Symmetry in plants: from Bravais' lattices to an explanation of Fibonacci phyllotaxis.

In 1837, fourteen years before publishing his seminal Etude sur la cristallographie Auguste Bravais and his brother Louis wrote an equally seminal work on the arrangement of leaves around the stem of a plant. In this paper, one of the very first truly bio-mathematical work, they introduce and analyze cylindrical lattices and conjecture that only those with the golden angle between successive leaves can exhibit the Fibonacci numbers of spirals predominant in plants. With the advent of the microscope, and following observations of the plants growing tips by Hofmeister, botanists Schwendener and van Iterson developed an accretion model of the plant structures. Their work use ideas of what we now would call renormalization of morphogenetic fronts to understand transitions between successive Fibonacci pairs. This gives rise to a simple explanation of the omni-presence of Fibonacci numbers in plants, which can be verified on digitized plant samples and with systematic computer simulations. This could inform crystallographers in their study of dislocations.

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Fibonacci Numbers, Golden Section, Golden Angle, Golden Rectangle and Golden Spiral

10.1017/9789384463137.008 ◽

2014 ◽

pp. 125-138

Author(s):

Asok Kumar Mallik

Keyword(s):

Golden Section ◽

Fibonacci Numbers ◽

Golden Angle ◽

Golden Rectangle

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The Rules of the Flock

The Rules of the Flock ◽

10.1093/oso/9780198853398.003.0006 ◽

2020 ◽

pp. 34-41

Author(s):

Helmut Satz

Keyword(s):

Low Temperature ◽

Mathematical Models ◽

Computer Simulations ◽

Empirical Studies ◽

Random Motion ◽

Spin Systems ◽

Behavior Patterns ◽

Seminal Work ◽

Flight Direction

Following the seminal work of T. Vicsek et al. (Budapest), mathematical models are formulated, based on next neighbor interactions (alignment of flight direction) leading to global correlations. Computer simulations of these models lead to behavior patterns very much like those observed in empirical studies of bird flocks. In particular, a transition from random motion to flock behavior is observed for sufficiently precise flight alignment, corresponding to sufficiently low temperature in spin systems.

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Dynamic MRI of swallowing: real-time volumetric imaging at 12 frames per second at 3 T

Magnetic Resonance Materials in Physics Biology and Medicine ◽

10.1007/s10334-021-00973-6 ◽

2021 ◽

Author(s):

Luuk Voskuilen ◽

Jasper Schoormans ◽

Oliver J. Gurney-Champion ◽

Alfons J. M. Balm ◽

Gustav J. Strijkers ◽

...

Keyword(s):

Image Quality ◽

Real Time ◽

Computer Simulations ◽

Dynamic Mri ◽

Clinical Problem ◽

Frame Rate ◽

Volumetric Imaging ◽

Golden Angle ◽

3D Acquisition ◽

Stack Of Stars

Abstract Objective Dysphagia or difficulty in swallowing is a potentially hazardous clinical problem that needs regular monitoring. Real-time 2D MRI of swallowing is a promising radiation-free alternative to the current clinical standard: videofluoroscopy. However, aspiration may be missed if it occurs outside this single imaged slice. We therefore aimed to image swallowing in 3D real time at 12 frames per second (fps). Materials and methods At 3 T, three 3D real-time MRI acquisition approaches were compared to the 2D acquisition: an aligned stack-of-stars (SOS), and a rotated SOS with a golden-angle increment and with a tiny golden-angle increment. The optimal 3D acquisition was determined by computer simulations and phantom scans. Subsequently, five healthy volunteers were scanned and swallowing parameters were measured. Results Although the rotated SOS approaches resulted in better image quality in simulations, in practice, the aligned SOS performed best due to the limited number of slices. The four swallowing phases could be distinguished in 3D real-time MRI, even though the spatial blurring was stronger than in 2D. The swallowing parameters were similar between 2 and 3D. Conclusion At a spatial resolution of 2-by-2-by-6 mm with seven slices, swallowing can be imaged in 3D real time at a frame rate of 12 fps.

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Fibonacci or quasi-symmetric phyllotaxis. Part I: why?

