Phi Packings: Density and Order in Particle Arrangements

• Phi Packings are configurations of particles or geometric motifs defined by the packing fraction φ, which measures the ratio of occupied to total space.
• They encompass methods ranging from exact dense periodic tetrahedron packings with analytical optimization to protocol-dependent, disordered jammed states.
• Research in φ-packings links density optimization and local structure metrics with novel approaches such as density–order maps and connections to Apollonian packings.

“Phi packings” (Editor’s term) are packings of particles, cells, or geometric motifs organized around the packing fraction $\phi$, i.e. the occupied volume or area divided by the total volume or area. In the literature represented here, $\phi$ serves several distinct roles: an exact objective in analytic constructions of dense periodic tetrahedron packings; a protocol-dependent observable in random, jammed, and frictional packings; a coordinate in density–order maps; and, in some two-dimensional theories, a quantity derivable directly from local cell statistics. A separate but related notation issue arises in the Apollonian literature, where uppercase $\Phi$ denotes a rank-4 indefinite Kac–Moody root system rather than packing fraction [1005.0011] [1101.1327] [2106.11774] [2102.02172].

1. Definitions and notational scope

The basic definition of packing fraction is consistent across the cited works, though adapted to particle geometry and dimension. For Platonic solids in experiment, $\phi$ is written as $\phi = N\,V_{\rm solid}/V_{\rm container}$ and reported as a mean over repeated trials together with its standard deviation [1008.1728]. For monodisperse hard spheres in a periodic box, $\phi = (N\,V_{\rm sphere})/V_{\rm box}$ with $V_{\rm sphere}=\pi D^3/6$ [1101.1327]. For regular tetrahedra in thermodynamic Monte Carlo studies, $\phi = N\,V_{\rm tetra}/V_{\rm cell}$ with $V_{\rm tetra}=a^3/(6\sqrt{2})$ [1012.5138].

In exact tetrahedron constructions, the definition is specialized to a four-particle periodic cell. For a regular tetrahedron of edge length $1$, $V_T=\sqrt{2}/12$, there are four tetrahedra per fundamental cell, the cell volume is $V_F=|\lambda_1\times\lambda_2\cdot\lambda_3|$, and hence
$$
\phi=\frac{4V_T}{V_F}=\frac{\sqrt{2}/3}{V_F}.
$$
Within the six-parameter family of Torquato and Jiao, $\lambda_1$, $\lambda_2$, $\lambda_3$, and the dimer offset are parameterized so that six face-to-face contacts are enforced analytically [1005.0011].

In two-dimensional circle packings, the same quantity is an area fraction. One formulation uses
$$
\phi=\frac{\sum_i \pi r_i^2}{L^2}
$$
for $N$ circles in a square box of side $L$ with periodic boundaries [2404.02316]. Another, developed for a protocol-independent theory of random disc packings, expresses $\phi$ through the cell-order distribution ${Q_k}$ as
$$
\phi=\frac{\pi(\bar k-2)}{8\sum_{k=3}^C Q_k\,\bar S_k},
$$
where $\bar k=\sum_k kQ_k$ and $\bar S_k$ is the mean area of a $k$-cell [2106.11774].

A potential misconception is purely notational. In “Apollonian Packings and Kac–Moody Root Systems,” the symbol $\Phi$ denotes a root system with Cartan matrix $a_{ii}=2$, $a_{ij}=-2$ for $i\neq j$; it is not a packing density. That paper instead studies an exponential generating function $Z(\mathbf s)$ attached to an Apollonian orbit [2102.02172].

2. Exact dense tetrahedron packings

The most developed exact $\phi$-theory in the present corpus concerns regular tetrahedra. Torquato and Jiao gave a six-parameter family of dense periodic packings with four particles per fundamental cell. The building block is a “dimer,” defined as two regular tetrahedra sharing a common face. The cell contains two such dimers, one at the origin and one at $c_1=(\eta_1,\eta_2,\eta_3)$, while the lattice is generated by $\lambda_1=(-\alpha_1,-\alpha_2,-\alpha_3)$, $\lambda_2=(\beta_1,-\beta_2,-\beta_3)$, and $\lambda_3=(\gamma_1,\gamma_2,\gamma_3)$. Solving the six face contacts yields $\gamma_i(\alpha,\beta)$ and $\eta_j(\alpha,\beta)$, and therefore an explicit $V_F(\alpha,\beta)$ and $\phi(\alpha,\beta)$ [1005.0011].

