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This talk focuses on plane tilings, how they have historically connected art and mathematics, and more recently have been connected to chemistry. What did the 2011 Nobel Prize in Chemistry have to do with medieval Islamic mosaic patterns? Bob tries to fit these pieces together.

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PAGE 2008 - Bonner

The document discusses the polygonal technique for deriving Islamic geometric patterns. It describes how patterns are created by placing a compass at key points on a tessellating polygonal grid. Various historical examples from the 11th-16th centuries demonstrating this technique are provided. Complex compound patterns combining different symmetrical grids, such as 10-fold and 12-fold, or 11-fold and 13-fold, are also examined.

The Why of Busyness - Pravin Mahajan

A brief summary of how our society has reached the point where being busy is valued so much that it interferes with the simple act of living.
This talk was presented at Convox 2015.

Islamic Geometry Pattern

The Great Mosque of Al-Kairouan
Tunisia

Parametric Envision to Islamic Geometric Patterns

The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.

Checking Out: Moving Forward During Times of Transition - Aaron Weinstein

In my field of psychology I am paid to focus on people’s behaviors. I have always been an acute observer of human interactions, and a few years ago, I began to notice patterns in my friends’ relationships in how they were able to deal with their emotions. I identified three variations in how people tend to react during times of transition and in today’s talk, I will guide you through my theory of checking out.
This talk was presented at Convox 2015.

Pattern Languages — An Approach to Holistic Knowledge Representation

Pattern Languages, developed by Christopher Alexander and his colleagues, are holistic manifestos for a given domain. This presentation provides an introduction to patterns and pattern languages and some hints for developing them.

PAGE 2008 - Bonner

The document discusses the polygonal technique for deriving Islamic geometric patterns. It describes how patterns are created by placing a compass at key points on a tessellating polygonal grid. Various historical examples from the 11th-16th centuries demonstrating this technique are provided. Complex compound patterns combining different symmetrical grids, such as 10-fold and 12-fold, or 11-fold and 13-fold, are also examined.

The Why of Busyness - Pravin Mahajan

A brief summary of how our society has reached the point where being busy is valued so much that it interferes with the simple act of living.
This talk was presented at Convox 2015.

Islamic Geometry Pattern

The Great Mosque of Al-Kairouan
Tunisia

Parametric Envision to Islamic Geometric Patterns

The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.

Checking Out: Moving Forward During Times of Transition - Aaron Weinstein

In my field of psychology I am paid to focus on people’s behaviors. I have always been an acute observer of human interactions, and a few years ago, I began to notice patterns in my friends’ relationships in how they were able to deal with their emotions. I identified three variations in how people tend to react during times of transition and in today’s talk, I will guide you through my theory of checking out.
This talk was presented at Convox 2015.

Pattern Languages — An Approach to Holistic Knowledge Representation

Pattern Languages, developed by Christopher Alexander and his colleagues, are holistic manifestos for a given domain. This presentation provides an introduction to patterns and pattern languages and some hints for developing them.

Analytical approach on design theories of christopher alexander

Notes on combination of form and pattern language to new concepts of complexity theory.
Dr. N. Mohajeri , Dr. Sh. Qom

Creating clay tile1

This document outlines the steps of a lesson plan for creating Islamic ceramic tiles. The objectives are to learn about patterns and colors in Islamic art, design a hexagon tile with outlines and colors, and add details to a clay tile. The lesson involves researching Islamic art examples, designing a symmetrical hexagon tile with patterns, and creating the tile out of clay using techniques like rolling out the shape, cutting it out, and applying designs and colors with glazes.

Four components of Islamic ornamentation

The document discusses the four main components of Islamic ornamentation: calligraphy, vegetal patterns, geometric patterns, and figural representations. It provides details on each: calligraphy is the most important element and can be used decoratively; vegetal patterns were adapted from other traditions; geometric patterns feature intricate combinations of simple shapes; and figural representations were used ornamentally despite restrictions on images. Across the four sections, it explores the origins, development, and uses of motifs in Islamic art.

Incredible islamic art (geometry and mathematics)

This document provides an introduction to geometry. It discusses how geometry is the mathematical study of shapes, figures, and positions in space. Some key topics covered include different types of shapes such as circles, triangles, squares, and three-dimensional solids. The document also discusses important historical figures that contributed to the field such as Plato, Euclid, and Archimedes. Additionally, it covers how geometry has been applied through Islamic geometric art and architecture.

