Self-Similar Space-Filling Packings in Three Dimensions

Baram, R. Mahmoodi; Herrmann, Hans J.

doi:10.1142/s0218348x04002549

Cited by 25 publications

(26 citation statements)

References 9 publications

(19 reference statements)

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“…The signicantly steeper power-law slope in this region (q À2.7 ) is close to what would be expected for close-packed spheres. 44 The persistence of the mass fractal scattering to q-values close to q max suggests that the fractal clusters grow self-similarly as gelation proceeds, unlike the low-f case where the fractal scaling changes continually throughout the gel transition.…”

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confidence: 98%

Homogeneous percolation versus arrested phase separation in attractively-driven nanoemulsion colloidal gels

et al. 2014

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We elucidate mechanisms for colloidal gelation of attractive nanoemulsions depending on the volume fraction (f) of the colloid. Combining detailed neutron scattering, cryo-transmission electron microscopy and rheological measurements, we demonstrate that gelation proceeds by either of two distinct pathways. For f sufficiently lower than 0.23, gels exhibit homogeneous fractal microstructure, with a broad gel transition resulting from the formation and subsequent percolation of droplet-droplet clusters.In these cases, the gel point measured by rheology corresponds precisely to arrest of the fractal microstructure, and the nonlinear rheology of the gel is characterized by a single yielding process. By contrast, gelation for f sufficiently higher than 0.23 is characterized by an abrupt transition from dispersed droplets to dense clusters with significant long-range correlations well-described by a model for phase separation. The latter phenomenon manifests itself as micron-scale "pores" within the droplet network, and the nonlinear rheology is characterized by a broad yielding transition. Our studies reinforce the similarity of nanoemulsions to solid particulates, and identify important qualitative differences between the microstructure and viscoelastic properties of colloidal gels formed by homogeneous percolation and those formed by phase separation.

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confidence: 98%

Homogeneous percolation versus arrested phase separation in attractively-driven nanoemulsion colloidal gels

et al. 2014

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“…Details on the construction algorithm and on the complete set of new configurations will be published elsewhere [8]. We give here only a qualitative description of this technique for the bichromatic packing.…”

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confidence: 99%

Space-Filling Bearings in Three Dimensions

2004

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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 can rotate without slip and with negligible torsion friction. PACS numbers: 46.55.+d, 61.43.Bn, 91.45.Dh Space-filling 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 significant heat. Space-filling bearings have also been used as toy models for turbulence and can also be used in mechanical devices [3]. Two dimensional spacefilling bearings have been shown to exist and a discrete infinity 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 importance to the physical applications.In this letter, we will report the discovery of a selfsimilar space-filling bearing in which an arbitrary chosen sphere can rotate around any axis and all the other spheres rotate accordingly without any sliding and with negligible torsion friction.In two dimensions, different classes of space-filling 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 sufficient 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 frustration.The situation in three dimensions is different 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 infinite number of interconnecting loops, we could construct unfrustrated configurations 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 different colors in such a way that no spheres of same color touch each other, as shown in Fig.1.No three dimensional space-filling bearing has been known up to now. The classical Apollonian packing is space-filling and self-similar but not a bearing since at least five colors are needed to assure different colors at each contact. This packing can be constructed in different ways [6,7]. By generalizing the inversion technique used in Ref.[6] to other Platonic Solids than the tetrahedron (the base of 3D Apollonian packing) we were able to construct five new packings including a bichromatic one. Details on the construction algorithm and on the complete set of new configurations will be published elsewhere [8]. We give here only a qualitative description of this technique for the bichr...

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“…Here, we show that for all the self-similar spacefilling packings constructed by inversive geometry of Refs. [19,[28][29][30][31], cuts along random hyperplanes generally have a fractal dimension of the one of the uncut packing minus one, what we prove analytically. Nevertheless, these packings are still heterogeneous fractals since cuts along special hyperplanes of a single packing show specific fractal dimensions.…”

Section: Introductionmentioning

confidence: 66%

“…1, which are generated using inversive geometry as in Refs. [19,[28][29][30][31] and which therefore are exactly self-similar. Reference [30] previously showed that the packing in Fig.…”

Section: Introductionmentioning

confidence: 99%

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Cutting Self-Similar Space-Filling Sphere Packings

Stäger

Herrmann

2018

Fractals

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Any space-filling packing of spheres can be cut by a plane to obtain a space-filling packing of disks. Here, we deal with space-filling packings generated using inversive geometry leading to exactly selfsimilar fractal packings. First, we prove that cutting along a random hyperplane leads in general to a packing with a fractal dimension of the one of the uncut packing minus one. Second, we find special cuts which can be constructed themselves by inversive geometry. Such special cuts have specific fractal dimensions, which we demonstrate by cutting a three-and a four-dimensional packing. The increase in the number of found special cuts with respect to a cutoff parameter suggests the existence of infinitely many topologies with distinct fractal dimensions.

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Self-Similar Space-Filling Packings in Three Dimensions

Abstract: We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bicromatic packing known up to now.

Cited by 25 publications

References 9 publications

Homogeneous percolation versus arrested phase separation in attractively-driven nanoemulsion colloidal gels

Homogeneous percolation versus arrested phase separation in attractively-driven nanoemulsion colloidal gels

Space-Filling Bearings in Three Dimensions

Cutting Self-Similar Space-Filling Sphere Packings

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