This six-parameter family contains several previously discovered dimer constructions as special cases. Imposing three additional contacts produces a three-parameter subfamily, and a further optimization on the residual nonoverlap boundary yields
$$
\phi_{\max}=\frac{4000}{4671}\approx 0.856347.
$$
The same analysis recovers the Kallus–Elser–Gravel family with $\phi(a)=100/(117-240a^2)$ and maximal value $12250/14319\approx 0.855506$. Torquato and Jiao stated that the value $4000/4671$ strongly suggests optimality among all packings with a four-particle basis, while explicitly leaving open whether it is maximal among all tetrahedron packings [1005.0011].

Chen, Engel, and Glotzer independently described the same density as the densest known packing within a three-parameter family of crystalline dimer packings. At the optimum, the unit cell is generated by
$$
a=\frac{3}{320}\langle 290,107,-7\rangle,\quad
b=\frac{3}{320}\langle -34,277,135\rangle,\quad
c=\frac{3}{320}\langle 94,-83,247\rangle,
$$
with offset
$$
d=\frac{1}{320}\langle 38,5,-25\rangle.
$$
They obtained $V_{\rm cell}=42039/1000$, tetrahedron volume $V_{\rm tetra}=9$ in their normalization, occupied volume $36$, and hence $\phi=4000/4671$. The resulting crystal has space group $P\bar 1$, contains two oppositely oriented dimers per cell, is analytically optimal within the three-parameter ansatz, and emerged overwhelmingly in isobaric Monte Carlo compressions for $N=4,8,12,16$ with periodic boundaries [1001.0586].

These exact dimer crystals were preceded and contrasted by other dense tetrahedron phases. Haji-Akbari and coauthors reported a dodecagonal quasicrystal compressible to $\phi=0.8324$, a crystalline approximant reaching $\phi=0.8503$, and a disordered jammed state at $\phi=0.7858$ when quasicrystal formation is suppressed. Their results showed that high $\phi$ for tetrahedra is not confined to simple crystals and can arise through entropy-driven assembly of complex motifs [1012.5138].

3. Random, jammed, and protocol-dependent packings

A central theme of $\phi$-packings is that high density is often protocol dependent rather than uniquely prescribed by geometry alone. In experiments on Platonic solids, fluidization and mechanical vibration were used as complementary preparation extremes for random loose packing and random close packing. For plastic tetrahedra the reported values were $\phi_{rlp}=0.51\pm0.01$ and $\phi_{rcp}=0.64\pm0.01$; for cubes, $0.54\pm0.01$ and $0.67\pm0.02$; for octahedra, $0.52\pm0.01$ and $0.64\pm0.01$; for dodecahedra, $0.51\pm0.01$ and $0.63\pm0.01$; and for icosahedra, $0.50\pm0.01$ and $0.59\pm0.01$. Both $\phi_{rlp}$ and $\phi_{rcp}$ peak at the cube and then decrease with number of faces. Higher interparticle friction in ceramic tetrahedra reduced $\phi$ relative to plastic tetrahedra under similar agitation protocols [1008.1728].

For frictionless monodisperse spheres, the notion of a unique random close packing density was directly challenged by Torquato, Jiao, and coauthors. Using the Lubachevsky–Stillinger algorithm for $\phi\gtrsim 0.64$ and a sequential-linear-programming algorithm for $\phi<0.64$, they generated collectively jammed disordered packings over the entire interval $\phi\in[0.6,\,0.74048\ldots]$, with narrow distributions centered at protocol-controlled values. Their conclusion was that there is no universal jamming point distinguishable solely by density or frequency of occurrence, and that packings should instead be characterized on a density–order-metric plane [1101.1327].

Protocol dependence is even sharper in frictional tetrahedra. Experimental packings prepared under isobaric conditions covered $\phi$ from approximately $0.469$ to $0.622$ using TAP, RAMP, and VIB protocols. The mean contact number $Z$ increased monotonically with $\phi$, but the $Z(\phi)$ relation depended on preparation history. The number of constraints per particle,
$$
C=3.0\,n_{F2F}+2.5\,n_{E2F}+1.5\,(n_{E2E}+n_{V2F}),
$$
also increased with $\phi$, and even the loosest packings had $C\approx 12$, roughly twice the six mechanical degrees of freedom of a rigid tetrahedron. These packings were therefore strongly hyperstatic rather than marginally jammed [1202.6272].