Geometry of Islamic Architecture

The document discusses the importance of geometry in Islamic architecture. Some key points:
- Geometry is one of the most important elements of Islamic art, with patterns constructed from basic shapes like circles, squares, stars and polygons.
- Repeating geometric patterns symbolize Allah's infinite nature and help demonstrate that the infinite can be found in small details.
- Basic geometric constructions can be used to create complex patterns using a straightedge and compass. Constructions of points like three, four, five and six are described.
- Examples like the Taj Mahal and Tomb of I'timad-ud-Daulah showcase the use of symmetry, right angles and geometric patterns in Islamic architectural design.

Redmodular

Este documento describe las redes modulares y cómo se usan para organizar el espacio bidimensional y tridimensional. Las redes modulares consisten en una cuadrícula de líneas que dividen el espacio en módulos iguales que se repiten. Los módulos pueden ser de diferentes formas y tamaños y pueden combinarse de varias maneras. Las redes modulares se usan comúnmente en el diseño y el arte para crear patrones y estructuras ordenadas.

Islamic Art And Geometric Design

This document provides materials for teaching about Islamic geometric art and pattern-making activities. It includes an introduction to geometric design in Islamic art, descriptions of selected artworks from the Metropolitan Museum of Art, and pattern-making activities that allow students to recreate Islamic geometric patterns using only a compass and straightedge. The activities are intended to spark interdisciplinary learning about math, art, culture and history.

Integralsm symbol

This document is a dialogue between Ni Suiti and Ki Algo, who represent the feminine and masculine unconscious aspects of the author. They are discussing the geometric pattern used as the symbol for the author's blog "Integralism". Ki Algo explains that the pattern is based on Penrose tiles, which can tile a plane in an aperiodic tessellation using just two shapes. This aperiodic tiling was significantly simplified over the years from over 20,000 tile shapes down to just two by mathematicians like Berger, Penrose, and others. Ki Algo further explains that similar aperiodic tiling patterns have since been discovered to occur naturally in quasicrystals. The dialogue continues with Ni Suiti learning that Islamic

BARBARAKERWINART_45pp

This document summarizes Barbara Kerwin's artistic career and body of work over 15 years. It describes her evolution from unconventional "constructed paintings" in graduate school to more traditional wall-mounted rectangular paintings. Several solo exhibitions are highlighted that explored themes like architecture, dreams, and film through variations on the rectangle format. The document also mentions a recent survey exhibition titled "Geometric Progressions" that curated Kerwin's work from the past 15 years and included a video interview.

Add Maths Project Work

Here are examples of evidence and assessments for different levels of student learning in this math lesson on polygons:
Exceeds Expectations:
- Content: Students can define different types of polygons (triangle, quadrilateral, etc.) and their properties without any aids. They can apply properties to solve multi-step problems.
- Language: Students actively participate in discussions, explaining their mathematical reasoning clearly and using precise vocabulary. They can answer higher-order thinking questions.
Meets Expectations:
- Content: Students can identify different polygons and state basic properties when using the word bank or textbook as a reference. They can solve one-step problems applying one property.
- Language: Students participate in discussions when called on, answering

Mathematics and Art

Mathematics and art have a long historical relationship. The Golden ratio, Geometric patterns, Fractals are all fascinating mathematical ideas that have inspired artists and architects for centuries, I am just exploring these ideas in this presentation

Mathematics in the Modern World

The document provides an overview of different types of fallacies in logic. It discusses semantic fallacies, which are errors due to ambiguity or incorrect construction of language. Examples of semantic fallacies given are equivocation, composition, and division. It also discusses material fallacies, which stem from issues with the subject matter itself. Examples of material fallacies provided are accident and confusing absolute and qualified statements. The document aims to define different logical fallacies and provide examples of each.

Forca Barca Math's quiz finals

This quiz was conducted for the students of our school BVM Eroor,Kerala......Hope u enjoy!!!! Please reply with comments......

