Table 6.

Parameters used in the 3D analysis and their meaning

Parameter	Meaning	General	Sphere	Rhombic dodecahedron
E	Energy function or Hamiltonian	J s+λ s(s−S)2+λ v(v−V)2
J	Edhesion energy (per contact length)	J	J	J
s	Cell surface	k s l 2	4π r 2	$8\sqrt{2}{l}^2$
v	Cell volume	k v l 3	$\frac{4}{3}\pi r^3$	$\frac{16}{3\sqrt{3}}{l}^3$
S	Rest surface area	$k_s{L}_s^2$	$4\pi R_s^2$	$8\sqrt{2}{L}_s^2$
V	Ttarget cell volume	$k_v{L}_v^3$	$\frac{4}{3}\pi R_v^3$	$\frac{16}{3\sqrt{3}}{L}_v^3$
λ s	Surface constraint	λ s	λ s	λ s
λ v	Volume constraint	λ v	λ v	λ v
l	Basic length scale	l	r (radius)	l
k s	Surface scaling factor	$\frac{s}{l^2}$	4π	$8\sqrt{2}$
k v	Volume scaling factor	$\frac{v}{l^3}$	$\frac{4}{3}\pi$	$\frac{16}{3\sqrt{3}}$
L s	Rest surface area, using basic length scale	$\sqrt{\frac{S}{k_s}}$	$R_s=\frac{1}{2}\sqrt{\frac{S}{\pi }}$	$\sqrt{\frac{S}{8\sqrt{2}}}$
L v	Target cell volume,using basic length scale	$\sqrt[3]{\frac{V}{k_v}}$	$R_v=\sqrt[3]{\frac{3V}{4\pi }}$	$\sqrt[3]{\frac{3\sqrt{3}V}{16}}$
E	Energy function or Hamiltonian, using basic length scale	${\mathit{\text{Jk}}}_s{l}^2+{\lambda }_s{(k_s{l}^2-k_s{L}_s^2)}^2+{\lambda }_v{(k_v{l}^3-k_v{L}_v^3)}^2$	$4\mathrm{\pi J}{r}^2+{\lambda }_s{(4\pi r^2-4\pi R_s^2)}^2+{\lambda }_v{(\frac{4}{3}\pi r^3-\frac{4}{3}\pi R_v^3)}^2$	$8\sqrt{2}J{l}^2$ $+{\lambda }_s{(8\sqrt{2}{l}^2-8\sqrt{2}{L}_s^2)}^2+{\lambda }_v{(\frac{16}{3\sqrt{3}}{l}^3-\frac{16}{3\sqrt{3}}{L}_v^3)}^2$
$\frac{\mathrm{\partial E}}{\mathrm{\partial l}}$	Energy variation per length change	$2k_{s}l\left(\gamma -\frac{3k_v}{2k_s}\mathrm{l\Pi }\right)$	4π r(2γ−r Π)	$16\sqrt{2}l\left(\gamma -\frac{l}{\sqrt{6}}\Pi \right)$
γ	Interfacial tension	$J+2k_s{\lambda }_s(l^2-L_s^2)$	$J+8\pi {\lambda }_s(r^2-R_s^2)$	$J+16\sqrt{2}{\lambda }_s(l^2-L_s^2)$
Π	Pressure	$-2k_v{\lambda }_v(l^3-L_v^3)$	$-\frac{8}{3}\pi {\lambda }_v(r^3-R_v^3)$	$-\frac{32}{3\sqrt{3}}{\lambda }_v(l^3-L_v^3)$
$\frac{\mathrm{\partial E}}{\mathrm{\partial l}}$	Energy variation per length change, full expansion	2k s(a l 5+b l 3−c l 2+τ l)	8π(a r 5+b r 3−c r 2+τ r)	$16\sqrt{2}\left(a{l}^5+b{l}^3-c{l}^2+\mathrm{\tau l}\right)$
a	Aggregate parameterin $\frac{\mathrm{\partial E}}{\mathrm{\partial l}}$ equation	$\frac{3k_v^2{\lambda }_v}{k_s}$	$\frac{4}{3}\pi {\lambda }_v$	$\frac{16}{9}\sqrt{2}{\lambda }_v$
