  We have discovered the existence of extremely polydisperse High
Internal-Phase-Ratio Emulsions (HIPE) in which the internal-phase droplets,
present at 95% volume fraction, remain spherical and organize themselves in the
available space according to Apollonian packing rules. Such Apollonian
emulsions are obtained from dispersing oil dropwise in water in the presence of
very little surfactant, and allowing them to evolve at rest for a week. The
packing structure of the droplets was confirmed through size distribution
measurements that evolved spontaneously towards power laws with the known
Apollonian exponents, as well as comparison of the structure factors of aged
HIPEs measured by Small-Angle X-ray Scattering with that of a numerically
simulated Random Apollonian Packing. Thanks to the perfect sphericity of the
droplets, Apollonian emulsions were found to display Newtonian ow even at such
extremely high volume fraction. We argue that these fascinating space-filling
assemblies of spherical droplets are a result of coalescence and fragmentation
processes obeying simple geometrical rules of conserving total volume and
sphericity, minimizing the elastic energy associated with interactions of
neighbouring droplets.

Soft Condensed Matter