In frictional sphere packings under shear, $\phi$ controls both yielding and elasticity. Metayer and coauthors prepared packings in the range $0.575\le \phi \le 0.628$ and found that the yield force increases exponentially with $\phi$, with a sharp change in slope at $\phi_c\simeq 0.595$. This same $\phi_c$ coincided with the onset of dilatancy and with a transition seen in penetration experiments, while analysis of force fluctuations showed that the shear modulus is significantly smaller than the bulk modulus [1012.5023].

4. Density, disorder, and local structure

Several of the cited works argue that $\phi$ alone is insufficient to characterize a packing. In jammed sphere packings, this point is formalized by the density–order-metric plane $(\phi,\Psi)$. One specific choice is the translational-order metric
$$
T=
\left|
\frac{\sum_{i=1}^{N_s}(n_i-n_i^{ideal})}{\sum_{i=1}^{N_s}(n_i^{FCC}-n_i^{ideal})}
\right|,
$$
with $T=0$ for a Poisson configuration and $T=1$ for perfect FCC. A bond-orientational metric such as
$$
Q_6=\left[\frac{4\pi}{13}\sum_{m=-6}^6 |\langle Y_{6m}\rangle|^2\right]^{1/2}
$$
may also be used. The reported order map shows that moving away from the MRJ regime near $\phi\approx 0.64$ in either direction entails increased order, higher contact number, and fewer rattlers [1101.1327].

For planar disc packings, disorder was built directly into the optimization problem. The cell-order distribution ${Q_k}$ records the fraction of $k$-cells in the planar contact graph, and the probability that a $k$-cell has at least two neighboring $k$-cells is
$$
R_{kk}=Q_k\Bigl[1-(1-Q_k)^k-kQ_k(1-Q_k)^{k-1}\Bigr].
$$
Imposing the disorder criteria $R_{33}\le 1/3$ and $R_{44}\le 1/4$ excludes large hexagonal and square-lattice patches. Under these constraints, the optimal COD is
$$
Q_3=0.562236,\quad Q_4=0.431815,\quad Q_5=0.005948,
$$
which yields the exact value
$$
\phi_{\rm RCP}=0.852525\ldots
$$
for the densest truly disordered packing in two dimensions [2106.11774].

A related geometric-structure strategy was developed for binary convex superdisks. There, the organizing principle is a Gaussian contact-angle distribution
$$
f(\theta)=\frac{1}{\sqrt{2\pi}\,\sigma(p)}
\exp!\left[-\frac{(\theta-\theta^\ast(p))^2}{2\sigma^2(p)}\right],
$$
combined with local two-particle cells whose packing fractions are averaged over contact types and angles:
$$
\phi(p;\alpha,x)=
\int
\Bigl[(1-x)\phi_{\rm local}^{(1)}(\theta)+x\phi_{\rm local}^{(2)}(\theta)\Bigr]
f(\theta)\,d\theta.
$$
For the simulated range $p\in[0.85,4.5]$, $\alpha\in[0.2,0.95]$, $x\in[0.05,0.95]$, the maximum deviation between theory and simulation was reported as under $1.5\%$ except for the most extreme limits. In the monodisperse circle limit, the theory gives $\phi_{\rm MRJ}(p=1)=0.834\ldots$ [1605.06716].

5. Bounds, extrapolations, and predictive formulas

In dense tetrahedron packing, exact constructions coexist with nonrigorous but informative upper-bound estimates. A classical insphere-based bound for general convex particles,
$$
\phi_{\max}\le \min!\left[\left(\frac{v_P}{v_S}\right)\left(\frac{\pi}{\sqrt{18}}\right),1\right],
$$
is useless for tetrahedra because it exceeds unity. Torquato and Jiao therefore adapted Rogers’ local-cluster idea. For a five-tetrahedron “wagon-wheel” cluster they obtained $\phi_W\approx 0.974857$, while a twenty-tetrahedron “icosahedron-like” cluster gave $\phi_I\approx 0.880755$. Since neither cluster tiles space, they regarded $\phi_I$ as the more plausible tight estimate and proposed the range
$$
0.856347\ldots \le \phi_{\max}\le 0.880755\ldots
$$
for the true maximal tetrahedron-packing density [1005.0011].

