers (Science 2011) contain dynamic covalent bonds (transesterification) whose exchange rate follows Arrhenius: \(k(T) = A \exp(-E_a/RT)\). Above the topology-freezing temperature \(T_v\), the network flows like glass yet retains elasticity at room temperature.

Healing kinetics

The empirical healing law:

\[ \eta(t) = \eta_\infty \left[ 1 - \exp(-k_h t) \right] \]

where \(\eta_\infty\) is the maximum attainable efficiency and \(k_h\) depends on chemistry, temperature, and diffusion across the crack interface.

6. Spider Silk: Recombinant Production

Nephila clavipes dragline silk has a tensile strength of 1.3 GPa and toughness of 350 MJ/m\(^3\) - combining strength rivaling steel with elasticity of rubber. The dominant protein, Major ampullate Spidroin 1 (MaSp1), contains repeating (GGAGQQ)/(poly-Ala) blocks that form beta-sheet nanocrystals dispersed in an amorphous matrix.

Recombinant expression systems

E. coli: heat-shock protein fusions push yields to 2 g/L (Xia et al., PNAS 2010)
Pichia pastoris: glycosylation pathways tolerate large repeats
Transgenic goats (BioSteel): silk proteins secreted in milk
Transgenic silkworms (Kraig Biocraft): heritable MaSp integration

Fiber spinning: biomimetic duct

In vivo, silk dope is 30-50 wt% protein in a disordered micellar state. The spider's spinning duct imposes: (1) pH drop from 7.6 to 5.7, (2) K\(^+\) removal, Na\(^+\) addition, (3) water extraction, (4) extensional flow. Together these trigger beta-sheet assembly. Artificial microfluidic spinnerets now replicate these conditions to produce recombinant fibers with 80% of natural strength.

\[ \sigma(\varepsilon) = E_{\text{amorph}} \varepsilon + E_{\text{cryst}} \phi_{\beta}(\varepsilon) \varepsilon \]

Composite stress-strain: amorphous matrix plus strain-aligned beta-sheet crystallites with volume fraction \(\phi_\beta(\varepsilon)\).

7. Simulation: DNA Origami Landscape

The simulation below plots the free-energy landscape of DNA origami folding during thermal annealing and the dependence of folding yield on cooling rate.

Python

script.py73 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ======================================================================
# DNA origami folding energy landscape
# A scaffold (M13 phage ~7249 nt) with staple strands. Folding is a
# nucleation-propagation process: seam formation -> staple binding.
# Model free energy along folding coordinate q (0 = random coil, 1 = folded)
# G(q) = nucleation barrier - DeltaH*q + kT*S(q)
# ======================================================================

q = np.linspace(0, 1, 500)

def folding_energy(q, DeltaH=-45.0, barrier=12.0, q_n=0.18, sigma=0.07):
    # Gaussian nucleation barrier at q_n
    nucleation = barrier * np.exp(-((q - q_n)**2)/(2*sigma**2))
    # Enthalpic drive from staple binding (linear in q)
    enthalpy = DeltaH * q
    # Configurational entropy cost for ordered intermediates
    S = -0.5 * (q*np.log(q + 1e-12) + (1-q)*np.log(1-q + 1e-12))
    return nucleation + enthalpy + 2.5 * S * (1 - q)

# Temperature series (annealing)
Temps = [65, 55, 45, 35]  # °C
colors = ['#d946ef', '#e879f9', '#f0abfc', '#fdf4ff']

fig, axes = plt.subplots(1, 2, figsize=(14, 5))
fig.patch.set_facecolor('#0a0014')

ax1 = axes[0]
for T, c in zip(Temps, colors):
    G = folding_energy(q, DeltaH=-45.0 * (1 - 0.005*(T-25)), barrier=12.0)
    ax1.plot(q, G, color=c, linewidth=2.3, label=f'T = {T} °C')

ax1.axhline(0, color='#666', linewidth=0.7, linestyle=':')
ax1.set_xlabel('Folding coordinate q', color='white', fontsize=11)
ax1.set_ylabel('Free energy G(q) [kcal/mol]', color='white', fontsize=11)
ax1.set_title('DNA origami folding landscape\nAnnealing from 65 to 35 °C lowers barrier', color='white', fontweight='bold')
ax1.legend(facecolor='#1a0a2e', edgecolor='#d946ef', labelcolor='white')
ax1.set_facecolor('#0a0014')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.15, color='#d946ef')
for sp in ax1.spines.values(): sp.set_color('#d946ef')

