## 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 Wolfram^{s} 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.

## Keywords

Cellular Automaton Iterate Function System Binomial Coefficient Sierpinski Gasket Black Cell## Preview

Unable to display preview. Download preview PDF.

## Reference

- 1.See figure 2.24 in chapter 2.Google Scholar
- 2.B. Bondarenko,
*Generalized Pascal Triangles and Pyramids: Their Fractals*,*Graphs and Applications*, Tashkent, Fan, 1990, in Russian.MATHGoogle Scholar - 3.
- 4.E. B. Dynkin and W. Uspenski:
*Mathematische Unterhaltungen**11*, VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.Google Scholar - 5.S. Wolfram,
*Geometry of binomial coefficients*, Amer. Math. Month. 91 (1984) 566–571.CrossRefGoogle Scholar - 6.For a more complete discussion see also M. Sved,
*Divisibility — With Visibility*, Mathematical lntclligencer 10, 2 (1988) 56–64.CrossRefGoogle Scholar - 7.See also figure 2.26 in chapter 2.Google Scholar
- 14.See section 9.2 in Chapter 9.Google Scholar
- 15.T. Toffoli, N. Margolus,
*Cellular Automata Machines: A New Environment For Modelling*, MIT Press, Cambridge, Mass., 1987.Google Scholar - 19.F. v. Haeseler, H.-O. Peitgen, G. Skordev,
*Pascal’s triangle*,*dynamical systems and attractors*,to appear in Ergodic Theory and Dynamical Systems.Google Scholar - 20.F. v. Haeseler, H.-O. Peitgen, G. Skordev,
*On the hierarchical and global structure of cellular automata and attractors of dynamical systems*,to appear.Google Scholar - 21.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.Google Scholar
- 22.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.^{24}Convergence is with respect to the Hausdorff metric.Google Scholar - 25.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.Google Scholar - 26.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.Google Scholar - 27.See section 5.9.Google Scholar
- 28.R. Mauldin, S. C. Williams,
*Hausdorff dimension in graph directed constructions*, Trans. Amer. Math. Soc. 309 (1988) 811–829.MathSciNetMATHCrossRefGoogle Scholar - 29.See J. Holte,
*A recurrence relation approach to fractal dimension in Pascal’s triangle*, International Congress of Mathematics, 1990.Google Scholar - 31.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.Google Scholar