Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

Grattan‐Guinness, I.

doi:10.4324/9780203014585

Cited by 26 publications

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“…We have noted that the proportion between stance and swing phase is close to ϕ , an irrational number (about 1.618034) already known in ancient Greece as “golden ratio” [18]. This number has been related to the problem reported by Euclid in III century BC to cut a given straight line so that the proportion between the shorter part to the longer one is the same as the longer part to the whole (see Section 2 for details) [18].…”

Section: Introductionmentioning

confidence: 99%

“…This number has been related to the problem reported by Euclid in III century BC to cut a given straight line so that the proportion between the shorter part to the longer one is the same as the longer part to the whole (see Section 2 for details) [18]. The Greek letter ϕ was chosen to indicate this number in honor of the sculptor Phidia, who supervised the construction of the Parthenon, the façade of which is a golden rectangle, that is, a rectangle having lengths in the proportion of ϕ [18, 19]. Henceforth, mathematicians, physicists, biologists, architects, and artists have been interested in the intrinsic symmetric properties of this number.…”

Section: Introductionmentioning

confidence: 99%

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The Golden Ratio of Gait Harmony: Repetitive Proportions of Repetitive Gait Phases

Iosa

Fusco

Marchetti

et al. 2013

BioMed Research International

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In nature, many physical and biological systems have structures showing harmonic properties. Some of them were found related to the irrational number ϕ known as the golden ratio that has important symmetric and harmonic properties. In this study, the spatiotemporal gait parameters of 25 healthy subjects were analyzed using a stereophotogrammetric system with 25 retroreflective markers located on their skin. The proportions of gait phases were compared with ϕ, the value of which is about 1.6180. The ratio between the entire gait cycle and stance phase resulted in 1.620 ± 0.058, that between stance and the swing phase was 1.629 ± 0.173, and that between swing and the double support phase was 1.684 ± 0.357. All these ratios did not differ significantly from each other (F = 0.870, P = 0.422, repeated measure analysis of variance) or from ϕ (P = 0.670, 0.820, 0.422, resp., t-tests). The repetitive gait phases of physiological walking were found in turn in repetitive proportions with each other, revealing an intrinsic harmonic structure. Harmony could be the key for facilitating the control of repetitive walking. Harmony is a powerful unifying factor between seemingly disparate fields of nature, including human gait.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Golden Ratio of Gait Harmony: Repetitive Proportions of Repetitive Gait Phases

Iosa

Fusco

Marchetti

et al. 2013

BioMed Research International

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“…1 With its 176 introductory articles the encyclopaedia [75] may be a useful guide, as it aspires to cover the history and philosophy of mathematics and logics up to around the 1930s; the contexts of all subjects and points raised in this article are covered.…”

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confidence: 99%

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

Grattan‐Guinness

2011

Log. Univers.

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A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: 'omnipresent' and 'multipresent' theories, and 'ubiquitous' notions that form dependent parts, or moments, of theories. The category of 'facets' is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. Mathematics Subject Classification (2000). Primary 00A30, 01A55, 01A60, 01A85, 03-03, 03-99; Secondary 00A35, 00A69, 00A71, 00A79, 03B10, 03C55, 03G05, 30-03, 31-03.Keywords. History and philosophy of pure and applied mathematics, history and philosophy of symbolic logics, parts and moments, generality of theories, theory change, monism and pluralism, sets and multisets, abstract algebras, metamathematics, model theory.

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“…Many physical and biological systems have structures that approximate Phi, an irrational number (about 1.618034), also known as the "golden ratio". The relation between Fibonacci numbers and the golden ratio is that the ratios of the successive numbers in the Fibonacci sequence converge on the golden ratio (Grattan-Guinness, 2002). Harmonic characteristics related to the golden ratio appear in the arrangement of leaves on a plant stem (Okabe, 2011), spiral structures of galaxies (Grattan-Guinness, 2002), pulse frequency of a star (Lindner et al, 2015), quantum phase transition (Coldea et al, 2010), nucleotide frequencies (Yamagishi & Shimabukuro, 2008), and cell (Staff et al, 2012) and shell (Gosling, 2008) growth.…”

Section: Introductionmentioning

confidence: 99%

“…The relation between Fibonacci numbers and the golden ratio is that the ratios of the successive numbers in the Fibonacci sequence converge on the golden ratio (Grattan-Guinness, 2002). Harmonic characteristics related to the golden ratio appear in the arrangement of leaves on a plant stem (Okabe, 2011), spiral structures of galaxies (Grattan-Guinness, 2002), pulse frequency of a star (Lindner et al, 2015), quantum phase transition (Coldea et al, 2010), nucleotide frequencies (Yamagishi & Shimabukuro, 2008), and cell (Staff et al, 2012) and shell (Gosling, 2008) growth. In human science, the golden ratio has been observed in body proportions (Ferring & Pancherz, 2008), bronchial airway segment bifurcations (Goldenberger, West, Dresselhaus, & Bhargava, 1985), gait phases of walking (Iosa et al, 2013), hair whorls (Paul, 2016), and aesthetic preference (Ricketts, 1982;Russell, 2000).…”

Section: Introductionmentioning

confidence: 99%

The Fibonacci Life Chart Method as a Predictor of Spiritual Experience

Sacco¹

2017

JEDP

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The Fibonacci Life Chart Method (FLCM) provides a framework linking development and spirituality. This study addressed the need for empirical research to test the hypotheses proposed by Sacco (Sacco, 2016). To address this problem, this study used case reports (N = 196) from the Alister Hardy Religious Experience Research Centre. The dynamical aspects of ages 11, 18, and 30 were examined as predictors of increased spiritual experience in adolescents and young adults. Results showed only ages 17 and 18 predicted a higher frequency of spiritual experience between ages 11 and 35. Age 18 was associated with a higher effect size (r = .27). This finding provides some empirical support for the FLCM as a predictor of spiritual experience, but not all hypotheses found support. Limitations to the study's design are discussed along with implications for future research.

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Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

Cited by 26 publications

References 0 publications

The Golden Ratio of Gait Harmony: Repetitive Proportions of Repetitive Gait Phases

The Golden Ratio of Gait Harmony: Repetitive Proportions of Repetitive Gait Phases

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

The Fibonacci Life Chart Method as a Predictor of Spiritual Experience

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