•

1 like•138 views

This paper analyzes the fractal properties of a one-dimensional Fibonacci chain in Fourier space. The authors find that plotting successive iterations of the Fibonacci chain reveals a cardioid shape, resembling the main pattern in the Mandelbrot set. This cardioid decreases in size following each iteration by approximately half the golden ratio. Minor modifications to the Fibonacci chain disrupt this pattern, showing it depends precisely on the substitution rules used to generate the chain. Only changing the ratio of segment lengths preserves the pattern's scaling. This fractal signature may correspond to quasiperiodic gearing in a geometric chuck that can draw similar cycloidal shapes.

Report

Share

Report

Share

Download to read offline

BT631-13-x-ray_crystallography_crystal_symmetry

The document discusses crystal structures and their characterization. It covers:
1. Crystals are composed of a repeating motif arranged in a three-dimensional lattice. The motif defines the structural unit and the lattice defines the geometric relationship between motifs.
2. Characterization of crystals includes determining their quality via X-ray diffraction patterns, estimating the number of molecules per unit cell using Matthews coefficient, and deriving the unit cell dimensions and space group from diffraction patterns.
3. Five basic plane lattices exist based on the geometry of the fundamental lattice cell: oblique, rectangular, square, hexagonal, and rectangular I. The combination of lattice symmetry and translation defines the plane crystal system and plane groups.

Group theory - Part -1

The document provides an introduction to group theory and symmetry elements in chemistry. It defines symmetry elements like proper axes of rotation, planes of symmetry, centers of inversion, and improper axes of rotation. It explains how symmetry operations like rotations and reflections combine to form point groups, which describe the symmetry of molecules. Examples of common point groups like C2v, C3v, D4h, and D6h are given with molecular structures that belong to each group. The document outlines the process of assigning point groups to molecules and includes examples of molecules from various groups.

Methods for Calculating Empires in Quasicrystals

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

Group theory questions and answers

This document discusses group theory and symmetry elements as they relate to several different molecules. It provides examples of identifying the point groups, symmetry elements like rotation axes and planes of inversion, and determining the representations of atomic and molecular orbitals for molecules like ammonia, acetone, ethanediol, propadiene, water, BH3, cyclopropenyl cation, butadiene, and trichlorborane. Worked examples are provided to demonstrate how to analyze symmetry properties and construct molecular orbital diagrams for various systems.

Gf3111931199

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

3.6 notes

This document contains theorems about perpendicular and parallel lines, including:
1) If two lines intersect to form congruent angles, then the lines are perpendicular.
2) If two lines are perpendicular, then they intersect to form four right angles.
3) If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
It also discusses finding the distance between a point and a line, and between two parallel lines using perpendicular segments and the distance formula.

Solid state chemistry

This document discusses different types of solids and their crystal structures. It describes crystalline solids as having long-range periodic atomic arrangements, while amorphous solids lack long-range order. Polycrystalline solids consist of many small crystallites. Common crystal structures include body-centered cubic, face-centered cubic, and hexagonal close-packed arrangements. Defects in crystal structures like point defects and dislocations are also summarized.

Crystal for teaching

1) Crystals exhibit flat faces that can be euhedral, subhedral, or anhedral depending on how perfectly the faces are crystallized. Crystals range in size from macrocrystalline to microcrystalline to nanocrystalline.
2) Crystal morphology is described by the intercepts of crystal faces on the crystallographic axes. Miller indices provide a notation for crystal directions and planes using these intercept ratios.
3) There are laws governing crystal symmetry and angles between faces that are constant for a given crystalline phase. Crystal structures can be obscured by distortions from the true symmetry.

BT631-13-x-ray_crystallography_crystal_symmetry

The document discusses crystal structures and their characterization. It covers:
1. Crystals are composed of a repeating motif arranged in a three-dimensional lattice. The motif defines the structural unit and the lattice defines the geometric relationship between motifs.
2. Characterization of crystals includes determining their quality via X-ray diffraction patterns, estimating the number of molecules per unit cell using Matthews coefficient, and deriving the unit cell dimensions and space group from diffraction patterns.
3. Five basic plane lattices exist based on the geometry of the fundamental lattice cell: oblique, rectangular, square, hexagonal, and rectangular I. The combination of lattice symmetry and translation defines the plane crystal system and plane groups.

Group theory - Part -1

The document provides an introduction to group theory and symmetry elements in chemistry. It defines symmetry elements like proper axes of rotation, planes of symmetry, centers of inversion, and improper axes of rotation. It explains how symmetry operations like rotations and reflections combine to form point groups, which describe the symmetry of molecules. Examples of common point groups like C2v, C3v, D4h, and D6h are given with molecular structures that belong to each group. The document outlines the process of assigning point groups to molecules and includes examples of molecules from various groups.

Methods for Calculating Empires in Quasicrystals

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

Group theory questions and answers

This document discusses group theory and symmetry elements as they relate to several different molecules. It provides examples of identifying the point groups, symmetry elements like rotation axes and planes of inversion, and determining the representations of atomic and molecular orbitals for molecules like ammonia, acetone, ethanediol, propadiene, water, BH3, cyclopropenyl cation, butadiene, and trichlorborane. Worked examples are provided to demonstrate how to analyze symmetry properties and construct molecular orbital diagrams for various systems.

Gf3111931199

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

3.6 notes

This document contains theorems about perpendicular and parallel lines, including:
1) If two lines intersect to form congruent angles, then the lines are perpendicular.
2) If two lines are perpendicular, then they intersect to form four right angles.
3) If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
It also discusses finding the distance between a point and a line, and between two parallel lines using perpendicular segments and the distance formula.

Solid state chemistry

This document discusses different types of solids and their crystal structures. It describes crystalline solids as having long-range periodic atomic arrangements, while amorphous solids lack long-range order. Polycrystalline solids consist of many small crystallites. Common crystal structures include body-centered cubic, face-centered cubic, and hexagonal close-packed arrangements. Defects in crystal structures like point defects and dislocations are also summarized.