HISTORY: Greek Architecture Codes (Part 1)

This document provides an overview of Greek architecture codes and principles. It discusses how the Greeks defined the world through mathematics, philosophy, and architecture using basic truths, patterns, and geometry. Key Greek mathematicians like Euclid and architects established principles of proportion and harmony that still influence design today, such as the golden ratio. The document also examines specific architectural orders developed by the Greeks, including the Doric and Ionic orders, describing their distinguishing characteristics and influence on Western architecture.

Trondheim small

This document discusses how using history can help popularize mathematics. It provides advantages of mathematicians and teachers learning history, such as better communication skills and understanding student difficulties. Teachers can use historical examples, problems, and biographies to make lessons more interesting. The document also discusses how history can improve the public image of math by showing it is a human endeavor and not dry/boring. It provides examples of how Croatia promotes math history awareness through publications, student projects, and curriculum.

Geometry

1. Euclid's Elements/Postulates - Euclid wrote a text titled 'Elements' in 300 BC which presented geometry through a small set of statements called postulates that are accepted as true. He was able to derive much of planar geometry from just five postulates, including the parallel postulate which caused much debate.
2. Euclid's Contribution to Geometry - Euclid is considered the "Father of Geometry" for his work Elements, which introduced deductive reasoning to mathematics. Elements influenced the development of the subject through its logical presentation of geometry from definitions and postulates.
3. Similar Triangles - Triangles are similar if they have the same shape but not necessarily the same

Golden ratio and golden rectangle

The document discusses the golden ratio and its applications. It provides background on the golden ratio's history in mathematics and its ubiquity across disciplines. Examples are given of the golden ratio's appearances in architecture like the Parthenon and pyramids, paintings by da Vinci, as well as relationships to shapes like the golden rectangle, triangle, and pentagram. The golden ratio is defined mathematically and its aesthetic appeal is noted.

Introduction to Crystallography

Crystallography is the branch of science that studies crystals, their growth and structure. It deals with crystals' external form, internal arrangement of atoms, and physical properties. Crystals have a regularly repeating internal structure. Miller indices are a notation system used to describe crystal planes and directions within a crystal lattice by three integers h, k, l. The law of rational indices states that the intercepts made by a crystal plane with the unit cell axes are inversely proportional to integers that become the Miller indices. Determining Miller indices involves finding intercepts with axes, converting to fractional coordinates, and taking reciprocals.

Presentation on the Euclid

Euclid of Alexandria lived around 300 BCE and wrote The Elements, the most influential and widely used mathematics textbook in history. The Elements collected, organized, and proved many geometric ideas and theorems that were known at the time. Euclid began with definitions, common notions, and postulates as foundations for geometry. He then used deductive reasoning to prove hundreds of theorems, establishing geometry as a logical science. While little is known about Euclid's life, his work The Elements has had an immense impact and remains highly influential to this day.

Maths Exploration

1. The document discusses a new technique called the "amplituhedron" developed by physicist Nima Arkani-Hamed and his research team to more easily calculate scattering amplitudes in quantum field theory.
2. The amplituhedron represents scattering processes geometrically rather than through thousands of Feynman diagrams. It allows the calculation of scattering amplitudes as the volume of the geometric shape.
3. The document provides an example of how the amplituhedron could represent the scattering of gluons, with each part of the shape corresponding to one gluon collision and the overall volume giving the scattering amplitude.

Geometry, Nature and Architecture

The document discusses the relationships between nature, geometry, and architecture. It provides examples of geometric patterns found in nature, such as hexagonal honeycomb structures and symmetrical flower patterns. Geometry has historically been used in architecture for spatial layout, decoration, and meeting structural requirements. Examples discussed include ancient Egyptian pyramids, Greek temples, Islamic tiling patterns, Chinese tulou structures, and Gothic cathedrals. Modern architects like Gaudi incorporated complex geometries in works like the Sagrada Familia, drawing inspiration from nature.

The Platonic Solids

The document discusses the five Platonic solids - geometric shapes that are the only possible convex regular polyhedra. These five shapes are the tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces). The Platonic solids have been known since antiquity and are notable for their symmetry, with all faces being the same regular polygon and all vertices having the same configuration. They can all be inscribed in a sphere and satisfy Euler's formula relating the number of vertices, edges, and faces.

