Maximum and Minimum Stable Random Packings of Platonic Solids

Jessica Baker and Arshad KudrolliDepartment of Physics, Clark University, Worcester, MA 01610

(Dated: May 31, 2018)

Motivated by the relation between particle shape and packing, we measure the volume frac-tion φ occupied by the Platonic solids which are a class of polyhedron with congruent sides,vertices and dihedral angles. Tetrahedron, cube, octahedron, dodecahedron, and icosahedronshaped plastic dice were fluidized or mechanically vibrated to find stable random loose packingφrlp = 0.51, 0.54, 0.52, 0.51, 0.50 and densest packing φrcp = 0.64, 0.67, 0.64, 0.63, 0.59, respectivelywith standard deviation ' ±0.01. We find that φ obtained by all protocols peak at the cube, whichis the only Platonic solid that can tessellate space, and then monotonically decrease with numberof sides. This overall trend is similar but systematically lower than the maximum φ reported forfrictionless Platonic solids, and below φrlp of spheres for the loose packings. Experiments withceramic tetrahedron were also conducted, and higher friction was observed to lead to lower φ.

PACS numbers: 61.43.-j;45.70.-n

I. INTRODUCTION

The packing of objects has long fascinated physicists,mathematicians, and the curious. While the centuriesold Kepler’s conjecture that the maximum packing ofspheres is π/

√18 ' 0.74 has been finally proven [1],

spheres tossed randomly into a jar and shaken, do notreach such high packing fractions, unless special proto-cols are used [2]. A random closed packed (RCP) vol-ume fraction φrcp = 0.6366 was found by mechanicallyvibrating a set of steel ball bearings [3]. In fact, workin the last decade has shown that random packing itselfis not unique, and there is a range of packing fractionswhich can be obtained for spherical particles with ran-dom order. These values are bounded at the upper endby φmrj = 0.64 [4], the so called maximally random closepacked state, and more tentatively at the lower end byrandom loose packing at φrlp = 0.55 [5, 6].

In contrast, much less is known for non-spherical par-ticles. It was only shown a few years ago that ran-domly packed prolate and oblate objects pack denserthan spherical particles [7]. The highest Bravais lat-tice packing of Platonic solids are also considered tobe the highest packings except in case of the tetrahe-drons [8, 9]. In the case of tetrahedrons, disorderedwagon-wheel packings were initially found to pack evenhigher. The maximum packing of tetrahedrons has sincebeen improved upon in rapid succession with differentapproaches [10, 11], and the current highest packing fortetrahedrons stands at 0.856347.. corresponding to dimerpackings of regular tetrahedrons [12–14]. As noted forspheres, maximum packing tend to be larger than poly-hedral packings which may be disordered or ordered whenbrought together randomly. For example, quasi-crystalswere observed with Monte Carlo simulation of tetrahe-drons by Haji-Akbari et al [15]. Experiments on randompacked tetrahedronal dice have been reported recently inRef. [16]. Volume fractions were said to be 0.76±.02 if theobserved packings were extrapolated to infinite systems,but the protocol by which the packings were prepared

(a) (b)

FIG. 1: (Color online) (a) A schematic diagram of the con-tainer which is vibrated vertically with an electromagneticshaker to obtain the mechanically shaken packings. (b) Aschematic diagram of the container used to obtain loose pack-ings using the fluidization protocol. The liquid is first injectedfrom below and the packing height measured after flow rateis turned off and the liquid drained.

was not clear.

In this paper, we experimentally investigate the pack-ings obtained with the Platonic solids using various ex-perimental preparation protocols including sequential ad-dition (unshaken), sequential addition with hand shak-ing, mechanical vibration, and fluidization. A questionwe also seek to address is if the packing fraction forfaceted particles approaches that for spheres from aboveor below in the limit of large number of faces. We alsotest the effect of the number of particles and the fric-tion between them on the packing. A further motivationfor our study comes from the fact that natural sand isoften faceted with sharp edges which can strongly influ-ence their packing density [17]. Platonic solids which areidealized faceted particles with congruent sides may be a

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Name Tetrahedron Cube Octahedron Dodecahedron Icosahedron Tetrahedron

Material Plastic Plastic Plastic Plastic Plastic Ceramic

Number of Faces 4 6 8 12 20 4

Dihedral Angle (radian) cos−1( 13) π

2cos−1(−1

3) cos−1(−

√5

5) cos−1(−

√5

3) cos−1( 1

3)

Solid Angle (steradian) cos−1( 2327

) π2

4 sin−1( 13) π − tan−1( 2

11) 2π − 5 sin−1( 2

3) cos−1( 23

27)

Vsolid (cm3) 1.56 4.0 2.4 4.0 3.5 1.7

TABLE I: (Color online) The properties of the Platonic solids studied. The volume for each kind of solid Vsolid were found byaveraging over 10 trials with water displacement measurements.

better starting point compared with smooth spheres tounderstand packing of rough particles.