Crystal for teaching

1) Crystals exhibit flat faces that can be euhedral, subhedral, or anhedral depending on how perfectly the faces are crystallized. Crystals range in size from macrocrystalline to microcrystalline to nanocrystalline.
2) Crystal morphology is described by the intercepts of crystal faces on the crystallographic axes. Miller indices provide a notation for crystal directions and planes using these intercept ratios.
3) There are laws governing crystal symmetry and angles between faces that are constant for a given crystalline phase. Crystal structures can be obscured by distortions from the true symmetry.

James DAVIES

This document contains the resume of James Davies. It outlines his education, including a Bachelor of Engineering from the University of Technology Sydney, as well as his professional experience as a Site Engineer at Hyne Timber and prior roles in research and manufacturing. It also lists his technical skills in areas like project management, process engineering, software development, and CAD programs. Personal details and referees are provided at the end.

Prevención de accidentes en la infancia

Este documento habla sobre la perspectiva de Emmi Pikler sobre la prevención de accidentes en la infancia. Pikler creía que los niños deben tener libertad de movimiento y no ser sobreprotegidos. Los niños aprenden mejor a evaluar sus propias capacidades y limitaciones a través de la experiencia. Los padres deben observar a los niños sin intervenir y crear un ambiente seguro donde los niños puedan explorar y descubrir por sí mismos.

Football, great grandpa, I …

The document is a story from a boy about his great grandfather who was an avid football fan. It describes an old photo of the boy's great grandfather in Germany 60 years ago. The boy has now played football for 4 years as part of a team with the motto "All for one, one for all". The story discusses their first football match and their first international match playing against a team from Norway, where they earned their first medals.

Revista a! 37 - Lo último en tecnología

El documento describe una exhibición en el Museo de la Ciudad que muestra más de 100 artefactos del hogar de entre 1940 y 1970. Durante ese período hubo una transformación tecnológica donde los aparatos incorporaron materiales más livianos y baratos como el aluminio, haciéndolos más económicos y accesibles. También la invención del transistor los hizo más pequeños, menos frágiles y de mejor calidad de sonido. La exhibición busca recordar los hábitos y estilos de vida de la época.

Exposicionambiental

El documento describe el primer encuentro de proyectos ambientales escolares en la zona 10 de Envigado. Presenta el proyecto ECOIDEAS del Colegio Manuel Mejía Vallejo, el cual se ha implementado durante los últimos cuatro años con un enfoque multidisciplinario y de emprendimiento. El proyecto aborda el manejo de residuos sólidos en la institución y promueve ideas de emprendimiento utilizando materiales reciclables. Su propósito es fomentar una cultura ambiental y convertir la escuela en una "Ecoescuela" a

Esams resume2015-1

This document is a resume for Esam Abdelgadir. It summarizes his career objective to obtain a leadership role in biotechnology manufacturing or process development. It then outlines his 17 years of experience in cell culture process development and manufacturing roles at various biotech companies, including optimizing cell culture processes, supervising researchers, implementing changes to processes and facilities, and training and mentoring associates. His experience includes work with bioreactors from lab scale to 1000L, disposable technologies, analytics, and validation.

49694929 instructivo-test-5-6

Este documento presenta las instrucciones para administrar el Test 5-6, el cual evalúa las habilidades fundamentales de los niños antes de comenzar la educación escolar formal. El test consta de 4 secciones que exploran el lenguaje, la percepción, las habilidades precálculo y la motricidad fina. Se instruye al evaluador sobre cómo aplicar cada sección del test de manera individualizada o en grupo, y cómo calificar los resultados para determinar el nivel de maduración del niño.

Dc machines 1

DC machines can operate from a DC source and either generate mechanical energy (DC generator) or convert mechanical energy to DC electricity (DC motor). A DC machine consists of a yoke, poles, field winding, armature core and winding, and commutator. In a DC motor, current flowing through the armature winding inside the magnetic field generated by the field winding experiences a rotational force, causing the armature to rotate. DC machines can be separately excited, self-excited, or series/compound excited. Common applications include DC motors used in industrial equipment and vehicles.

Adulto mayor

Este documento proporciona instrucciones para realizar una entrevista a un adulto mayor que aborde aspectos biopsicosociales. La entrevista debe incluir una ficha inicial con datos demográficos y de salud, una entrevista semidirigida sobre aspectos físicos, psicológicos y de inserción social, y un análisis final sobre la aplicación de teorías del apego y desapego y la utilidad de la entrevista para la formación médica del estudiante. El objetivo es investigar las características del adulto mayor

Dcmachines 2016

The document describes the construction and operation of DC machines. It contains the following key points:
1. A DC machine consists of a stator with poles that produce a magnetic field and a rotor with coils connected by a commutator. The interaction between the magnetic field and current in the rotor coils causes the rotor to rotate.
2. In a DC motor, the stator poles are supplied with DC current to generate a magnetic field. The rotor coils are also supplied with DC current, causing the rotor to rotate due to Lorentz forces. The commutator reverses the current to maintain rotation.
3. In a DC generator, a mechanical input rotates the rotor coils in the stator's

Pengertian Rukun Solat

Pendidikan Islam Tahun 4 KSSR

Dc machines 1

The document discusses DC machines which can operate as either motors or generators depending on the direction of power flow. A DC machine consists of a static electromagnetic or permanent magnetic field and a rotating armature. The field produces a magnetic medium and the armature produces voltage and torque under the action of the magnetic field. An advantage of DC motors is their speed is easy to control over a wide range, while DC generators are now quite rare. Most DC machines have AC voltages and currents internally but produce DC outputs via a commutator mechanism.

Músculos y huesos del miembro superior

El documento describe los huesos del miembro superior humano, que se divide en 4 partes: cintura escapular, brazo, antebrazo y mano. En total contiene 32 huesos. Describe cada hueso y su localización en detalle. También menciona los principales músculos de cada parte del miembro superior y sus acciones.

