Generalization of space-filling bearings to arbitrary loop size
Journal of Physics A: Mathematical and General
A model for space-filling packing of discs rolling sliplessly on each other is presented. This model is an extension of the model presented by Herrmann et al to a basic loop size of an arbitrary even number of discs. This model might have application in geophysics, dense granular flows and turbulence. We show the existence and uniqueness of such constructions, analytically, for loop size up to eight, and numerically for bigger basic loop sizes, and make precise the algorithm to construct them.
... to construct them. A classification of the solution is also given and extensive studies of the fractal dimension are performed, showing that it varies considerably with the base loop size. distinguish between the 'Hausdorff dimension' and the 'box-counting dimension' and suppose them equal. For further discussion of this issue see  .) This packing presents exactly the opposite characteristic of what we are looking for, since every closed loop has an odd number of bearings, hence the Apollonian packing is completely 'locked'. The existence of strip filling with discs without rotational frustration, baptized 'spacefilling bearings' (SFBs), was proven in  using a method based on iteratively applying conformal transformations: reflections, translations and inversions. It was shown that there are two different families, denoted by families 1 and 2 (F1 and F2, respectively), each one having different realizations parametrized by two positive integers (n 1 , n 2 ). The fractal dimension of these families was also numerically estimated to lie in the range between 1.306 and 1.520 [1, 7] . This idealized gauge modelling permitted a theoretical estimate of properties such as faultgauge size distributions as measured in    , and remanent magnetization in rocks issued from cataclastic shear flow  . Nevertheless, the study in  was limited to SFBs with a 'basic loop' size of four elements: i.e., constructions in which all closed loops of the bearings are, at least, four discs in size (this term will be defined more precisely below). The aim of this work is to prove the existence and study the properties of SFBs with a basic loop size larger than four discs. We will first introduce a variant of the study presented in  which is more adequate for the general case and that will lead us to conclude the existence of SFBs with a larger, even number of bearings in their basic loop. We will show that, as for the four-fold loop case, for each given basic loop size, there exists two topologically different families of SFBs, each one having an infinite number of different realizations described by two integers. This will be followed by a study of the properties for the different possible solutions, especially their fractal dimension. The layout of this paper is as follows: in section 2 we show the method used for the construction of general loop size SFBs. In particular, we discuss the working assumptions (section 2.1), the notation used (section 2.2), the choice of the transformations (section 2.3), the conditions that the transformations must fulfil (see section 2.4 with complementary mathematical details given in the appendix) and how the construction algorithm is effectively implemented utilizing a computer (section 2.5). In section 3 we discuss the issue of the fractality of the constructed set, and in section 4 we present some examples of general loop size SFBs we have calculated and study their size distribution. Section 5 contains a discussion of the results in the light of the experimental work presented in    13] . A more detailed version of the work presented here can be found in  in which the working assumptions and the choice of the transformations (sections 2.1 and 2.3, respectively) are discussed extensively. Construction of a generalized base loop size SFB Working assumptions