Space-filling bearings in three dimensions

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres ca

a r X i v :c o n d -m a t /0312460v 1 18 D e c 2003

Space-?lling bearings in three dimensions

R.Mahmoodi Baram ?and H.J.Herrmann ?

Institute for Computational Physics,University of Stuttgart Pfa?enwaldring 27,70569Stuttgart,Germany

N.Rivier ?

LDFC,Universit´e Louis Pasteur,3,rue de l’Universit´e F 67084Strasbourg cedex

(Dated:February 2,2008)

We present the ?rst space-?lling bearing in three dimensions.It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres can rotate without slip and with negligible torsion friction.

PACS numbers:46.55.+d,45.70.-n,61.43.Bn,91.45.Dh

Space-?lling bearings have been introduced in several contexts,such as in explaining the so-called seismic gaps [1,2]of geological faults in which two tectonic plates slide against each other with a friction much less than the expected one,without production of earthquakes or of any signi?cant heat.Space-?lling bearings have also been used as toy models for turbulence and can also be used in mechanical devices [3].Two dimensional space-?lling bearings have been shown to exist and a discrete in?nity of realizations has been constructed [4,5].The remaining question still open is:Do they also exist in three dimensions?This question is of fundamental im-portance to the physical applications.

In this letter,we will report the discovery of a self-similar space-?lling bearing in which an arbitrary cho-sen sphere can rotate around any axis and all the other spheres rotate accordingly without any sliding and with negligible torsion friction.

In two dimensions,di?erent classes of space-?lling bearings of disks have been constructed in Refs.[4,5]by requiring the loops to have an even number of disks,since in two dimensions this is obviously a necessary and su?cient condition for disks to be able to rotate without any slip.Successive disks must rotate,in alternation,clockwise and counter-clockwise in order to avoid frus-tration.

The situation in three dimensions is di?erent from two dimensions in two ways;The axes of rotation need not be parallel,and the centers of spheres in a loop may not lie all in the same plane.As a result,even in an isolated odd loop spheres could rotate without friction.But,as we will see,in the packings with an in?nite number of interconnecting loops,we could construct unfrustrated con?gurations of rotating spheres when all loops have an even number of spheres.Such a packing is bichromatic ,i.e.,one can color the spheres using only two di?erent colors in such a way that no spheres of same color touch each other,as shown in Fig.1.

No three dimensional space-?lling bearing has been known up to now.The classical Apollonian packing is space-?lling and self-similar but not a bearing since at

FIG.1:Bichromatic packing of spheres with octahedral sym-metry.No two spheres of same color touch each other.The image on the left shows the initial con?guration and the ?rst generation of inserted spheres.

least ?ve colors are needed to assure di?erent colors at each contact.This packing can be constructed in di?er-ent ways [6,7].By generalizing the inversion technique used in Ref.[6]to other Platonic Solids than the tetrahe-dron (the base of 3D Apollonian packing)we were able to construct ?ve new packings including a bichromatic one.Details on the construction algorithm and on the complete set of new con?gurations will be published else-where [8].We give here only a qualitative description of this technique for the bichromatic packing.

Let us consider ?lling a sphere of unit radius.The ?lling procedure is initialized by placing seven initial spheres on the vertices and the center of a regular oc-tahedron inside the unit sphere,so that the spheres on the vertices do not touch each other but touch the unit sphere and the one in the center.Further spheres are inserted by inversion [9]such that this topology can be preserved on all scales,imposing that,all spheres are on vertices or centers of (deformed)octahedra.

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres ca

2 The inversions are made iteratively with respect to

nine inversion spheres[13]:One inversion sphere is con-

centric with the unit sphere,and is perpendicular to the

six initial spheres on the vertices of the octahedron[14].

Inversion around it,therefore,leaves the vertex spheres

invariant,and maps the unit sphere onto the central

sphere.The other eight inversion spheres are associated

to each face of the octahedron,that is,they are perpen-

dicular to the unit sphere and to the three spheres on the

vertices of that face.The inversion around each of these

inversion spheres maps the unit sphere and spheres of

the corresponding face into themselves(that is,it gives

no new sphere),and the other four initial spheres into

four new spheres within the space between the face and

the unit sphere.

Figure2shows the plane cut through the centers of

the unit sphere and four initial spheres.Dashed circles

are cuts of the inversion spheres.Sphere S:0,1,···is

mapped,by the inversion sphere shown by a thick dashed

line,onto sphere S′.The inversion around this sphere

gives no new images of spheres1and2.In the?rst it-

eration we make all possible inversions which give new

and smaller spheres.In the next iterations,the newly-

generated spheres are mapped to smaller spheres.For

example,sphere0′is mapped(by the central inversion

sphere)onto0′′.In this way,the remaining empty space

is?lled in the limit of in?nite iterations while the bichro-

matic topology of the contacts is preserved.

Using this algorithm,the con?guration of initial

spheres which gives a bichromatic packing is unique.

Strictly speaking,it is shown that the only value for radii

of the initial spheres which leads to the bichromatic pack-

ing without(partial)overlapping of generated spheres is

√

(

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres ca

3 Putting Eq.(2)into Eq.(3)we?nd the relation between

the angular velocities of the?rst and third sphere:

R3 ω3=R1 ω1+α12?r12?α23?r23.(4)

In general,we can relate the angular velocities of the?rst

and j th spheres of an arbitrary chain of spheres in no-slip

contacts by:

R j ωj=(?1)j?1R1 ω1+j?1

i=1

(?1)j?iαi,i+1?r i,i+1.(5)

As long as the chain is open,the spheres can rotate with-out slip with the angular velocities given by Eq.(5)and no restrictions onαi,i+1.But,for a loop of n spheres in contact,spheres j and j+n are identical,so that

R1 ω1=(?1)n R1 ω1+

n

i=1

(?1)n?i+1αi,i+1?r i,i+1.(6)

A similar equation holds for every sphere j=1,···,n in the loop.

Although for a single loop there are many solutions of Eq.(6),not all will serve our purpose.In a packing, each sphere belongs to a very large number of loops and

all loops should be consistent and avoid frustration.In other words,the angular velocity obtained for a sphere as a member of one loop should be the same as being a member of any other loop.

If the loop contains an even number n of spheres, Eq.(6)becomes a relation between the hitherto arbitrary coe?cients of connectionαi,i+1,

n

i=1

(?1)iαi,i+1?r i,i+1=0.(7) Using the fact that the loop is geometrically closed:

n

i=1

(R i+R i+1)?r i,i+1=0,(8) a solution for Eq.(7)is

αi,i+1=c(?1)i(R i+R i+1),(9) where c is an arbitrary constant.Putting this in Eq.(5),

yields the angular velocities

ωj=

1

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres ca

4

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