Abstract
Being introduced to the Pascal triangle for the first time, one might think that this mathematical object was a rather innocent one. Surprisingly it has attracted the attention of innumerable scientists and amateur scientists over many centuries. One of the earliest mentions (long before Pascal’s name became associated with it) is in a Chinese document from around 1303.1 Boris A. Bondarenko,2 in his beautiful recently published book, counts several hundred publications which have been devoted to the Pascal triangle and related problems just over the last two hundred years. Prominent mathematicians as well as popular science writers such as Ian Stewart,3 Evgeni B. Dynkin and Wladimir A. Uspenski,4 and Stephen Wolframs have devoted articles to the marvelous relationship between elementary number theory and the geometrical patterns found in the Pascal triangle. In chapter 2 we introduced one example: the relation between the Pascal triangle and the Sierpinski gasket.
Mathematics is often defined as the science of space and number [...] It was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns.
Lynn Arthur Steen, 1988
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Reference
See figure 2.24 in chapter 2.
B. Bondarenko, Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs and Applications, Tashkent, Fan, 1990, in Russian.
Stewart, Game, Set, and Math, Basil Blackwell, Oxford, 1989.
E. B. Dynkin and W. Uspenski: Mathematische Unterhaltungen 11, VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.
S. Wolfram, Geometry of binomial coefficients, Amer. Math. Month. 91 (1984) 566–571.
For a more complete discussion see also M. Sved, Divisibility — With Visibility, Mathematical lntclligencer 10, 2 (1988) 56–64.
See also figure 2.26 in chapter 2.
See section 9.2 in Chapter 9.
T. Toffoli, N. Margolus, Cellular Automata Machines: A New Environment For Modelling, MIT Press, Cambridge, Mass., 1987.
F. v. Haeseler, H.-O. Peitgen, G. Skordev, Pascal’s triangle, dynamical systems and attractors,to appear in Ergodic Theory and Dynamical Systems.
F. v. Haeseler, H.-O. Peitgen, G. Skordev, On the hierarchical and global structure of cellular automata and attractors of dynamical systems,to appear.
E. E. Kummer, Über Ergänzungssätze zu den allgemeinen Reziprozitätsgesetzen, Journal für die reine und angewandte Mathematik 44 (1852) 93–146. For the result relevant to our discussion see pages 115–116.
t is related to several published criteria, like the one in I. Stewart, Game, Set, and Math,Basil Blackwell, Oxford, 1989, which Stewart attributes to Edouard Lucas following Gregory J. Chaitin’s book Algorithmic Information Theory,Cambridge University Press, 1987.24Convergence is with respect to the Hausdorff metric.
n this regard we also refer to S. J. Willson, Cellular automata can generate fractals,Discrete Applied Math. 8 (1984) 91–99. who studied limit sets of linear cellular automata via resealing techniques.
Sketching some recent work from F. v. Haeseler, H.-O. Peitgen, G. Skordev, Pascal’s triangle, dynamical systems and attractors,to appear in Ergodic Theory and Dynamical Systems.
See section 5.9.
R. Mauldin, S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829.
See J. Holte, A recurrence relation approach to fractal dimension in Pascal’s triangle, International Congress of Mathematics, 1990.
A. W. M. Dress, M. Gerhardt, N. I. Jaeger, P. J. Plath, H. Schuster, Some proposals concerning the mathematical modelling of oscillating heterogeneous catalytic reactions on metal surfaces. In L. Rensing and N. I. Jaeger (eds.), Temporal Order, Springer-Verlag, Berlin, 1984.
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© 1992 Springer Science+Business Media New York
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Peitgen, HO., Jürgens, H., Saupe, D. (1992). Pascal’s Triangle: Cellular Automata and Attractors. In: Chaos and Fractals. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4740-9_9
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