Acta Societatis Botanicorum Poloniae ◽

10.5586/asbp.3533 ◽

2016 ◽

Vol 85 (4) ◽

Cited By ~ 2

Author(s):

Christophe Golé ◽

Jacques Dumais ◽

Stéphane Douady

Keyword(s):

Fibonacci Numbers ◽

Companion Paper ◽

Disk Accretion ◽

Elementary Model ◽

Data Analyses ◽

Biological Studies ◽

Accretion Model ◽

Square Packing ◽

Plant Data

The study of phyllotaxis has focused on seeking explanations for the occurrence of consecutive Fibonacci numbers in the number of helices paving the stems of plants in the two opposite directions. Using the disk-accretion model, first introduced by Schwendener and justified by modern biological studies, we observe two distinct types of solutions: the classical Fibonacci-like ones, and also more irregular configurations exhibiting nearly equal number of helices in a quasi-square packing, the quasi-symmetric ones, which are a generalization of the whorled patterns. Defining new geometric tools allowing to work with irregular patterns and local transitions, we provide simple explanations for the emergence of these two states within the same elementary model. A companion paper will provide a wide array of plant data analyses that support our view.

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A High Resolution Study of G.P. Zones in Aluminum - 4.2 at % Silver

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100055904 ◽

1982 ◽

Vol 40 ◽

pp. 722-723 ◽

Cited By ~ 1

Author(s):

R. Gronsky

Keyword(s):

Electron Microscopy ◽

Transmission Electron Microscopy ◽

High Resolution ◽

Computer Simulations ◽

Structural Characteristics ◽

X Ray Diffraction ◽

X Ray ◽

Transmission Electron ◽

Resolution Study

The phenomenon of clustering in Al-Ag alloys has been extensively studied since the early work of Guinierl, wherein the pre-precipitation state was characterized as an assembly of spherical, ordered, silver-rich G.P. zones. Subsequent x-ray and TEM investigations yielded results in general agreement with this model. However, serious discrepancies were later revealed by the detailed x-ray diffraction - based computer simulations of Gragg and Cohen, i.e., the silver-rich clusters were instead octahedral in shape and fully disordered, atleast below 170°C. The object of the present investigation is to examine directly the structural characteristics of G.P. zones in Al-Ag by high resolution transmission electron microscopy.

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Multislice calculations for quasi-periodic and non-periodic objects

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100173820 ◽

1990 ◽

Vol 48 (4) ◽

pp. 138-139

Author(s):

R. Herrera ◽

A. Gómez

Keyword(s):

Electron Diffraction ◽

Computer Simulations ◽

Periodic Table ◽

Amorphous Materials ◽

Space Group ◽

Essential Step ◽

Shape Effects ◽

Atomic Scattering ◽

Diffraction Patterns ◽

Periodic Continuation

Computer simulations of electron diffraction patterns and images are an essential step in the process of structure and/or defect elucidation. So far most programs are designed to deal specifically with crystals, requiring frequently the space group as imput parameter. In such programs the deviations from perfect periodicity are dealt with by means of “periodic continuation”.However, for many applications involving amorphous materials, quasiperiodic materials or simply crystals with defects (including finite shape effects) it is convenient to have an algorithm capable of handling non-periodicity. Our program “HeGo” is an implementation of the well known multislice equations in which no periodicity assumption is made whatsoever. The salient features of our implementation are: 1) We made Gaussian fits to the atomic scattering factors for electrons covering the whole periodic table and the ranges [0-2]Å−1 and [2-6]Å−1.

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Computer simulations for the TEM bend contour technique

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100163137 ◽

1996 ◽

Vol 54 ◽

pp. 134-135

Author(s):

Vladimir Yu. Kolosov ◽

Anders R. Thölén

Keyword(s):

Computer Simulations ◽

Real Space ◽

Advantages And Disadvantages ◽

Short Overview ◽

Diffraction Contrast ◽

Beam Method ◽

Convergent Beam ◽

Contour Technique

In this paper we give a short overview of two TEM applications utilizing the extinction bend contour technique (BC) giving the advantages and disadvantages; especially we consider two areas in which the BC technique remains unique. Special attention is given to an approach including computer simulations of TEM micrographs.BC patterns are often observed in TEM studies but are rarely exploited in a serious way. However, this type of diffraction contrast was one of the first to be used for analysis of imperfections in crystalline foils, but since then only some groups have utilized the BC technique. The most extensive studies were performed by Steeds, Eades and colleagues. They were the first to demonstrate the unique possibilities of the BC method and named it real space crystallography, which developed later into the somewhat similar but more powerful convergent beam method. Maybe, due to the difficulties in analysis, BCs have seldom been used in TEM, and then mainly to visualize different imperfections and transformations.

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