Golden, , and π Flowers: A Spiral Story

Naylor, Michael

doi:10.1080/0025570x.2002.11953126

Cited by 17 publications

(20 citation statements)

References 1 publication

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“…The integral ∞ 0 t x−1 e −t dt satisfies also the difference equation (18) which can be shown by integration by parts. Hence ∞ 0 t n−1 e −t dt = (n − 1)!.…”

Section: The Gamma Function γmentioning

confidence: 85%

“…2 Thus we have proven (21) for 0 ≤ x ≤ 1. For x > 1 condition (21) follows by repeated application of the difference equation (18). 2 For the estimate of the last limit one could also use Stirling's formula, see [15].…”

Section: The Gamma Function γmentioning

confidence: 99%

“…In [18] a spiral of seeds in the sun flower is described, where in that paper the exciting properties of the discrete points of the spirals are exhibited. For these points we get the recursion formula z n+1 = 1 + 1 n + 1 · e ib · z n , n ∈ N 0 , z 0 = 1, with a fixed real b.…”

Section: Sun Flower Spiralmentioning

confidence: 99%

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Normal solutions of difference equations, Euler?s functions and spirals

Gronau¹

2004

Aequationes Math.

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We consider the linear first order difference equationfor the unknown function f : I ⊆ R → C. We present, under some conditions on the given function α, a product representation of a specific solution of the considered difference equation that can be understood as the normal or principal solution in the concept of N. E. Nörlund, F. John, W. Krull and others. This normal solution is characterized by its asymptotic behavior near infinity. In that way we get a characterization of e.g. the gamma function, the q-gamma function and of several classical spirals.

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“…The integral ∞ 0 t x−1 e −t dt satisfies also the difference equation (18) which can be shown by integration by parts. Hence ∞ 0 t n−1 e −t dt = (n − 1)!.…”

Section: The Gamma Function γmentioning

confidence: 85%

Section: The Gamma Function γmentioning

confidence: 99%

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Normal solutions of difference equations, Euler?s functions and spirals

Gronau¹

2004

Aequationes Math.

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“…GA spiral arrays are obtained by a simple generation rule, expressed in polar coordinates r and  , first proposed by Vogel in order to approximate the complex arrangements of florets in the sunflower head in two spatial dimensions [25,[27][28][29]:…”

Section: Plasmon-enhanced Large-angle Scattering In Ga Spiral Arraysmentioning

confidence: 99%

Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells

Trevino¹,

Forestiere²,

Martino³

et al. 2012

Opt. Express

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We report on the design, fabrication and measurement of ultra-thin film Silicon On Insulator (SOI) Schottky photo-detector cells with nanostructured plasmonic arrays, demonstrating broadband enhanced photocurrent generation using aperiodic golden angle spiral geometry. Both golden angle spiral and periodic arrays of various center-to-center particle spacing were investigated to optimize the photocurrent enhancement. The primary photocurrent enhancement region is designed for the spectral range 600nm-950nm, where photon absorption in Si is inherently poor. We demonstrate that cells coupled to spiral arrays exhibit higher photocurrent enhancement compared to optimized periodic gratings structures. The findings are supported through coupled-dipole numerical simulations of radiation diagrams and finite difference time domain simulations of enhanced absorption in Si thin-films.

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“…Logarithmic spirals such as golden spirals (with a growth factor of φ = 1.6180339887, the golden ratio) are ubiquitous in nature and are seen in arrangements of leaves, seeds, pinecones and many different arrangements in nature. 1 Such spiral patterns are frequently observed and utilized in a variety of phenomena including galaxies, biological organisms, as well as turbulent flows. 2 Many have attributed mysticism to the presence of origins -the golden ratio was called τ (tau), the symbol of life, and was also known to ancient Egyptians (The ankh) and Hindus.…”

Section: Introductionmentioning

confidence: 99%

Spirals, Scalp Whorls and Skin Biomechanics: Nature’s Own Design For Expansion

Paul

2016

Preprint

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This paper began as an exercise in curiosity -logarithmic spiral designs abound in nature --in galaxies, flowers, pinecones and on human scalps as whorls. Why are humans the only primates to have whorls on the scalp? Is the formation of scalp whorls mechanical or genetic? A mechanical theory has long been postulated-the mechanical theory suggests that hair whorl patterning is determined by the tension on the epidermis during rapid expansion of the cranium while the hair follicle is growing downwards -however, this has never before been experimentally proven conclusively. We found, that under certain conditions, we were able to experimentally recreate spirals on the scalp to demonstrate that the basis of scalp whorls is indeed mechanical -and that logarithmic spirals are indeed nature's own design for rapid expansion of organic tissues. Given our experiments only created whorls when certain conditions were satisfied (and not in others), they have given us great insight into the mechanical formation of skin whorls and the physiology of skin stretch. We believe that these findings will lead to many more advances in understanding skin dynamics and indeed the behavior of any living tissue when confronted by stretch. As a corollary, the application of the results of these studies have led us to the discovery of a new surgical technique for closure of scalp defects using the golden spiral pattern, and this will be the subject of a separate paper.

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Golden, , and π Flowers: A Spiral Story

Cited by 17 publications

References 1 publication

Normal solutions of difference equations, Euler?s functions and spirals

Normal solutions of difference equations, Euler?s functions and spirals

Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells

Spirals, Scalp Whorls and Skin Biomechanics: Nature’s Own Design For Expansion

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