Abstract
Let us ask if it is possible to tile a plane with wheels rolling on each other such that all the area is covered with wheels. This rather exotic question can arise in various contexts. One could imagine the wheels to be eddies on the surface of an incompressible fluid and then ask if the fluid motion can be totally decomposed into stable eddies. Or, one could think of mechanical roller bearings between two moving surfaces, like two tectonic plates, and then ask if one can completely fill the space between the rolling cylinders with other rolling cylinders such that no cylinder excerces any frictional work on another one. The question we are asking is, in fact, geometrical.
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References
McCann, W., S. Nishenko, L. Sykes, J. Krause, Pageoph 117, 1082 (1979); C. Lomnitz, Bull. Seism. Soc. Am. 72, 1441 (1982).
Sammis, C., G. King, R. Biegel, Pageoph 125, 777 (1987).
Mandelbrot, B.B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
Boyd, D.W., Mathematica 20, 170 (1973); see also ref. 3.
Falconer, J.K., The Geometry of Fractal Sets, Cambridge Univ. Press, 1985; L.R. Ford, Automorphic Functions, Chelsen Publ., 1929.
Herrmann, H.J., G. Mantica, D. Bessis, Phys. Rev. Lett. 65, 3223 (1990).
Manna, S.S., H.J. Herrmann, preprint.
for a review see G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987) or A. Coniglio, L. de Arcangelis and H.J. Herrmann, Physica A 157, 21 (1989).
Huber, G., private communication.
Manna, S.S., T. Vicsek, preprint.
Bessis, D., S. Demko, Comm. in Math. Phys., preprint.
Batchelor, G.K., Theory of homogeneous turbulence, Cambridge Univ. Press, 1982.
Kauffman, S.A., J. Theor. Biol. 22, 437 (1969).
Derrida, B., Y. Pomeau, Europhys Lett. 1, 45 (1986); B. Derrida, D. Stauffer, Europhys Lett. 2, 739 (1986).
da Silva, L.R., H.J. Herrmann, J. Stat. Phys. 52, 463 (1988).
Wolfram, S., Theory and Application of Cellular Automata, World Scientific, Singapore, 1986; E. Bienenstock, F. Fogelman-Soulié, G. Weisbuch (eds), Disordered Systems and Biological Organization, Springer, NATO ASI Series F, 1986.
Herrmann, H.J., in Nonlinear Phenomena in Complex Systems, A.N. Proto (ed), Elsevier, Amsterdam, 1989.
Pomeau, Y., J. Phys. A 17, L415 (1984); H.J. Herrmann, J. Stat. Phys. 45, 145 (1986).
Zabolitzky, J.G., H.J. Herrmann, J. Comp. Phys. 76, 426 (1988); S.C. Glotzer, D. Stauffer, S. Sastry, Physica A 164, 1 (1990).
Stauffer, D., in Computer Simulation Studies in Condensed Matter Physics II, D.P. Landau, K.K. Mon and H.-B. Schüttler (eds), Springer, Heidelberg, 1990.
de Arcangelis, L., A. Coniglio, Europhys Lett. 7, 113 (1988).
Hilhorst, H.J., M. Nijmeijer, J. Physique 48, 185 (1987).
Golinelli, O., B. Derrida, J. Phys. A 22, L939 (1989).
Mariz, A.M., J. Phys. A 23, 979 (1990).
Stanley, H.E., D. Stauffer, J. Kertész, H.J. Herrmann, Phys. Rev. Lett. 59, 2326 (1987).
Costa, U., J. Phys. A 20, L583 (1987).
Le Caër, G., J. Phys.A 22, L647 (1989) and Physica A 159, 329 (1989).
Derrida, B., G. Weisbuch, Europhys Lett. 4., 657 (1987).
Coniglio, A., L. de Arcangelis, H.J. Herrmann, N. Jan, Europhys. Lett. 8, 315 (1989).
Ashkin, J., E. Teller, Phys. Rev. 64, 178 (1943).
Parisi, G., Nucl. Phys. B 180, 378 (1981).
Guenoun, P., F. Perrot, D. Beysens, Phys. Rev. Lett. 63, 1152 (1989).
de Arcangelis, L., H.J. Herrmann, A. Coniglio, J. Phys. A 23, L265 (1990).
Neumann, A.U., B. Derrida, J. Physique 49, 1647 (1988).
see e.g. G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987).
de Arcangelis, L., A. Coniglio, H.J. Herrmann, Europhys Lett. 9, 749 (1989).
Campbell, I.A., L. de Arcangelis, Europhys Lett. …
de Arcangelis, L., H.J. Herrmann, A. Coniglio, J. Phys. A 22, 4659 (1989).
Boissin, N., H.J. Herrmann, J. Phys. A 24, L43 (91).
de Arcangelis, L., A. Coniglio, preprint; A. Coniglio, in Correlations and Connectivity in Constrained Geometries, H.E. Stanley and N. Ostrowsky (eds), Kluwer, Dordrecht, 1990.
da Cruz, H.R., U.M.S. Costa and E.M.F Curado, J. Phys. A 22, L651 (1989).
Mariz, A.M., H.J. Herrmann, L. de Arcangelis, J. Stat. Phys. 59, 1043 (1990).
Stauffer, D., Physica A 162, 27 (1990).
Mariz, A.M., H.J. Herrmann, J. Phys. A 22, L1081 (1989); O. Gollinelli, Physica A 167, 736 (1990).
Golinelli, O., B. Derrida, J. Physique 49, 1663 (1988).
Manna, S.S., J. Physique 51, 1261 (1990).
Barber, M.N., B. Derrida, J. Stat. Phys. 51, 877 (1988).
Miranda, E., preprint HLRZ 82/90.
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Herrmann, H.J. (1992). Two Chosen Examples for Fractals: One Deterministic, the Other Random. In: Goles, E., Martínez, S. (eds) Statistical Physics, Automata Networks and Dynamical Systems. Mathematics and Its Applications, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2578-9_4
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