5 estructura del programa de formacion cívica 3°

Este documento presenta la estructura y propósitos del programa de Formación Cívica y Ética para tercer grado de educación básica en México. Describe cuatro objetivos para los estudiantes, los principios que orientan la asignatura, sus tres ejes formativos, ámbitos de aprendizaje y ocho competencias cívicas y éticas que se desarrollan de manera progresiva.

X-Ray Topic.ppt

X-ray diffraction is a technique used to analyze the crystal structure of materials. It works by firing X-rays at a crystalline sample and observing the scattered rays, which are determined by the positions of atoms in the crystal structure. Bragg's law describes the conditions under which constructive interference of X-rays scattered from crystal planes occurs, producing a diffraction pattern. Analysis of diffraction patterns can provide information about a material's lattice structure, symmetry, and chemical composition. Miller indices are used to describe crystallographic planes and directions within a crystal lattice.

Metallic phase with long range phys rev lett.53.1951

Metallic Phase with Long-Range Orientational Order and No Translational Symmetry
D. Shechtman and I. Blech
Department of Materials Engineering, Israel Institute of Technology Technion, 3200 Haifa, Israel
and
D. Gratias
Centre d'Etudes de Chimie Metallurgique, Centre National de la Recherche Scientiftque, F-94400 Vitry, France
and
J. W. Cahn
Center for Materials Science, National Bureau ofStandards, Gaithersburg, Maryland 20760
(Received 9 October 1984)
We have observed a metallic solid (Al —14-at. /o-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.

2012 NComms - Geometrically-locked vortex lattices in semiconductor quantum f...

The document summarizes research on generating and observing stable vortex lattices in polariton condensates in semiconductor microcavities. Researchers pumped three spots in a triangular geometry and observed the formation of a honeycomb lattice containing up to 100 vortices and antivortices extending over tens of microns. Numerical simulations matched the experimental observations and showed the lattice forms due to ferromagnetic coupling between the condensates at each pumped spot, with the phase locked by the imposed triangular geometry. The vortex lattice was stable for many minutes and highly sensitive to the optically imposed geometry.

A_I_Structure.pdf

This document provides an overview of solid state structure and crystallography. It discusses the types of solids, order in solids, symmetry in crystals including translation symmetry and the resulting crystal systems in 2D and 3D. It then covers the Bravais lattices that arise in 2D and 3D from the crystal systems and additional translational symmetries. Key concepts covered include Miller planes, X-ray diffraction including Bragg's law and structure factor, and common crystal structure motifs. The document is intended as a learning aid for students taking a chemistry of materials course.

99995069.ppt

Crystal structure determination uses X-ray diffraction to analyze the arrangement of atoms in crystals. X-rays are diffracted by the periodic lattice of a crystal in predictable ways. Bragg's law describes the conditions under which constructive interference occurs between X-rays reflected from different crystal lattice planes, producing intense diffracted beams. By measuring the angles and intensities of these diffracted beams, researchers can determine the size and shape of the unit cell and deduce the positions of atoms within the cell. The reciprocal lattice formalism relates diffraction phenomena to the periodicity of the crystal lattice.

Quantum chaos in clean many-body systems - Tomaž Prosen

This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.

Fourier transform in X-ray crystallography .ppt

The Fourier transform connects the diffraction pattern of X-rays scattered by a crystal to the electron density distribution within the crystal's unit cell. The structure factors obtained from diffraction intensities represent the Fourier transform of the electron density. However, the phases of the structure factors are lost during data collection, posing the phase problem. Techniques like isomorphous replacement, multiple wavelength anomalous diffraction, and molecular replacement aim to determine the missing phase information and allow calculation of electron density maps revealing a crystal's structure.

SRS to study Polytype NNN relationships

This document discusses the use of synchrotron edge topography to study the relationships between different polytypes in silicon carbide (SiC). It provides background on polytypism in SiC, including different theories proposed to explain its occurrence. The document explains that synchrotron radiation is well-suited to study polytypes due to its high resolution. It describes how edge topography allows determining the spatial extent of disorder between polytypes and deducing their neighborhood relationships by superimposing Laue reflections. The technique reveals finer features at polytype boundaries like long period polytypes and one-dimensionally disordered layers.

James DAVIES

This document contains the resume of James Davies. It outlines his education, including a Bachelor of Engineering from the University of Technology Sydney, as well as his professional experience as a Site Engineer at Hyne Timber and prior roles in research and manufacturing. It also lists his technical skills in areas like project management, process engineering, software development, and CAD programs. Personal details and referees are provided at the end.

Prevención de accidentes en la infancia

Este documento habla sobre la perspectiva de Emmi Pikler sobre la prevención de accidentes en la infancia. Pikler creía que los niños deben tener libertad de movimiento y no ser sobreprotegidos. Los niños aprenden mejor a evaluar sus propias capacidades y limitaciones a través de la experiencia. Los padres deben observar a los niños sin intervenir y crear un ambiente seguro donde los niños puedan explorar y descubrir por sí mismos.

Football, great grandpa, I …

The document is a story from a boy about his great grandfather who was an avid football fan. It describes an old photo of the boy's great grandfather in Germany 60 years ago. The boy has now played football for 4 years as part of a team with the motto "All for one, one for all". The story discusses their first football match and their first international match playing against a team from Norway, where they earned their first medals.

Revista a! 37 - Lo último en tecnología

El documento describe una exhibición en el Museo de la Ciudad que muestra más de 100 artefactos del hogar de entre 1940 y 1970. Durante ese período hubo una transformación tecnológica donde los aparatos incorporaron materiales más livianos y baratos como el aluminio, haciéndolos más económicos y accesibles. También la invención del transistor los hizo más pequeños, menos frágiles y de mejor calidad de sonido. La exhibición busca recordar los hábitos y estilos de vida de la época.

