Another look at parastichies

Yeatts, F. Richard

doi:10.1016/s0025-5564(97)00015-1

Cited by 5 publications

(4 citation statements)

References 2 publications

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“…It is to search s for points where two parastichies are very close to be orthogonal. This relation which is deduced from results given by Yeatts [20] is: s = abs[ fufu+1 4γuγu+1 ] where γ u , related to approximants of τ , is γ u = 2π(λf u − f u−1 ). This relation give real values for s very close to f2u+1 4π but clearly different (for instance s ′ b ≃ 127.067 compared to s b ≃ 127.085).…”

Section: Grain Boundariesmentioning

confidence: 96%

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Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry

Sadoc

Rivier

Charvolin

2012

Acta Crystallogr A Found Crystallogr

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Phyllotaxis, the search for the most homogeneous and dense organizations of small disks inside a large circular domain, was first developed to analyze arrangements of leaves or florets in plants. Then it has become an object of study not only in botany, but also in mathematics, computer simulations and physics. Although the mathematical solution is now well known, an algorithm setting out the centers of the small disks on a Fermat spiral, the very nature of this organization and its properties of symmetry remain to be examined. The purpose of this paper is to describe a phyllotactic organization of points through its Voronoi cells and Delaunay triangulation and to refer to the concept of defects developed in condensed matter physics. The topological constraint of circular symmetry introduces an original inflation-deflation symmetry taking the place of the translational and rotational symmetries of classical crystallography.

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Section: Grain Boundariesmentioning

confidence: 96%

“…When Voronoi cells are more clearly hexagons two distances are close to ð3=5 1=2 Þ 1=2 ' 2:053. The behaviour of the distance between the two points defined by s and s þ f u as a function of s is obtained using a relation given by Yeatts (1997):…”

Section: Distances Between First Neighboursmentioning

confidence: 99%

Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry

Sadoc

Rivier

Charvolin

2012

Acta Crystallogr A Found Crystallogr

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“…In the grains where Voronoi cells are more clearly hexagons two distances are close to 3π √ 5 ≃ 2.053. The distance between the two points defined by s and s + f u as a function of s is obtained using the relation (see [4,14]):…”

Section: Metric Propertiesmentioning

confidence: 99%

“…2.053. The distance between the two points defined by s and s + f u as a function of s is obtained using the relation (see [10,19]):…”

Section: Metric Propertiesmentioning

confidence: 99%

Phyllotaxis on surfaces of constant Gaussian curvature

Sadoc

Charvolin

Rivier

2013

J. Phys. A: Math. Theor.

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A close packed organization with circular symmetry of a large number of small discs on a plane is obtained when the centres of the discs are distributed according to the algorithm of phyllotaxis. We study here the distributions obtained on surfaces of constant Gaussian curvatures, positive for the sphere or negative for the hyperbolic plane. We examine how the properties of homogeneity, isotropy and self-similarity typical of the distribution on the plane, and resulting from the presence of circular grain boundaries with quasicrystalline sequences, are affected by the curvature of the bearing surface. The quasicrystalline sequences of the grain boundaries appear indeed to be structural invariants, but the widths of the grains they separate vary differently with the curvature of the surface. The self similarity of the whole organization observed on the plane is therefore lost on the hyperbolic plane and the sphere. The evolutions of the local order within the grains show no differences except on the equatorial belt of the sphere where the isotropy is decreased owing to the symmetry of this finite surface around its equator.

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Phyllotaxis

Hellwig

Neukirchner

2009

Math. Semesterber.

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Another look at parastichies

Cited by 5 publications

References 2 publications

Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry

Phyllotaxis: a non-conventional crystalline solution to packing efficiency in situations with radial symmetry

Phyllotaxis on surfaces of constant Gaussian curvature

Phyllotaxis

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