# Panel 2: Yield vs annealing rate
rates = np.logspace(-2, 2, 100)  # °C / min
# Yield peaks at moderate cooling; too fast -> kinetic trap; too slow -> thermal damage
yield_f = np.exp(-0.03 * rates) * (1 - np.exp(-8 / (rates + 0.1)))
yield_f = yield_f / yield_f.max() * 0.92

ax2 = axes[1]
ax2.semilogx(rates, 100*yield_f, color='#d946ef', linewidth=2.5)
ax2.fill_between(rates, 0, 100*yield_f, color='#d946ef', alpha=0.18)
ax2.axvline(1.0, color='#fbbf24', linestyle='--', linewidth=1.2)
ax2.text(1.2, 20, 'Optimal ~1 °C/min', color='#fbbf24', fontsize=10)
ax2.set_xlabel('Cooling rate [°C / min]', color='white', fontsize=11)
ax2.set_ylabel('Folding yield (%)', color='white', fontsize=11)
ax2.set_title('DNA origami yield vs cooling rate', color='white', fontweight='bold')
ax2.set_facecolor('#0a0014')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.15, color='#d946ef')
for sp in ax2.spines.values(): sp.set_color('#d946ef')

plt.tight_layout()
plt.savefig('output.png', facecolor='#0a0014', dpi=110, bbox_inches='tight')
plt.close()
print('DNA origami analysis complete.')
print(f'Scaffold length: 7249 nt (M13mp18)')
print(f'Typical staples: 200-250, each 32 nt')
print(f'Typical folding yield at 1 C/min: ~80-90%')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Simulation: Nacre Crystal Growth

Four panels: classical nucleation vs growth regime, tablet-thickness toughness optimum (Gao 2003), tablet growth kinetics and a cross-material toughness bar chart.

Python

script.py100 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ======================================================================
# Nacre (mother-of-pearl) crystal growth: aragonite tablets (CaCO3)
# inside a chitin + silk-like protein matrix.
# Classical nucleation theory: J = A exp(-DeltaG*/kT)
# Crystal growth rate depends on supersaturation sigma = ln(IAP/Ksp)
# ======================================================================

sigma = np.linspace(0.05, 3.0, 400)  # supersaturation

def nucleation_rate(sigma, A=1e30, B=18.0):
    # J = A exp(-B / sigma^2); B ~ 16*pi*gamma^3*v^2 / (3 (kT)^3)
    return A * np.exp(-B / (sigma**2 + 1e-8))

def growth_rate(sigma, k=1.2e-6, n=1.7):
    return k * sigma**n  # m/s (screw dislocation + 2D nucleation)

J = nucleation_rate(sigma)
R = growth_rate(sigma)

# Sea urchin spine: single calcite crystal with 0.1 wt% proteins,
# fracture toughness 10x synthetic calcite (~1.5 vs 0.15 MPa m^0.5)
# Nacre: 95% aragonite, 5% organic; Kic ~ 5-8 MPa m^0.5

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0a0014')

# Panel 1: Nucleation vs growth regime
ax1 = axes[0, 0]
ax1.semilogy(sigma, J, color='#d946ef', linewidth=2.5, label='Nucleation rate J')
ax2b = ax1.twinx()
ax2b.plot(sigma, R*1e9, color='#2dd4bf', linewidth=2.5, label='Growth rate R [nm/s]')
ax1.axvspan(0.1, 0.6, color='#2dd4bf', alpha=0.15, label='Biomineralization window')
ax1.set_xlabel('Supersaturation sigma', color='white')
ax1.set_ylabel('Nucleation rate J [#/m^3 s]', color='#d946ef')
ax2b.set_ylabel('Growth rate R [nm/s]', color='#2dd4bf')
ax1.set_title('Classical nucleation theory for CaCO3\nNacre operates at low sigma (slow, controlled growth)', color='white', fontweight='bold')
ax1.set_facecolor('#0a0014')
ax1.tick_params(colors='white')
ax2b.tick_params(colors='white')
ax1.grid(True, alpha=0.15)
for sp in ax1.spines.values(): sp.set_color('#d946ef')

# Panel 2: Hierarchy (tablets and interface stress)
ax2 = axes[0, 1]
tablet_th = np.linspace(100, 1500, 300)  # nm
# Brick-and-mortar toughness model (Gao 2003):
# critical tablet length L_c ~ rho * t * Sigma_M / tau_i
# For aragonite: Sigma_M ~ 0.4 GPa, tau_interface ~ 50 MPa, aspect ratio ~ 10
toughness = 2.5 + 4.5 * np.tanh((500 - tablet_th)/200) + 0.3*np.random.RandomState(2).randn(len(tablet_th))*0.0
ax2.plot(tablet_th, toughness, color='#f0abfc', linewidth=2.5)
ax2.axvspan(200, 500, color='#fbbf24', alpha=0.2, label='Biological optimum\n(200-500 nm)')
ax2.set_xlabel('Aragonite tablet thickness [nm]', color='white')
ax2.set_ylabel('Fracture toughness K_Ic [MPa m^0.5]', color='white')
ax2.set_title('Nacre toughness vs tablet thickness\nGao (2003) flaw-tolerant size', color='white', fontweight='bold')
ax2.legend(facecolor='#1a0a2e', edgecolor='#d946ef', labelcolor='white')
ax2.set_facecolor('#0a0014')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.15)
for sp in ax2.spines.values(): sp.set_color('#d946ef')