For random close packing of circles with arbitrary size distributions, Meer, Galoustian, de Falco Manuel, and Weeks found that $\phi_{rcp}$ is determined to high accuracy by the polydispersity $\delta$ and skewness $S$ of the radius distribution, provided no hexagonal domains intrude. Their empirical representation is
$$
\phi_{rcp}(\delta,S)=\phi_1(\delta)-[\phi_1(\delta)-\phi_0]\exp{-[\ln X]^2},
$$
with $\phi_0=0.840$, $X=[S-S_1(\delta)+p(\delta)]/p(\delta)$, and
$$
\phi_1(\delta)=\frac{\phi_0+\delta^2\phi_0}{1+\delta^2\phi_0}.
$$
Simulation data for bidisperse and bi-Gaussian mixtures were reported to collapse on this master curve to within $\pm 0.002$, while the appearance of hexagonal order can raise $\phi$ by up to $\sim 0.02$ above the fit [2404.02316].

Ultradense disordered ellipse packings provide a further caution against equating near-crystalline $\phi$ with crystalline order. Using a Lubachevsky–Stillinger-like growth algorithm combined with biased SWAP Monte Carlo and transient degrees of freedom, Mailman and collaborators reported jammed ellipse packings with maximum $\phi_J(1.30)=0.9034$, only $0.4\%$ below the monodisperse-ellipse crystal density $\phi_{\rm xtal}=\pi/(2\sqrt{3})\simeq 0.9069$. Over $1.25\lesssim \alpha \lesssim 1.40$, they found $\phi_J(\alpha)>0.995\,\phi_{\rm xtal}$, yet the packings remained disordered; for $1.3\lesssim \alpha \lesssim 2.0$, the coordination number also stayed within $0.5\%$ of the isostatic value $Z_{\rm iso}=6$ [2409.19196].

6. Generalizations and alternative frameworks

The $\phi$-packing viewpoint extends beyond tetrahedra, spheres, and discs to broader one-parameter and manifold-based families. For truncated triangular bipyramids, Jiao and coauthors studied $\phi_{\max}(t)$ as a function of truncation parameter $t$ and identified eight distinct dense periodic packings, with regime boundaries marked by discontinuities in $\phi'{\max}(t)$ or, once, in $\phi''{\max}(t)$. Two four-particle packings, $P_{4}1$ and $P_{4}2$, occur for $t_3\approx 0.5010\le t\le t_5\approx 0.5777$, and the transition at $t_4\approx 0.5321$ is “second-order” in the sense that $\phi$ and $\phi'$ are continuous while $\phi''$ jumps. In self-assembly simulations, only one of the eight densest packings assembled, while the previously reported degenerate dodecagonal quasicrystal remained stable up to $t=0.45$ [1304.3147].

A different generalization derives packing theory from phyllotaxis, Markoff theory, and products of linear forms. In this framework, one studies density-stable lattices under diagonal volume-preserving deformations and obtains the exact lower bounds
$$
\Delta'_2=\frac{\pi}{2\sqrt{5}}\approx 0.7025,\qquad
\Delta'_3=\frac{\sqrt{3}\,\pi}{14}\approx 0.3890.
$$
These constants underlie almost uniformly distributed point sets on real analytic Riemannian surfaces in local diagonal charts and also a three-dimensional analogue of the Vogel spiral inside a ball. This suggests a broader interpretation of packing fraction as a local density control parameter on manifolds, not only in Euclidean particle packings [2106.12333].

Finally, the algebraic theory of Apollonian packings introduces a distinct but historically adjacent meaning of “$\Phi$ packings.” Here an Apollonian circle packing is realized as a Weyl orbit of a rank-4 indefinite Kac–Moody root system $\Phi$, with generating function
$$
Z(\mathbf s)=\sum_{w\in W} e^{-\langle w\mathbf c,\mathbf s\rangle}.
$$
The series converges exactly on the Tits cone of $\Phi$, and the boundary of that cone inherits the geometric structure of the Apollonian packing. A plausible implication is that not all “packing” theories organized around Greek-letter notation concern density optimization; some instead encode packing geometry through automorphic and Lie-theoretic structures [2102.02172].