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(,( )\I\IL". I \RII '. Readers' comments rDering substantialthetJretical and practicalcontributions to issues that have beenraised in textspublished in Leonardo are welcomed. The Editorsreservethe right to edit and sIwrtenletters. Letu!rsshould be written in English and sentto the Main Editorial Office. COMMENT ON "ON THE AEsTHETICS OF SIERPINSKI GASKETS FORMED FROM LARGE PASCAL'S TRIANGLES" Almost anyone can study the (Pascal's) triangle and discover moreproperties, but it is unlikely that they will benew. -Martin Gardner Pascal's triangle is almost universally famous, although it should more properly be called Khayyam's triangle after the Persian mathematicianphilosopher -poet who, in all probability , was its true discoverer. But such are the vagaries of history that even the Encyclopaedia Britannica does not mention it in the biographies of either Blaise Pascal or Omar Khayyam; I will, with no disrespect to Pascal, call it Khayyam's triangle here. Contrary to Gardner's assertion, new results keep on being discovered; indeed, a multitude of papers in the Fibonacci (btarterly attest to the richness of the morphology of this triangle . This is also amply borne out in Pickover's paper (Leonardo 23, No.4, 411-417, 1990). Attention in this connection is drawn to a recently published monograph by Bondarenko [1]. This work is as yet available only in the Russian language; it is devoutly wished that arrangements will be made for the publication of an English translation by some appropriate learned society in the near future. As Pickover has mentioned, computer graphics tools are very well suited for discovering the esthetics of Khayyam's triangle. In particular, he has shown the symmetries of the triangle modulo N, where N > 1 is a positive integer. In this respect, it must be mentioned that Khayyam's triangle modulo P, where P > 1 is prime, is a deterministic fractal with a well-defined similarity dimension [2]; a paper by Long is highly recommended [3]. Even more interesting are the generalizations [4,5] of Khayyam 's triangle, which yield intricate patterns when computed modulo N [6]. As a result of Pickover's paper, it is expected that readers will find exploration of Khayyam's triangle and its generalizations rewarding. I conclude with the following thought: While Khayyam's triangle modulo 1 contains only zeroes, the triangle modulo 00 contains all numbers except zeroes! AkhIesb Lakhtakia Department of Engineering Science and Mechanics Pennsylvania State University 227 Hammond Building University Park, PA 16802 U.SA References I. B. A. Bondarenko, Generalized Pascal's Triangles and Pyramids: Fractals, Graphs and Applirations (Tashkent: Acad. Nauk Uzbek., 1990). 2. N. S. Holter, A. Lakhtakia, V. K. Varadan. V. V. Varadan and R. Messier. "On a New Class of Planar Fractals: The Pascal-Sierpinski Gasket", journal ofPhysics A 19 (1986) pp. 1753-1759. 3. C. T. Long, "Pascal's Triangle Modulo P", Fibonacei Q!JartFrl~ 19 (1981) pp. 458-463. 4. G. N. Philippou, "Fibonacci Polynomials of Order K and Probability Distributions of Order K". Ph.D. diss.. University of Patras, Patras, Greece, 1984. 5. G. N-, Philippou and C. Georghiou, "FibonacciType Polynomials and Pascal Triangles of Order K", Fibonacci Q!Jarll!rly23, (1985) pp. 100-105. 6. A. Lakhtakia, R. Messier, V. K. Varadan and V. V. Varadan, "Fractal Structures Derivable from the Generalisations of the Pascal Triangle",journal ofPhysics A 20. (1987) pp. 1735-1738. COMMENT ON "How I CAME TO PAINT THE CRAB NEBULA: THE DEVELOPMENT OF COSMIC THEMES IN My OIL PAINTINGS" The underlying theme of Berta Golahny's Crab Nebula painting (Leonardo 23, No.4, 363-365,1990), and two later paintings on the same subject, is creation-specifically, a modern, twentieth-eentury understanding of the inception and continuing life of the universe from a scientific point of view. It is particularly appropriate that Golahny's painting of the Crab Nebula grew out of her Genesis series based on the biblical seven days of Creation. In seeking to make the Fourth Day ofCreation, the creation of the sun, moon and stars, more convincing, the artist consulted photographs of the cosmos and the latest discoveries in space exploration . This led directly to a new, scientific interpretation of creation in the phenomenon of a pulsing, exfoliating nebula, tinged with astral, saturated colors...


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