# Panel 3: Time evolution of crystal volume
t = np.linspace(0, 72, 500)  # hours
V = np.zeros_like(t)
V[0] = 1e-21  # m^3 seed
for i in range(1, len(t)):
    V[i] = V[i-1] + growth_rate(0.35) * V[i-1]**(2/3) * (t[i]-t[i-1])*3600 * 1e-6
ax3 = axes[1, 0]
ax3.plot(t, V*1e15, color='#d946ef', linewidth=2.5)
ax3.set_xlabel('Time [h]', color='white')
ax3.set_ylabel('Tablet volume [fL]', color='white')
ax3.set_title('Aragonite tablet growth kinetics\n(sigma = 0.35, controlled by organic matrix)', color='white', fontweight='bold')
ax3.set_facecolor('#0a0014')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.15)
for sp in ax3.spines.values(): sp.set_color('#d946ef')

# Panel 4: Comparison chart
ax4 = axes[1, 1]
materials = ['Calcite\n(geologic)', 'Aragonite\n(geologic)', 'Sea urchin\nspine', 'Nacre\n(abalone)', 'Tooth enamel\n(hydroxyapatite)']
kic = [0.2, 0.3, 1.5, 6.5, 0.9]
colors_bar = ['#6b7280', '#6b7280', '#2dd4bf', '#d946ef', '#fbbf24']
ax4.bar(materials, kic, color=colors_bar, edgecolor='white', linewidth=0.8)
ax4.set_ylabel('Fracture toughness K_Ic [MPa m^0.5]', color='white')
ax4.set_title('Biomineral vs geological carbonates\n(log-scale toughness amplification)', color='white', fontweight='bold')
ax4.set_facecolor('#0a0014')
ax4.tick_params(colors='white')
ax4.grid(True, alpha=0.15, axis='y')
for sp in ax4.spines.values(): sp.set_color('#d946ef')

plt.tight_layout()
plt.savefig('output.png', facecolor='#0a0014', dpi=110, bbox_inches='tight')
plt.close()
print('Nacre / biomineralization simulation complete.')
print('Key finding: organic matrix + templating + low supersaturation -> 20x tougher than geologic aragonite.')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

9. Simulation: Self-Healing Kinetics

Healing efficiency vs time for four chemistries, multi-cycle durability, and the animal wound-healing phase template that inspires the field.

Python

script.py72 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ======================================================================
# Self-healing polymer kinetics
# White et al. (2001, Nature): microcapsule DCPD + Grubbs catalyst
# Healing efficiency eta(t) = 1 - exp(-k*t)
# ======================================================================

t = np.linspace(0, 48, 500)  # hours

# Different chemistries
systems = {
    'DCPD / Grubbs (microcapsule)': {'k': 0.22, 'eta_max': 0.75, 'color': '#d946ef'},
    'Vitrimer (exchange bonds)':    {'k': 0.09, 'eta_max': 0.95, 'color': '#2dd4bf'},
    'Diels-Alder thermoreversible': {'k': 0.15, 'eta_max': 0.88, 'color': '#fbbf24'},
    'Hydrogen-bond supramolecular':  {'k': 0.7,  'eta_max': 0.55, 'color': '#f472b6'},
}

fig, axes = plt.subplots(1, 3, figsize=(18, 5))
fig.patch.set_facecolor('#0a0014')

ax1 = axes[0]
for name, p in systems.items():
    eta = p['eta_max'] * (1 - np.exp(-p['k'] * t))
    ax1.plot(t, 100*eta, color=p['color'], linewidth=2.3, label=name)
ax1.set_xlabel('Healing time [h]', color='white')
ax1.set_ylabel('Healing efficiency eta (%)', color='white')
ax1.set_title('Self-healing kinetics across chemistries', color='white', fontweight='bold')
ax1.legend(facecolor='#1a0a2e', edgecolor='#d946ef', labelcolor='white', fontsize=9)
ax1.set_facecolor('#0a0014')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.15)
for sp in ax1.spines.values(): sp.set_color('#d946ef')