II. EXPERIMENTAL METHOD

A. Materials

The Platonic solids are a class of convex polyhedronswith faces of congruent polygons and the same numberof faces meeting at each vertex. These conditions leadto congruent faces, dihedral angles and solid angles (Ta-ble I). There are only 5 shapes that belong to this cate-gory: the 4 sided tetrahedron, 6 sided cube, 8 sided oc-tahedron, 12 sided icosahedrons and 20 sided dodecahe-dron. The actual particles studied in our experiment areplastic dice (density ρ = 1.16 g/cm3) which have slightlyrounded edges with properties listed in Table I. To esti-mate the actual volume fractions occupied by the solidsused, we measured the volume using water displacementtechnique. We thus assume that we are calculating thevolume fraction of a Platonic solid which is in between thecircumscribed and inscribed limit because of the roundededges. To understand the effect of the rounding further,one has to obtain the distribution of contacts that involvevertices, edges, and the distribution of contact angles,which is beyond the scope of our technique. The predom-inance of these kind of contacts may indicate that we aresystematically over estimating the volume fraction. Onthe other hand predominance of face-face contacts maylead to an underestimate. While it is possible some ofthese effects offset each other, it is difficult estimate thenet error without detailed understanding on contacts.

Ceramic tetrahedrons (ρ = 1.63 g/cm3) were also usedto compare number of particles and effects of frictionon packing fractions. The two kinds of materials haveslightly different coefficient of friction (µplastic = 0.375,µceramic = 0.480), which is measured using a tilted planeand finding the angle at which particles begin to slidepast each other. Our system is athermal and therefore en-ergy has to be supplied externally to rearrange the solids.

B. Packing preparation protocols

We use four different random packing protocols to pre-pare the packing of Platonic solids in cylindrical con-tainers with a semi-hemispherical bottom boundary tominimize surface area. We found this shape best suitedto determine volume fractions accurately because eventhough a spherical container has a low surface to volumeratio, it is practically difficult to fill a particle under anoverarching surface.

1. Sequential addition

In the first packing protocol, the particles are addedsequentially at a random location from a height of abouta few times the particle size. This ensured that the par-ticles land in a stable configuration without significantlymoving the particles which were already in place in thepacking.

A thin plate is placed on top after the packing is pre-pared and the average height is noted. Using this height,the total volume of the container Vcontainer is obtainedto determine the volume fraction occupied by the solidsφ = NVsolid/Vcontainer, where N is the number of parti-cles added. This packing protocol was repeated 10 timesfor each kind of solid to determine the mean packing frac-tion and standard deviation.

2. Hand shaken

In the second method, the container is shaken by handafter two layers of particles are added to the containerso that particles rattle and have an opportunity to rear-range. After all the particles are added, φ is determined(as above) from the measured height of the packing. Aswe will see this protocol leads to relatively higher φ.

3

(a) (b) (c) (d) (e)

FIG. 2: (Color online) Images of typical disordered packings observed for (a) tetrahedrons, (b) cubes, (c) octahedrons, (d)dodecahedron, and (e) icosahedron. The images shown correspond to the top surface of the packing obtained after sequentialaddition of particles.

3. Mechanically shaken

Because it is impossible to shake the particles system-atically by hand, we also built an experimental system inwhich the container is rigidly attached to an electromag-netic shaker (see schematic diagram shown in Fig. 1(a))similar to previous systems used to study random closepacking of spheres [3]. The shaker is connected to a func-tion generator allowing us to apply a prescribed vibrationfrequency f and acceleration strength Γ. After initiallyplacing the particles randomly inside the container, andvibrating over various lengths of times, the height of thepacking is recorded after the vibration is turned off toensure that the packing obtained is stable. The volumefraction φ is then obtained as described in the unshakencase. This protocol gave rise to the highest packing frac-tions that we observed.

We first performed measurement with experimentsover 120 minutes of vibration, and found that the pack-ing fractions increased rapidly initially by about 5% ina few minutes and then did not vary significantly. Sub-sequently, we performed and measure φ after 10 minutesof applied vibration. While slow evolution of volumefraction over prolonged periods of vibrations has beenobserved with spherical particles [18], it appears thatPlatonic solids, which are faceted, get frustrated morequickly because they cannot rotate and roll as easily inplace as spherical particles. We also tested the frequencydependence of obtained φ and found that a peak wasobserved at f ∼ 50 Hz. Because we are primarily in-terested in this method to obtain the maximum possibledensity using random agitations, we report here data forf ∼ 50 Hz and 5g where g is the acceleration due to grav-ity. We present the frequency dependence of the observedpacking fraction later in Sec. IV C.

