Book•
Beyond Measure: A Guided Tour Through Nature, Myth and Number
01 Jan 2002-
TL;DR: In the context of fractals, Chaos, plant growth and other Dynamical Systems: Self-Referential Systems Nature's Number System Number: Gray Code and the Towers of Hanoi Gray Code, Sets, and Logic Chaos Theory: A Challenge to Predictability Fractals Chaos and Fractals as discussed by the authors.
Abstract: Essays in Geometry and Number as They Arise in Nature, Music, Architecture and Design: The Spiral in Nature and Myth The Vortex of Life Harmonic Law The Projective Nature of the Musical Scale The Music of the Spheres Tangrams and Amish Quilts Linking Proportions, Architecture, and Music A Secret of Ancient Geometry The Hyperbolic Brunes Star The Hidden Pavements of the Laurentian Library Measure in Megalithic Britain The Flame-hand Letters of the Hebrew Alphabet Concepts Described in Part I Reappear in the Context of Fractals, Chaos, Plant Growth and Other Dynamical Systems: Self-Referential Systems Nature's Number System Number: Gray Code and the Towers of Hanoi Gray Code, Sets, and Logic Chaos Theory: A Challenge to Predictability Fractals Chaos and Fractals The Golden Mean Generalizations of the Golden Mean -- I Generalizations of the Golden Mean -- Il Polygons and Chaos Growth of Plants: A Study in Number Dynamical Systems.
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TL;DR: The bivariate Fibonacci and Lucas p –polynomials are studied and some properties of these polynomials are obtained.
51 citations
TL;DR: In this article, the m-extension of the Fibonacci and Lucas p-numbers (p⩾ ǫ 0 is integer and m à > 0 is real number) is defined and continuous functions for the m extension of these numbers are obtained using the generalized Binet formulas.
Abstract: In this article, we define the m-extension of the Fibonacci and Lucas p-numbers (p ⩾ 0 is integer and m > 0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers (p = 1, m = 1), Pell and Pell–Lucas numbers (p = 1, m = 2), Fibonacci and Lucas p-numbers (m = 1), Fibonacci m-numbers (p = 1), Pell and Pell–Lucas p-numbers (m = 2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden (p, m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.
38 citations
TL;DR: In this article, the authors find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point.
Abstract: In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pictures are drawn by using the parametric equations we found.
37 citations
TL;DR: The evidence presented here shows thatphyllotactic whorls of leaf homologues are not positioned in Fibonacci patterns, and the consensus starting to emerge from different subdisciplines in the phyllotaxis literature supports the alternative perspective that phyllOTactic patterns arise from local inhibitory interactions among the existing primordia already positioned at the shoot apex.
18 citations
TL;DR: In this paper, the authors discussed the terminology behind batik crafting and showed the aspects of self-similarity in its ornaments, and showed how the harmony of traditional crafting and modern computation could bring us a more creative aspects of the beautiful harmony inherited in the aesthetic aspects of batik craft.
Abstract: The paper discusses the terminology behind batik crafting and showed the aspects of self-similarity in its ornaments. Even though a product of batik cannot be reduced merely into its decorative properties, it is shown that computation can capture some interesting aspects in the batik-making ornamentation. There are three methods that can be exploited to the generative batik, i.e.: using fractal as the main source of decorative patterns, the hybrid batik that is emerged from the acquisition of L-System Thue-Morse algorithm for the harmonization within the grand designs by using both fractal images and traditional batik patterns, and using the random image tessellation as well as previous tiling algorithms for generating batik designs. The latest can be delivered by using a broad sources of motifs and traditionally recognized graphics. The paper concludes with certain aspects that shows how the harmony of traditional crafting and modern computation could bring us a more creative aspects of the beautiful harmony inherited in the aesthetic aspects of batik crafting.
10 citations
Cites background from "Beyond Measure: A Guided Tour Throu..."
...Somehow, the beauty in art and culturally growed aesthetics could also be interestingly discussed in the sense of fractal geometry (cf. Kappraff, 2002 & Malkevitch, 2003)....
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