Exposicionambiental

El documento describe el primer encuentro de proyectos ambientales escolares en la zona 10 de Envigado. Presenta el proyecto ECOIDEAS del Colegio Manuel Mejía Vallejo, el cual se ha implementado durante los últimos cuatro años con un enfoque multidisciplinario y de emprendimiento. El proyecto aborda el manejo de residuos sólidos en la institución y promueve ideas de emprendimiento utilizando materiales reciclables. Su propósito es fomentar una cultura ambiental y convertir la escuela en una "Ecoescuela" a

Esams resume2015-1

This document is a resume for Esam Abdelgadir. It summarizes his career objective to obtain a leadership role in biotechnology manufacturing or process development. It then outlines his 17 years of experience in cell culture process development and manufacturing roles at various biotech companies, including optimizing cell culture processes, supervising researchers, implementing changes to processes and facilities, and training and mentoring associates. His experience includes work with bioreactors from lab scale to 1000L, disposable technologies, analytics, and validation.

49694929 instructivo-test-5-6

Este documento presenta las instrucciones para administrar el Test 5-6, el cual evalúa las habilidades fundamentales de los niños antes de comenzar la educación escolar formal. El test consta de 4 secciones que exploran el lenguaje, la percepción, las habilidades precálculo y la motricidad fina. Se instruye al evaluador sobre cómo aplicar cada sección del test de manera individualizada o en grupo, y cómo calificar los resultados para determinar el nivel de maduración del niño.

Dc machines 1

DC machines can operate from a DC source and either generate mechanical energy (DC generator) or convert mechanical energy to DC electricity (DC motor). A DC machine consists of a yoke, poles, field winding, armature core and winding, and commutator. In a DC motor, current flowing through the armature winding inside the magnetic field generated by the field winding experiences a rotational force, causing the armature to rotate. DC machines can be separately excited, self-excited, or series/compound excited. Common applications include DC motors used in industrial equipment and vehicles.

Adulto mayor

Este documento proporciona instrucciones para realizar una entrevista a un adulto mayor que aborde aspectos biopsicosociales. La entrevista debe incluir una ficha inicial con datos demográficos y de salud, una entrevista semidirigida sobre aspectos físicos, psicológicos y de inserción social, y un análisis final sobre la aplicación de teorías del apego y desapego y la utilidad de la entrevista para la formación médica del estudiante. El objetivo es investigar las características del adulto mayor

Dcmachines 2016

The document describes the construction and operation of DC machines. It contains the following key points:
1. A DC machine consists of a stator with poles that produce a magnetic field and a rotor with coils connected by a commutator. The interaction between the magnetic field and current in the rotor coils causes the rotor to rotate.
2. In a DC motor, the stator poles are supplied with DC current to generate a magnetic field. The rotor coils are also supplied with DC current, causing the rotor to rotate due to Lorentz forces. The commutator reverses the current to maintain rotation.
3. In a DC generator, a mechanical input rotates the rotor coils in the stator's

Pengertian Rukun Solat

Pendidikan Islam Tahun 4 KSSR

Dc machines 1

The document discusses DC machines which can operate as either motors or generators depending on the direction of power flow. A DC machine consists of a static electromagnetic or permanent magnetic field and a rotating armature. The field produces a magnetic medium and the armature produces voltage and torque under the action of the magnetic field. An advantage of DC motors is their speed is easy to control over a wide range, while DC generators are now quite rare. Most DC machines have AC voltages and currents internally but produce DC outputs via a commutator mechanism.

Músculos y huesos del miembro superior

El documento describe los huesos del miembro superior humano, que se divide en 4 partes: cintura escapular, brazo, antebrazo y mano. En total contiene 32 huesos. Describe cada hueso y su localización en detalle. También menciona los principales músculos de cada parte del miembro superior y sus acciones.

5 estructura del programa de formacion cívica 3°

Este documento presenta la estructura y propósitos del programa de Formación Cívica y Ética para tercer grado de educación básica en México. Describe cuatro objetivos para los estudiantes, los principios que orientan la asignatura, sus tres ejes formativos, ámbitos de aprendizaje y ocho competencias cívicas y éticas que se desarrollan de manera progresiva.

James DAVIES

James DAVIES

Prevención de accidentes en la infancia

Prevención de accidentes en la infancia

Football, great grandpa, I …

Football, great grandpa, I …

Revista a! 37 - Lo último en tecnología

Revista a! 37 - Lo último en tecnología

Exposicionambiental

Exposicionambiental

Esams resume2015-1

Esams resume2015-1

49694929 instructivo-test-5-6

49694929 instructivo-test-5-6

Dc machines 1

Dc machines 1

Adulto mayor

Adulto mayor

Dcmachines 2016

Dcmachines 2016

Pengertian Rukun Solat

Pengertian Rukun Solat

Dc machines 1

Dc machines 1

Músculos y huesos del miembro superior

Músculos y huesos del miembro superior

5 estructura del programa de formacion cívica 3°

5 estructura del programa de formacion cívica 3°

X-Ray Topic.ppt

X-ray diffraction is a technique used to analyze the crystal structure of materials. It works by firing X-rays at a crystalline sample and observing the scattered rays, which are determined by the positions of atoms in the crystal structure. Bragg's law describes the conditions under which constructive interference of X-rays scattered from crystal planes occurs, producing a diffraction pattern. Analysis of diffraction patterns can provide information about a material's lattice structure, symmetry, and chemical composition. Miller indices are used to describe crystallographic planes and directions within a crystal lattice.

Metallic phase with long range phys rev lett.53.1951

Metallic Phase with Long-Range Orientational Order and No Translational Symmetry
D. Shechtman and I. Blech
Department of Materials Engineering, Israel Institute of Technology Technion, 3200 Haifa, Israel
and
D. Gratias
Centre d'Etudes de Chimie Metallurgique, Centre National de la Recherche Scientiftque, F-94400 Vitry, France
and
J. W. Cahn
Center for Materials Science, National Bureau ofStandards, Gaithersburg, Maryland 20760
(Received 9 October 1984)
We have observed a metallic solid (Al —14-at. /o-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.

2012 NComms - Geometrically-locked vortex lattices in semiconductor quantum f...

The document summarizes research on generating and observing stable vortex lattices in polariton condensates in semiconductor microcavities. Researchers pumped three spots in a triangular geometry and observed the formation of a honeycomb lattice containing up to 100 vortices and antivortices extending over tens of microns. Numerical simulations matched the experimental observations and showed the lattice forms due to ferromagnetic coupling between the condensates at each pumped spot, with the phase locked by the imposed triangular geometry. The vortex lattice was stable for many minutes and highly sensitive to the optically imposed geometry.