4. Fluidization

In the final protocol used to prepare the packing, wefirst fill the particles inside a container with a hemispher-ical mesh at the bottom through which water can be in-

jected with rates high enough to fully fluidize the plasticdice (see Fig. 1(b)). A flow rate of 1.6× 104 cm3/min isapplied for 3 minutes to fully agitate the system, and thenthe flow rate is slowly reduced to zero so that particlesslowly settle layer by layer with low relative velocities.The water is then completely drained from the systemand the height of the packing is measured to obtain φ.This set up and method is similar to that used to obtainthe limit of random loose packing in spheres [6], and wealso find the lowest φ for the Platonic solids among allprotocols we attempted.

III. OBSERVED PACKINGS

Figure 2 shows an image corresponding to typical pack-ings observed for each of the Platonic solids. In this case,200 particles were added inside a container with a 9.3 cmdiameter using the sequential addition protocol and theimage was taken of the top layer which is also similar toany intermediate layer. The particles appear to be allrandomly located relative to each other. (We quantifythe disorder in the packing later in Sec. IV D using thevariance of the orientation of the face of the polyhedronrelative to the vertical axis.) The packings prepared us-ing the other protocols appear similarly random.

The packing fractions for each of the Platonic solidsobtained using the protocols described in Section II Bare listed in Table. II. As expected, the fluidized proto-col creates the least dense packing which we consider asφrlp , and the mechanical shaker produces the most densepacking which we consider as φrcp. Further we note thatthe sequential addition (unshaken) and hand shaken pro-tocols produce intermediate packings. In all cases φ varysimilarly and peak at the cube and then decrease withincreasing number of sides.

To understand the trends in the observed packings,we have plotted the mean values for each kind of solidand protocol in Fig. 3 along with the maximum known φwhich have been reported [8, 14]. Interestingly, we findthat the data follows the same trend as the theoreticalmaximum φ, but is systematically lower because of the

4

Shape φSequentialAddition φHandShaken φMechanicallyShaken φFluidization

Tetrahedon (Plastic) 0.54± 0.01 0.62± 0.02 0.64± 0.01 0.51± 0.01

Cube (Plastic) 0.57± 0.01 0.66± 0.02 0.67± 0.02 0.54± 0.01

Octahedron (Plastic) 0.57± 0.01 0.62± 0.01 0.64± 0.01 0.52± 0.01

Dodecahedron (Plastic) 0.56± 0.01 0.60± 0.01 0.63± 0.01 0.51± 0.01

Isocahedron (Plastic) 0.53± 0.01 0.57± 0.01 0.59± 0.01 0.50± 0.01

Tetrahedon (Ceramic) 0.48± 0.02 0.59± 0.01 - -

TABLE II: The mean packing fractions (± one standard deviation) observed for each of the Platonic solids using the fourprotocols.

0.4

0.5

0.6

0.7

0.8

0.9

1

0 4 8 12 16 20 24

φ

Number of faces

Theoretical MaximumSphere- FCC

Mechanically ShakenSphere - MRJ

Fluidized

Sphere - RLP

FIG. 3: (Color online) The packing fraction for the Platonicsolids compared with the maximum known packing for thatsolid. The observed random packings follow the same trend asthe maximum packings. The maximum value for tetrahedronis from Ref. [14], and for the remaining Platonic solids arefrom Ref. [8]. The maximum packing along with the RCPand RLP value for spheres is also plotted for comparison.

disordered nature of their packing. While lower φ thanmaximum can be anticipated, it is somewhat unexpectedthat the trend for disordered packing reflects the maxi-mum packing.

IV. SUPPLEMENTARY MEASUREMENTS

A. Effect of Friction

Because the particles we use have friction, and thepresence of friction reduces the minimum number of par-ticle contacts needed for static equilibrium from 3 to 2in three dimensions, it is important to consider its effecton packing. In the case of spheres, friction is known toaffect packing fractions especially at the loose packinglimit [19]. Needing fewer neighbors implies the parti-cles are further apart and therefore the packing can beless dense. It is possible for this reason that we observea lower dense close packing value for tetrahedrons com-pared with simulations which found 0.6817 with friction-less tetrahedrons using a relaxation algorithm [20]. Toexplore this further, we compare packings of the plastictetrahedrons with the more frictional ceramic tetrahe-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 200 400 600 800 1000 1200 1400

Number of Particles

Hand ShakenSequential Addition

φ

FIG. 4: (Color online) The volume fraction is not observed tovary significantly beyond 200 as the number of tetrahedronsare increased by an order of magnitude. The lines are a guideto the eye.

drons. We find that for both the unshaken and handshaken protocols (see Table II), the larger the coefficientof friction, the lower the packing fraction with a 9% and5% lower packing fraction for the ceramic case.