# Panel 2: Crack closure over cycles
cycles = np.arange(1, 11)
dcpd = 0.75 * np.exp(-0.25*(cycles-1))  # capsules deplete
vitrimer = np.ones_like(cycles, dtype=float) * 0.92
ax2 = axes[1]
ax2.plot(cycles, 100*dcpd, 'o-', color='#d946ef', linewidth=2.3, label='Microcapsule DCPD')
ax2.plot(cycles, 100*vitrimer, 's-', color='#2dd4bf', linewidth=2.3, label='Vitrimer (extrinsic)')
ax2.set_xlabel('Damage cycle #', color='white')
ax2.set_ylabel('Healing efficiency (%)', color='white')
ax2.set_title('Multi-cycle healing\nCapsule systems deplete; vitrimers heal indefinitely', color='white', fontweight='bold')
ax2.legend(facecolor='#1a0a2e', edgecolor='#d946ef', labelcolor='white')
ax2.set_facecolor('#0a0014')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.15)
for sp in ax2.spines.values(): sp.set_color('#d946ef')

# Panel 3: Analogy to animal wound healing (4 phases)
phases = ['Haemostasis\n(min)', 'Inflammation\n(hours)', 'Proliferation\n(days)', 'Remodeling\n(weeks)']
x = np.array([0.05, 1, 5, 20])
intensity = np.array([1.0, 0.85, 0.6, 0.3])
ax3 = axes[2]
ax3.bar(phases, intensity, color=['#f472b6', '#fbbf24', '#2dd4bf', '#d946ef'], edgecolor='white')
ax3.set_ylabel('Relative bioactivity', color='white')
ax3.set_title('Animal wound-healing phases\n(template for bio-inspired polymers)', color='white', fontweight='bold')
ax3.set_facecolor('#0a0014')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.15, axis='y')
for sp in ax3.spines.values(): sp.set_color('#d946ef')

plt.tight_layout()
plt.savefig('output.png', facecolor='#0a0014', dpi=110, bbox_inches='tight')
plt.close()
print('Self-healing polymer kinetics complete.')
print('Key scaling: eta(t) = eta_max * (1 - exp(-k t))')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

10. Engineering Applications

DNA nanomachines

DNA origami used as drug-delivery nanocages that open in response to aptamer-recognized cancer markers (Douglas et al., Science 2012).

Nacre-mimetic ceramics

Freeze-casting Al\(_2\)O\(_3\)/PMMA composites achieves\(K_{Ic}\) of 30 MPa m\(^{1/2}\) - 300x nacre itself (Munch et al., Science 2008).

Vitrimer coatings

Automotive clear-coats that re-flow at 130 °C to erase scratches. Commercial since 2019 in select Japanese OEMs.

Microlure recombinant silk

AMSilk GmbH Biosteel fiber used in Adidas Futurecraft BioFabric shoes and biomedical sutures - biocompatible, 100% degradable.

Key design principles from biology

Hierarchical structure across 6+ length scales
High organic/inorganic interface area for crack deflection
Templated nucleation under low supersaturation
Reversible bonds enabling damage tolerance
Ambient processing (aqueous, neutral pH, 20-37 °C)

References

Rothemund, P.W.K. (2006). Folding DNA to create nanoscale shapes and patterns. Nature, 440, 297-302.
Douglas, S.M. et al. (2009). Self-assembly of DNA into nanoscale three-dimensional shapes. Nature, 459, 414-418.
White, S.R. et al. (2001). Autonomic healing of polymer composites. Nature, 409, 794-797.
Montarnal, D., Capelot, M., Tournilhac, F., Leibler, L. (2011). Silica-like malleable materials from permanent organic networks. Science, 334, 965-968.
Gao, H., Ji, B., Jager, I.L., Arzt, E., Fratzl, P. (2003). Materials become insensitive to flaws at nanoscale. PNAS, 100, 5597-5600.
Meyers, M.A. et al. (2008). Biological materials: structure and mechanical properties. Progress in Materials Science, 53, 1-206.
Xia, X.X. et al. (2010). Native-sized recombinant spider silk protein produced in metabolically engineered E. coli. PNAS, 107, 14059-14063.
Kroger, N. & Poulsen, N. (2008). Diatoms - from cell wall biogenesis to nanotechnology. Annu. Rev. Genet., 42, 83-107.
Aizenberg, J. et al. (2005). Skeleton of Euplectella: structural hierarchy from the nanoscale to the macroscale. Science, 309, 275-278.
Nelson, D.R. (2002). Defects and Geometry in Condensed Matter Physics. Cambridge University Press.

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