Analytical approach on design theories of christopher alexander

Notes on combination of form and pattern language to new concepts of complexity theory.
Dr. N. Mohajeri , Dr. Sh. Qom

Creating clay tile1

This document outlines the steps of a lesson plan for creating Islamic ceramic tiles. The objectives are to learn about patterns and colors in Islamic art, design a hexagon tile with outlines and colors, and add details to a clay tile. The lesson involves researching Islamic art examples, designing a symmetrical hexagon tile with patterns, and creating the tile out of clay using techniques like rolling out the shape, cutting it out, and applying designs and colors with glazes.

Four components of Islamic ornamentation

The document discusses the four main components of Islamic ornamentation: calligraphy, vegetal patterns, geometric patterns, and figural representations. It provides details on each: calligraphy is the most important element and can be used decoratively; vegetal patterns were adapted from other traditions; geometric patterns feature intricate combinations of simple shapes; and figural representations were used ornamentally despite restrictions on images. Across the four sections, it explores the origins, development, and uses of motifs in Islamic art.

Incredible islamic art (geometry and mathematics)

This document provides an introduction to geometry. It discusses how geometry is the mathematical study of shapes, figures, and positions in space. Some key topics covered include different types of shapes such as circles, triangles, squares, and three-dimensional solids. The document also discusses important historical figures that contributed to the field such as Plato, Euclid, and Archimedes. Additionally, it covers how geometry has been applied through Islamic geometric art and architecture.

Geometry of Islamic Architecture

The document discusses the importance of geometry in Islamic architecture. Some key points:
- Geometry is one of the most important elements of Islamic art, with patterns constructed from basic shapes like circles, squares, stars and polygons.
- Repeating geometric patterns symbolize Allah's infinite nature and help demonstrate that the infinite can be found in small details.
- Basic geometric constructions can be used to create complex patterns using a straightedge and compass. Constructions of points like three, four, five and six are described.
- Examples like the Taj Mahal and Tomb of I'timad-ud-Daulah showcase the use of symmetry, right angles and geometric patterns in Islamic architectural design.

Redmodular

Este documento describe las redes modulares y cómo se usan para organizar el espacio bidimensional y tridimensional. Las redes modulares consisten en una cuadrícula de líneas que dividen el espacio en módulos iguales que se repiten. Los módulos pueden ser de diferentes formas y tamaños y pueden combinarse de varias maneras. Las redes modulares se usan comúnmente en el diseño y el arte para crear patrones y estructuras ordenadas.

Islamic Art And Geometric Design

This document provides materials for teaching about Islamic geometric art and pattern-making activities. It includes an introduction to geometric design in Islamic art, descriptions of selected artworks from the Metropolitan Museum of Art, and pattern-making activities that allow students to recreate Islamic geometric patterns using only a compass and straightedge. The activities are intended to spark interdisciplinary learning about math, art, culture and history.

Analytical approach on design theories of christopher alexander

Analytical approach on design theories of christopher alexander

Creating clay tile1

Creating clay tile1

Four components of Islamic ornamentation

Four components of Islamic ornamentation

Incredible islamic art (geometry and mathematics)

Incredible islamic art (geometry and mathematics)

Geometry of Islamic Architecture

Geometry of Islamic Architecture

Redmodular

Redmodular

Islamic Art And Geometric Design

Islamic Art And Geometric Design

Integralsm symbol

This document is a dialogue between Ni Suiti and Ki Algo, who represent the feminine and masculine unconscious aspects of the author. They are discussing the geometric pattern used as the symbol for the author's blog "Integralism". Ki Algo explains that the pattern is based on Penrose tiles, which can tile a plane in an aperiodic tessellation using just two shapes. This aperiodic tiling was significantly simplified over the years from over 20,000 tile shapes down to just two by mathematicians like Berger, Penrose, and others. Ki Algo further explains that similar aperiodic tiling patterns have since been discovered to occur naturally in quasicrystals. The dialogue continues with Ni Suiti learning that Islamic

BARBARAKERWINART_45pp

This document summarizes Barbara Kerwin's artistic career and body of work over 15 years. It describes her evolution from unconventional "constructed paintings" in graduate school to more traditional wall-mounted rectangular paintings. Several solo exhibitions are highlighted that explored themes like architecture, dreams, and film through variations on the rectangle format. The document also mentions a recent survey exhibition titled "Geometric Progressions" that curated Kerwin's work from the past 15 years and included a video interview.