A_I_Structure.pdf

This document provides an overview of solid state structure and crystallography. It discusses the types of solids, order in solids, symmetry in crystals including translation symmetry and the resulting crystal systems in 2D and 3D. It then covers the Bravais lattices that arise in 2D and 3D from the crystal systems and additional translational symmetries. Key concepts covered include Miller planes, X-ray diffraction including Bragg's law and structure factor, and common crystal structure motifs. The document is intended as a learning aid for students taking a chemistry of materials course.

99995069.ppt

Crystal structure determination uses X-ray diffraction to analyze the arrangement of atoms in crystals. X-rays are diffracted by the periodic lattice of a crystal in predictable ways. Bragg's law describes the conditions under which constructive interference occurs between X-rays reflected from different crystal lattice planes, producing intense diffracted beams. By measuring the angles and intensities of these diffracted beams, researchers can determine the size and shape of the unit cell and deduce the positions of atoms within the cell. The reciprocal lattice formalism relates diffraction phenomena to the periodicity of the crystal lattice.

Quantum chaos in clean many-body systems - Tomaž Prosen

This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.

Fourier transform in X-ray crystallography .ppt

The Fourier transform connects the diffraction pattern of X-rays scattered by a crystal to the electron density distribution within the crystal's unit cell. The structure factors obtained from diffraction intensities represent the Fourier transform of the electron density. However, the phases of the structure factors are lost during data collection, posing the phase problem. Techniques like isomorphous replacement, multiple wavelength anomalous diffraction, and molecular replacement aim to determine the missing phase information and allow calculation of electron density maps revealing a crystal's structure.

SRS to study Polytype NNN relationships

This document discusses the use of synchrotron edge topography to study the relationships between different polytypes in silicon carbide (SiC). It provides background on polytypism in SiC, including different theories proposed to explain its occurrence. The document explains that synchrotron radiation is well-suited to study polytypes due to its high resolution. It describes how edge topography allows determining the spatial extent of disorder between polytypes and deducing their neighborhood relationships by superimposing Laue reflections. The technique reveals finer features at polytype boundaries like long period polytypes and one-dimensionally disordered layers.

lecture4.pdf

This document discusses the process of X-ray crystallography structure determination. It begins with a review of Miller planes and structure factors. It then covers how phases are determined, including molecular replacement, isomorphous replacement using heavy atoms, and anomalous dispersion techniques like SAD and MAD that utilize anomalous scattering. The steps of structure determination are outlined, involving calculating electron density maps from phases and building/refining atomic models. Key concepts like structure factors, anomalous scattering, and using differences in Friedel pairs to determine heavy atom positions are also explained.

Xray diff pma

This document provides an overview of key concepts in X-ray crystallography, including:
1) There are 14 Bravais lattices that describe the repetitive symmetry within crystal structures. Bravais demonstrated that there are only 14 ways in which repetitive symmetry can occur in 3D crystal lattices.
2) A unit cell is the smallest repeating unit that describes the symmetry and dimensions of the crystal structure. There are 7 crystal systems that classify unit cells based on their angles and dimensions.
3) Bragg's law describes the conditions under which X-ray diffraction from crystal planes results in constructive interference. This law is key to solving crystal structures using single crystal X-ray diffraction techniques.

Grain boundaries

This document discusses two approaches to describing the mechanical behavior of grain boundaries: the continuous Frank-Bilby approach and the discrete Read-Shockley approach. It focuses on the discrete approach, explaining Bollmann's model of intrinsic dislocations that form periodic networks at grain boundaries according to the misorientation angle between crystals. Primary intrinsic dislocations account for the deviation from a single crystal structure and have Burgers vectors of the crystal lattice. Their spacing decreases with increasing misorientation angle according to the Read-Shockley formula. Examples are provided for low-angle tilt and twist grain boundaries.

Grain boundaries

1) There are two approaches to describing grain boundaries mechanically: a continuous approach using dislocation density tensors and a discrete approach considering arrays of dislocations.
2) The continuous Frank-Bilby equation relates the Burgers vector density necessary to achieve compatibility at the interface between crystals to the affine transformations relating their lattices.
3) The discrete Read-Shockley model links the misorientation angle and spacing of edge dislocations forming low-angle symmetrical tilt grain boundaries. It predicts the intergranular energy as a function of angle.

MRS poster (5)

This document summarizes research on tuning the electronic structure of tin and lead halide perovskites through atomic layering, epitaxial strain, and structural distortions. Key findings include:
1) Atomic layering of perovskites, either through organic linkers or in Ruddlesden-Popper phases, increases band gap due to quantum confinement effects.
2) Epitaxial strain of up to 3% and induced structural distortions can modify electronic structure and shift band gaps by over 1eV.
3) Atomic layering, strain, and distortions effectively tune band gaps throughout the visible spectrum for applications in solar cells.

Linear-Algebra-Friedberg-Insel-Spence-4th-E.pdf

This document provides an index of symbols and terms used in linear algebra. It defines over 100 symbols ranging from basic matrix notation like Aij to more advanced concepts like the generalized eigenspace. The index serves as a reference for symbols introduced and used throughout a linear algebra text. It covers topics such as matrices, linear transformations, vector spaces, eigenvalues, and inner product spaces. Providing concise definitions of important mathematical terms is useful for readers to efficiently learn the language of linear algebra.

A Chiral Icosohedral QC and it's mapping to an E8 QC by Klee Irwin and Fang Fang

This document introduces a chiral icosahedral quasicrystalline structure called the Fibonacci Icosagrid (FIG) QC and maps it unexpectedly to a quasicrystalline structure derived from the mathematical group E8. The FIG QC embeds two cross-sectional quasicrystalline structures from the E8 QC perfectly. Additionally, the FIG QC, E8 QC cross-sections, and a 20-tetrahedron cluster all share a deep relationship involving their construction using a Golden Angle rotation. This novel mapping between the FIG and E8-derived QCs may provide a new approach for unifying physics theories.

