Scissors Congruence, the Golden Ratio and Volumes in Hyperbolic 5-Space

Kellerhals, Ruth

doi:10.1007/s00454-012-9397-5

Cited by 8 publications

(8 citation statements)

References 13 publications

(27 reference statements)

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“…One approach is based on Schläfli's formula (2.3), suitably adapted for hyperbolic five-polytopes. It allows us to express vol 5 ( 5 ) as a sum of simple integrals over dilogarithmic functions as follows (see [19] and [22,Sec. 4.2]).…”

Section: Volumes Of Some Hyperbolic Coxeter Polytopesmentioning

confidence: 99%

Hyperbolic Orbifolds of Minimal Volume

Kellerhals

2014

Comput. Methods Funct. Theory

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We provide a survey of hyperbolic orbifolds of minimal volume, starting with the results of Siegel in two dimensions and with the contributions of Gehring, Martin and others in three dimensions. For higher dimensions, we summarise some of the most important results, due to Belolipetsky, Emery and Hild, by discussing related features such as hyperbolic Coxeter groups, arithmeticity and consequences of Prasad's volume, as well as canonical cusps, crystallography and packing densities. We also present some new results about volume minimisers in dimensions six and eight related to Bugaenko's cocompact arithmetic Coxeter groups.

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Section: Volumes Of Some Hyperbolic Coxeter Polytopesmentioning

confidence: 99%

Hyperbolic Orbifolds of Minimal Volume

Kellerhals

2014

Comput. Methods Funct. Theory

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“…The golden ratio, denoted by , is an irrational number given by = (1 + √ 5)/2 [1]. The paper by Ackermann [2] may likely be the earliest literature on the golden ratio in a mathematics journal in English in 1895, but it attracted and has attracted the interest of scientists and engineers in various fields of sciences and engineering, ranging from chemistry to computer science; see, for example, [1], Benassi [3], Putz [4], Orita et al [5], Perez [6], Hassaballah et al [7], Kellerhals [8], Henein et al [9], Hurtley [10], Coldea et al [11], Affleck [12], Jones et al [13], Kaygn et al [14], Cervantes et al [15], Chebotarev [16], Benavoli et al [17], Manikantan et al [18], Assimakis et al [19], Good [20], Davis and Jahnke [21], Totland [22], Moufarrège [23], Boeyens [24], Iñiguez et al [25], Andrews and Zhang [26], Hofri and Rosberg [27], Itai and Rosberg [28], Cassandras and Julka [29], and Tanackov et al [30], just to mention a few.…”

Section: Instructionmentioning

confidence: 99%

Golden Ratio Phenomenon of Random Data Obeying von Karman Spectrum

Zhao

2013

Mathematical Problems in Engineering

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von Karman originally deduced his spectrum of wind speed fluctuation based on the Stokes-Navier equation. Taking into account, the practical issues of measurement and/or computation errors, we suggest that the spectrum can be described from the point of view of the golden ratio. We call it the golden ratio phenomenon of the von Karman spectrum. To depict that phenomenon, we derive the von Karman spectrum based on fractional differential equations, which bridges the golden ratio to the von Karman spectrum and consequently provides a new outlook of random data following the von Karman spectrum in turbulence. In addition, we express the fractal dimension, which is a measure of local self-similarity, using the golden ratio, of random data governed by the von Karman spectrum.

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“…as well as the results related to its series representations, has fascinated quite a number of mathematicians since the first half of the XVII century [57,58], both for its theoretical implications and practical applications [32,67]. Indeed, everything began when in 1644 the Italian mathematician Pietro Mengoli proposed the famous Basel problem in mathematical analysis, which also has relevance to number theory.…”

Section: Introductionmentioning

confidence: 99%

A single parameter Hermite–Padé series representation for Apéry’s constant

Soria-Lorente¹,

Berres²

2020

NNTDM

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Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for ζ (3) which only depends on one single integer parameter. This is accomplished by deducing a Hermite-Padé approximation problem using ideas of Sorokin (1998). As a consequence we get a new recurrence relation for the approximation of ζ(3) as well as a corresponding new continued fraction expansion for ζ(3), which do no reproduce Apéry's phenomenon, i.e., though the approaches are different, they lead to the same sequence of Diophantine approximations to ζ (3). Finally, the convergence rates of several series representations of ζ(3) are compared.

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Scissors Congruence, the Golden Ratio and Volumes in Hyperbolic 5-Space

Cited by 8 publications

References 13 publications

Hyperbolic Orbifolds of Minimal Volume

Hyperbolic Orbifolds of Minimal Volume

Golden Ratio Phenomenon of Random Data Obeying von Karman Spectrum

A single parameter Hermite–Padé series representation for Apéry’s constant

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