b	Aggregate parameterin $\frac{\mathrm{\partial E}}{\mathrm{\partial l}}$ equation	2k s λ s	8π λ s	$16\sqrt{2}{\lambda }_s$
c	Aggregate parameterin $\frac{\mathrm{\partial E}}{\mathrm{\partial l}}$ equation	$\frac{3k_v^2{\lambda }_v{L}_v^3}{k_s}$	$\frac{4}{3}\pi {\lambda }_v{R}_v^3$	$\frac{16}{9}\sqrt{2}{\lambda }_v{L}_v^3$
τ	Length-independent component of interfacial tension	$J-2k_s{\lambda }_s{L}_s^2$	$J-8\pi {\lambda }_s{R}_s^2$	$J-16\sqrt{2}{\lambda }_s{L}_s^2$
ϕ	$\frac{b}{6a}$	$\frac{k_s^2{\lambda }_s}{9k_v^2{\lambda }_v}$	$\frac{{\lambda }_s}{{\lambda }_v}$	$\frac{3{\lambda }_s}{2{\lambda }_v}$
ψ	$\frac{c}{8a}$	$\frac{L_v^3}{8}$	$\frac{R_v^3}{8}$	$\frac{L_v^3}{8}$
ν (when ϕ>0)	$\frac{\tau }{12a{\varphi }^2}$	$\frac{\left(J-2k_s{\lambda }_s{L}_s^2\right)9k_v^2{\lambda }_v}{4k_s^3{\lambda }_s^2}$	$\frac{\left(J-8\pi {\lambda }_s{R}_s^2\right){\lambda }_v}{16\pi {\lambda }_s^2}$	$\frac{\left(J-16\sqrt{2}{\lambda }_s{L}_s^2\right){\lambda }_v}{48\sqrt{2}{\lambda }_s^2}$
μ(when ϕ>0)	$\frac{\psi }{{\varphi }^{\frac{3}{2}}}$	$\frac{27k_v^3{\lambda }_v^{\frac{3}{2}}{L}_v^3}{8k_s^3{\lambda }_s^{\frac{3}{2}}}$	$\frac{R_v^3{\lambda }_v^{\frac{3}{2}}}{8{\lambda }_s^{\frac{3}{2}}}$	$\frac{L_v^3{\lambda }_v^{\frac{3}{2}}}{6\sqrt{6}{\lambda }_s^{\frac{3}{2}}}$
ν ′(when ϕ=0)	$\frac{\tau }{12a}$	$\frac{{\mathit{\text{Jk}}}_s}{36k_v^2{\lambda }_v}$	$\frac{J}{16\pi {\lambda }_v}$	$\frac{3J}{64\sqrt{2}{\lambda }_v}$
μ ′(when ϕ=0)	ψ	$\frac{L_v^3}{8}$	$\frac{R_v^3}{8}$	$\frac{L_v^3}{8}$
Bifurcation 1 (γ(l ∗)=0)	Transition from negative to positive interfacial tension at equilibrium	$\nu =-\frac{9k_v^2{\lambda }_v{L}_v^2}{2k_s^2{\lambda }_s}$	$\nu =-\frac{{\lambda }_v{R}_v^2}{2{\lambda }_s}$	$\nu =-\frac{{\lambda }_v{L}_v^2}{3{\lambda }_s}$
		ν ′=0	ν ′=0	ν ′=0
Bifurcation 2 (pseudo-transcritical)	Transition of l ∗=0 from unstable to stable	ν=0	ν=0	ν=0
		ν ′=0	ν ′=0	ν ′=0
Bifurcation 3 (fold)	Transition from 2 to 0 non-trivial equilibria	ν=f(μ)(μ−f(μ)), where $f\left(\mu \right)=sinh\left(\frac{1}{3}\text{arcsinh}\left(\mu \right)\right)$	ν=f(μ)(μ−f(μ))	ν=f(μ)(μ−f(μ))
		${\nu }^{\prime }=\frac{{\mu {}^{\prime }}^{\frac{4}{3}}}{2^{\frac{2}{3}}}$	${\nu }^{\prime }=\frac{{\mu {}^{\prime }}^{\frac{4}{3}}}{2^{\frac{2}{3}}}$	${\nu }^{\prime }=\frac{{\mu {}^{\prime }}^{\frac{4}{3}}}{2^{\frac{2}{3}}}$