An03302430245

This document presents a study of five dimensional string cosmological models with bulk viscosity in general relativity. It investigates three cases of the field equations and determines solutions for each. For case I, the solution reduces the model to Minkowski spacetime. For case II, the solution yields a model that expands from a big bang and stops expanding at a finite time, with the volume increasing over time. The energy density is only due to bulk viscosity and Lyra geometry. For case III, three specific solutions are presented: one that reduces to a flat 5D spacetime; one that contracts/expands depending on constants; and one that follows power law inflation. Physical properties like expansion, shear, and density are also analyzed for each case.

jNVU Presentation on atomic and laser.ppsx

The document discusses X-ray diffraction and the structure factor in determining crystal structures. It provides background on how X-rays were first used by Max Von Laue and William Bragg to diffract off crystal planes. Bragg's law relates the wavelength of X-rays to the diffraction pattern. The structure factor describes the amplitude and phase of diffracted waves from crystal lattice planes and is essential for determining 3D electron density and crystal structures. Methods for X-ray diffraction including Laue, rotating crystal, and powder are described. Expressions for calculating the structure factor of simple cubic, body-centered cubic, and face-centered cubic unit cells are also provided.

Curv encyclopedia-fadili

This document discusses curvelets and ridgelets, which are geometric image representations designed to sparsely represent anisotropic elements like lines and curves. Ridgelets are based on applying a 1D wavelet transform to slices of the Radon transform of an image, representing lines as point singularities. Curvelets have elongated elements at multiple scales and orientations following a parabolic scaling law. Both provide more compact representations of edges than wavelets and have applications in image processing tasks like denoising, detection, and compression. The document outlines definitions and implementations of the continuous and discrete ridgelet and curvelet transforms.

modelling polymers under tension

The document discusses modelling polymers under tension. It introduces various polymer models including freely jointed chain, freely rotating chain, Gaussian chain and worm like chain models. It then describes the Lagowski Noolandi model in more detail. The document also discusses force-extension curves and how they can provide insight into polymer unfolding under tension. Finally, it presents results comparing force-extension curves for homopolymers and random block copolymers under varying parameters such as stiffness and composition.

Instantons in 1D QM

This document is Scott Shermer's master's thesis on instantons and perturbation theory in the 1-D quantum mechanical quartic oscillator. It begins by reviewing the harmonic oscillator and perturbation theory. It then discusses non-perturbative phenomena like instantons and Borel resummation. The focus is on obtaining the ground state energy of the quartic oscillator Hamiltonian using both perturbative and non-perturbative techniques, and addressing ambiguities that arise for negative coupling.

X-Ray Topic.ppt

X-Ray Topic.ppt

Metallic phase with long range phys rev lett.53.1951

Metallic phase with long range phys rev lett.53.1951

2012 NComms - Geometrically-locked vortex lattices in semiconductor quantum f...

2012 NComms - Geometrically-locked vortex lattices in semiconductor quantum f...

A_I_Structure.pdf

A_I_Structure.pdf

99995069.ppt

99995069.ppt

Quantum chaos in clean many-body systems - Tomaž Prosen

Quantum chaos in clean many-body systems - Tomaž Prosen

Fourier transform in X-ray crystallography .ppt

Fourier transform in X-ray crystallography .ppt

SRS to study Polytype NNN relationships

SRS to study Polytype NNN relationships

lecture4.pdf

lecture4.pdf

Xray diff pma

Xray diff pma

Grain boundaries

Grain boundaries

Grain boundaries

Grain boundaries

MRS poster (5)

MRS poster (5)

Linear-Algebra-Friedberg-Insel-Spence-4th-E.pdf

Linear-Algebra-Friedberg-Insel-Spence-4th-E.pdf

A Chiral Icosohedral QC and it's mapping to an E8 QC by Klee Irwin and Fang Fang

A Chiral Icosohedral QC and it's mapping to an E8 QC by Klee Irwin and Fang Fang

An03302430245

An03302430245

jNVU Presentation on atomic and laser.ppsx

jNVU Presentation on atomic and laser.ppsx

Curv encyclopedia-fadili

Curv encyclopedia-fadili

modelling polymers under tension

modelling polymers under tension

Instantons in 1D QM

Instantons in 1D QM

LEARNING TO LIVE WITH LAWS OF MOTION .pptx

CLASS 11 PHYSICS PPT

Lattice Defects in ionic solid compound.pptx

lattice of ionic solid

Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...

Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Music and Medieval History

Introduction_Ch_01_Biotech Biotechnology course .pptx

ntroduction_Ch_01_Biotech

seed production, Nursery & Gardening.pdf

This presentation offers a general idea of the structure of seed, seed production, management of seeds and its allied technologies. It also offers the concept of gene erosion and the practices used to control it. Nursery and gardening have been widely explored along with their importance in the related domain.

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

Centrifugation is a powerful technique used in laboratories to separate components of a heterogeneous mixture based on their density. This process utilizes centrifugal force to rapidly spin samples, causing denser particles to migrate outward more quickly than lighter ones. As a result, distinct layers form within the sample tube, allowing for easy isolation and purification of target substances.

Compositions of iron-meteorite parent bodies constrainthe structure of the pr...

Magmatic iron-meteorite parent bodies are the earliest planetesimals in the Solar System,and they preserve information about conditions and planet-forming processes in thesolar nebula. In this study, we include comprehensive elemental compositions andfractional-crystallization modeling for iron meteorites from the cores of five differenti-ated asteroids from the inner Solar System. Together with previous results of metalliccores from the outer Solar System, we conclude that asteroidal cores from the outerSolar System have smaller sizes, elevated siderophile-element abundances, and simplercrystallization processes than those from the inner Solar System. These differences arerelated to the formation locations of the parent asteroids because the solar protoplane-tary disk varied in redox conditions, elemental distributions, and dynamics at differentheliocentric distances. Using highly siderophile-element data from iron meteorites, wereconstruct the distribution of calcium-aluminum-rich inclusions (CAIs) across theprotoplanetary disk within the first million years of Solar-System history. CAIs, the firstsolids to condense in the Solar System, formed close to the Sun. They were, however,concentrated within the outer disk and depleted within the inner disk. Future modelsof the structure and evolution of the protoplanetary disk should account for this dis-tribution pattern of CAIs.

Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...

We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.

Signatures of wave erosion in Titan’s coasts

The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

CSIR UGC NET CONTENT

Farming systems analysis: what have we learnt?.pptx

Presentation given at the official farewell of Prof Ken Gillet at Wageningen on 13 June 2024

Methods of grain storage Structures in India.pdf

•Post-harvestlossesaccountforabout10%oftotalfoodgrainsduetounscientificstorage,insects,rodents,micro-organismsetc.,
•Totalfoodgrainproductioninindiais311milliontonnesandstorageis145mt.InIndia,annualstoragelosseshavebeenestimated14mtworthofRs.7,000croreinwhichinsectsaloneaccountfornearlyRs.1,300crores.
•InIndiaoutofthetotalproduction,about30%ismarketablesurplus
•Remaining70%isretainedandstoredbyfarmersforconsumption,seed,feed.Hence,growerneedstoragefacilitytoholdaportionofproducetosellwhenthemarketingpriceisfavourable
•TradersandCo-operativesatmarketcentresneedstoragestructurestoholdgrainswhenthetransportfacilityisinadequate

Post translation modification by Suyash Garg

overview of PTM helps to the students who wants to clear their basics about it.

AJAY KUMAR NIET GreNo Guava Project File.pdf

AJAY KUMAR NIET GreNo Guava Project PDF File

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.

Analysis of Polygenic Traits (GPB-602)

Quantitative genetics – study of inheritance of quantitative characters or polygenic characters in experimental populations.

the fundamental unit of life CBSE class 9.pptx

CBSE class 9 the fundamental unit of life

LEARNING TO LIVE WITH LAWS OF MOTION .pptx

LEARNING TO LIVE WITH LAWS OF MOTION .pptx

Lattice Defects in ionic solid compound.pptx

Lattice Defects in ionic solid compound.pptx

Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...

Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Introduction_Ch_01_Biotech Biotechnology course .pptx

Introduction_Ch_01_Biotech Biotechnology course .pptx

seed production, Nursery & Gardening.pdf

seed production, Nursery & Gardening.pdf

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

Module_1.In autotrophic nutrition ORGANISM

Module_1.In autotrophic nutrition ORGANISM

Compositions of iron-meteorite parent bodies constrainthe structure of the pr...

Compositions of iron-meteorite parent bodies constrainthe structure of the pr...

Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...

Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...

acanthocytes_causes_etiology_clinical sognificance-future.pptx

acanthocytes_causes_etiology_clinical sognificance-future.pptx

Signatures of wave erosion in Titan’s coasts

Signatures of wave erosion in Titan’s coasts

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

Farming systems analysis: what have we learnt?.pptx

Farming systems analysis: what have we learnt?.pptx

Methods of grain storage Structures in India.pdf

Methods of grain storage Structures in India.pdf

Post translation modification by Suyash Garg

Post translation modification by Suyash Garg

AJAY KUMAR NIET GreNo Guava Project File.pdf

AJAY KUMAR NIET GreNo Guava Project File.pdf

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

Analysis of Polygenic Traits (GPB-602)

Analysis of Polygenic Traits (GPB-602)