B. System size

Because the plastic dice were expensive, the cost be-came prohibitive to test with a larger number of dice.However, ceramic tetrahedrons used in grinding mediawere relatively inexpensive and we were able to test thenumber dependence in this case. In Fig. 4, we plotthe measured packing fraction for numbers ranging fromaround 100 to 1200 in proportionately larger containers.Here we used the sequential addition and hand shakenprotocol (the particle weight was too high to use ourshaker and fluidization experimental setup in this case.)We find that after N = 200, φ remains constant withinexperimental error and therefore we believe that φ re-ported in Table II are representative of even larger pack-ings and surface effects are small.

C. Driving frequency dependence

We measured φ for each of the Platonic solids as a func-tion of frequency while keeping the driving strength Γ/f

5

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

20 30 40 50 60 70 80 90 100 110

φ

Frequency (Hz)

TetrahedronCubeOctahedronDodecahedronIcosahedron

FIG. 5: (Color online) The observed packing fraction versusdriving frequency.

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24

σ2(e

xper

imen

t) /σ

2 (ran

dom

)

Number of Faces

Sequential AdditionMechanically Shaken

FIG. 6: (Color online) The ratio of the variance of observedprojected areas and that of random projected areas. The datais closer to 1 which indicates random packing rather than tozero, which would correspond to an ordered system.

constant (see Fig. 5). A peak is observed at f = 50 Hz.The appearance of peak in frequency can be explainedas follows. At the lowest frequencies, the particles aretossed up which appears to result in lower packing frac-tions. Whereas, at high frequency particles do not ap-pear to receive sufficient energy to rearrange from initialpackings formed after particles are added.

D. Characterization of randomness of packing

In order to parameterize the randomness in the pack-ings, we use the images of the packing obtained from thetop (as shown in Fig. 2) and measure the projected areaof the largest visible face. All the projected areas foundlie above a minimum (0.333, 0.408, 0.577, 0.795, 0.795 forthe 4,6,8,12 and 20 sided polyhedron respectively) whichis dependent on the geometry of the polyhedron becausethe projected area is normalized with the area of the facewhen laying normal to the perpendicular. Because of thenormalization, the maximum projected area is 1. We firstsample 100 projected area for each Platonic solid andpreparation technique. Then we calculate the varianceamong these projected areas (see Fig. 6). For a period-

ically ordered system, all the particles would have thesame orientation, and therefore the same projected areafor the top face. This implies that the variance of theseareas would be zero. As a point of comparison, we findthe variance of 100 random numbers generated in theallowed range for each shape. This variance representsthat of a highly random system. We find the variance ofthe projected areas for all shapes except the mechanicallyshaken cube to be closer to the random number variancethan it is to the zero variance for an ordered orientation(see Fig. 6). As cubes have a strong tendency to align,the variance of the projected areas lies slightly below 0.5and closer to the ordered extreme for the mechanicallyshaken case. From this we conclude that in most caseswe have disordered or random packings.

V. COMPARISON WITH SPHERE PACKING

In Fig. 3, we also plot the maximum φ, φrcp and φrlp forspheres along those for the Platonic solids. The spherecan be considered as a limit of polyhedral shaped par-ticles as the number of sides goes to infinity, and givesa context to understand the observed packing fractions.As has been noted previously, φmax for all the Platonicsolids exceed that for spheres, consistent with Ulam’sconjecture that convex particles pack to a greater φ thanspheres. Examining the known maximum packing struc-ture for Platonic solids [8, 14] it appears that faces alignnear contacts to give the greater φ.

However for dense packings, φmrj for spheres lies belowonly the cube and octahedron, but as the trend decreasesfrom the octahedron to the dodecahedron and icosahe-drons, it appears that the sphere value would now beapproached from below. Finally, in considering the looseextreme, we find that all the Platonic solids pack lessdense than φrlp for spheres. This leads us to conjecturethat all convex particles in the random loose packed con-figuration pack less dense than a corresponding packingof spheres.

VI. SUMMARY

In summary, we have obtained a wide range of pack-ing fractions which depend on the number of faces andedges of the packing solid. The values systematically de-pend on the protocols used to create these random pack-ings. We hypothesize that the limiting values we reportcorrespond approximately to random loose packing andrandom dense packing for these shapes. Interestingly, wefind the overall trend similar to that for the maximumpacking fraction. In contrast with the maximum pack-ings fraction which always exceed that for spheres, therandom packings for Platonic solids with large numberof sides pack looser than for spheres. In closing, we notethat our packings of idealized polyhedral particles cre-ated with random protocols may give better insight into

6

true packings of faceted particles such as sand found innature.

Acknowledgments

The work was supported by the National Science Foun-dation under Grant No. DMR-0605664. We appreciate

comments on the manuscript by Salvatore Torquato.

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