the fundamental unit of life CBSE class 9.pptx

the fundamental unit of life CBSE class 9.pptx

- 1. The Unexpected Fractal Signatures in Fibonacci chain Fang Fang, Raymond Aschheim and Klee Irwin Quantum Gravity Research, Los Angeles, CA, U.S.A. E-mail: fang@QuantumGravityResearch.org Abstract. Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = φ. The corresponding pointwise dimension is 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to the change in the Fibonacci order but not to the L/S ratio. 1. Introduction Quasicrystals possess exotic and sometimes anomalous properties that have interested the scientiﬁc community since their discovery by Shechtman in 1982 [1]. Of particular interest in this manuscript is the self similar property of quasicrystals that makes them fractal. Historically, research on this aspect of quasicrystalline property has revolved around spectral and wave function analysis [2, 3, 4]. Mathematical investigation [5, 6] of the geometric structure of quasicrystals is less well represented in the literature than experimental work. In this paper, a new framework for analyzing the fractal nature of quasicrystals is introduced. Speciﬁcally, the fractal properties of a one-dimensional Fibonacci chain and its derivatives are studied in the complex Fourier space. The results may also be found in two and three dimensional quasicrystals that can be constructed using a network of one dimensional Fibonacci chains [7]. 2. The fractal signature of the Fibonacci chain in Fourier space The Fibonacci chain is a quasiperiodic sequence of short and long segments where the ratio between the long and the short segment is the golden ratio [8]. It is an important 1D quasicrystal that uniquely removes the arbitrary closeness in the non-quasicrystalline grid space of a quasicrystal and convert it into a quasicrystal as well [7]. The Fourier representation of a Fibonacci chain is given as follows: zs = 1 √ n n r=1 ure 2πi(r−1)(s−1) n , (1) where ur = (φ − 1)( (r + 1)φ − rφ − 2) + φ = (φ − 1)( (r + 1)φ − rφ ) + (2 − φ), r = 1, 2, ...n (2)
- 2. is a Fibonacci chain of length n with units of length φ and 1. Here φ = √ 5+1 2 is the golden ratio. Note that the Fibonacci chain can also be generated using substitution rules [8] where m iterations of the substitution method produces a Fibonacci chain of length equal to the mth Fibonacci number. Consecutive iterations of the Fibonacci chain are plotted in the Fourier space in Fig. 1. The main cardioid (heart shape) in the Mandelbrot set [9] appeared at the center of each plot. It is known that the denominators of the periods of the circular bulbs at sequential scales in the Mandelbrot set follow the Fibonacci number sequence [10]. The plots reveal that the size of the cardioid decreases with successive iterations, and the scaling factor for the size change is approximately one half of the golden ratio. Moreover, the orientation of the cardioid undergoes mirror ﬂips with every odd and even numbered iteration. Upon magniﬁcation, the fractal nature becomes apparent near the real axis. The fractal dimension1 of the complex Fourier representation of the Fibonacci chain is approximately 0.7. Figure 1. The Fourier space representation of a Fibonacci chain is shown here using the substitution method with diﬀerent iterations, starting with 20. The horizontal axis represents the real part and the vertical axis represents the imaginary part of the Fourier coeﬃcients. 3. The derivatives of the Fibonacci chain in Fourier space It is tempting to conjecture that this kind of Fractal signature will appear in any ﬁnite portion of the Fibonacci chain or a Fibonacci chain that is nearly perfect (a Fibonacci chain that is generated strictly from the substitution rules mentioned earlier). A series of tests has been conducted to verify this conjecture. The results are surprising. Figure 5 shows the results of the Fourier space of the Fibonacci chain of 26th iteration with various modiﬁcations, mostly minor modiﬁcations, for example, small truncation from the head or tail of the chain, scrambling the order of a very small part of the chain or changing of the L/S ratio. These results show that the 1 pointwise dimension [11]
- 3. Figure 2. a) The Mandelbrot set, c) the fractal structure in the Fourier space of a Fibonacci chain with 25 iterations and b) overlays a and c together. Figure 3. Appearance of the fractal pattern at the 34th iteration of the Fibonacci chain. fractal pattern, especially the cardioid shape, is very sensitive to any modiﬁcation except the changing the L/S ratio, in which case only the scaling of the fractal pattern changes. In other words, the fractal pattern is a direct result of the substitution rule (or inﬂation/deﬂation rule) which gives the Fibonacci ordering of the L and S. The exact value of the L and S does not matter for the pattern. Any little breakdown of the rule will result in a non-prefect-closure of the cardioid therefore a complete diﬀerent pattern. It is similar to the butterﬂy eﬀect in chaos theory. Its worth noting that derivative patterns resemble the cycloidal pattern drawn by a geometric chuck. Figure 6 show pictures taken from a book published 100 years ago by Bazley [12]. Figure 6a is a picture of a geometric chuck Figure 6b-d are a few cycloids created by the geometric chuck with diﬀerent settings on the chuck. The fractal signature of the perfect Fibonacci chain may correspond to some perfect quasiperioidic gearing between the plates of the geometric chuck.
- 4. Figure 4. Pointwise dimension of the of the Fibonacci chain in Fourier space. 4. Summary This paper reports a novel fractal analysis of a Fibonacci chain in Fourier space by computing the pointwise dimensions and visual inspection of the Fourier pattern. The cardioid structure in the Mandelbrot set is observed for the case of the one dimensional Fibonacci chain, with alternating mirror symmetry according to odd or even instances of the substitution iteration that generates the Fibonacci chains. The Fourier pattern is very sensitive to how strictly the chain follows the substitution rules. Any small variation in that will completely change the pattern. In comparison, the change in the L/S ratio has no inﬂuence in the Fourier pattern other than a change in scaling.
- 5. Figure 5. The Fourier space representation of a Fibonacci chain of 26th iteration, with a) the ﬁrst segment removed; b) the ﬁrst two segments removed; c) the ﬁrst three segments removed; d) the ﬁrst six segments removed, e) the ﬁrst seven segments removed, f) the ﬁrst segment length replaced with 0; g) no modiﬁcation; h) the last segment removed, i) the last two segments remove; j) the last 46367 segments removed; k) the last 46368 segments removed and the Fibonacci chain of 26th iteration is truncated to the Fibonacci chain of 25th iteration; l) the 1st segment L replaced with S; m) the last two segments ﬂipped order; n) the order of the last ﬁve segments scrambled; o) the order of the last ten segments scrambled; p) the order of the last 100 segments scrambled, and the comparison between q) the original Fibonacci chain and r) the chain with modiﬁed L/S ratio where L/S = 2.
- 6. Figure 6. a) A picture of a geometric chuck; b-d) Cycloids generated by the geometric chuck with diﬀerent settings.
- 7. 5. References [1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Physical Review Letters, vol. 53, no. 20, p. 1951, 1984. [2] U. Grimm and M. Schreiber, “Energy spectra and eigenstates of quasiperiodic tight-binding hamiltonians,” arXiv:cond-mat/0212140, 2002. [3] P. Repetowicz, U. Grimm, and M. Schreiber, “Exact eigenstates of tight-binding hamiltonians on the penrose tiling,” arXiv:cond-mat/9805321, 1998. [4] N. Mac´e, A. Jagannathan, and F. Pi´echon, “Fractal dimensions of the wavefunctions and local spectral measures on the ﬁbonacci chain,” arXiv:1601.00532, 2016. [5] P. Ramachandrarao, A. Sina, and D. Sanyal, “On the fractal nature of Penrose tiling,” Current Science, vol. 79, no. 3, pp. 364–367, 2000. [6] V. Yudin and Y. Karygina, “Fractal images of quasicrystals as an example of Penrose lattice,” Crystallogrphy reports, vol. 46, no. 6, pp. 922–926, 2001. [7] F. Fang and K. Irwin, “An icosahedral quasicrystal as a golden modiﬁcation of the icosagrid and its connection to the e8 lattice,” arXiv:1511.07786, 2015. [8] D. Levine and P. J. Steinhardt, “Quasicrystals. i. deﬁnition and structure,” Physical Review B, vol. 34, no. 2, p. 596, 1986. [9] A. Douady, J. H. Hubbard, and P. Lavaurs, “Etude dynamique des polynˆomes complexes,” Universit´e de Paris-Sud, D´ep. de Math´ematique, 1984. [10] R. L. Devaney, http://math.bu.edu/DYSYS/FRACGEOM2, 2016. [11] S. H. Strogatz, Nonlinear Dynamics and Chaos with applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, 1994. [12] T. S. Bazley, Index to Geometric Chuck. Waterlow and Sons, 1875.