Add Maths Project Work

Here are examples of evidence and assessments for different levels of student learning in this math lesson on polygons:
Exceeds Expectations:
- Content: Students can define different types of polygons (triangle, quadrilateral, etc.) and their properties without any aids. They can apply properties to solve multi-step problems.
- Language: Students actively participate in discussions, explaining their mathematical reasoning clearly and using precise vocabulary. They can answer higher-order thinking questions.
Meets Expectations:
- Content: Students can identify different polygons and state basic properties when using the word bank or textbook as a reference. They can solve one-step problems applying one property.
- Language: Students participate in discussions when called on, answering

Mathematics and Art

Mathematics and art have a long historical relationship. The Golden ratio, Geometric patterns, Fractals are all fascinating mathematical ideas that have inspired artists and architects for centuries, I am just exploring these ideas in this presentation

Mathematics in the Modern World

The document provides an overview of different types of fallacies in logic. It discusses semantic fallacies, which are errors due to ambiguity or incorrect construction of language. Examples of semantic fallacies given are equivocation, composition, and division. It also discusses material fallacies, which stem from issues with the subject matter itself. Examples of material fallacies provided are accident and confusing absolute and qualified statements. The document aims to define different logical fallacies and provide examples of each.

Forca Barca Math's quiz finals

This quiz was conducted for the students of our school BVM Eroor,Kerala......Hope u enjoy!!!! Please reply with comments......

HISTORY: Greek Architecture Codes (Part 1)

This document provides an overview of Greek architecture codes and principles. It discusses how the Greeks defined the world through mathematics, philosophy, and architecture using basic truths, patterns, and geometry. Key Greek mathematicians like Euclid and architects established principles of proportion and harmony that still influence design today, such as the golden ratio. The document also examines specific architectural orders developed by the Greeks, including the Doric and Ionic orders, describing their distinguishing characteristics and influence on Western architecture.

Trondheim small

This document discusses how using history can help popularize mathematics. It provides advantages of mathematicians and teachers learning history, such as better communication skills and understanding student difficulties. Teachers can use historical examples, problems, and biographies to make lessons more interesting. The document also discusses how history can improve the public image of math by showing it is a human endeavor and not dry/boring. It provides examples of how Croatia promotes math history awareness through publications, student projects, and curriculum.

Geometry

1. Euclid's Elements/Postulates - Euclid wrote a text titled 'Elements' in 300 BC which presented geometry through a small set of statements called postulates that are accepted as true. He was able to derive much of planar geometry from just five postulates, including the parallel postulate which caused much debate.
2. Euclid's Contribution to Geometry - Euclid is considered the "Father of Geometry" for his work Elements, which introduced deductive reasoning to mathematics. Elements influenced the development of the subject through its logical presentation of geometry from definitions and postulates.
3. Similar Triangles - Triangles are similar if they have the same shape but not necessarily the same

Golden ratio and golden rectangle

The document discusses the golden ratio and its applications. It provides background on the golden ratio's history in mathematics and its ubiquity across disciplines. Examples are given of the golden ratio's appearances in architecture like the Parthenon and pyramids, paintings by da Vinci, as well as relationships to shapes like the golden rectangle, triangle, and pentagram. The golden ratio is defined mathematically and its aesthetic appeal is noted.

Introduction to Crystallography

Crystallography is the branch of science that studies crystals, their growth and structure. It deals with crystals' external form, internal arrangement of atoms, and physical properties. Crystals have a regularly repeating internal structure. Miller indices are a notation system used to describe crystal planes and directions within a crystal lattice by three integers h, k, l. The law of rational indices states that the intercepts made by a crystal plane with the unit cell axes are inversely proportional to integers that become the Miller indices. Determining Miller indices involves finding intercepts with axes, converting to fractional coordinates, and taking reciprocals.

Presentation on the Euclid

Euclid of Alexandria lived around 300 BCE and wrote The Elements, the most influential and widely used mathematics textbook in history. The Elements collected, organized, and proved many geometric ideas and theorems that were known at the time. Euclid began with definitions, common notions, and postulates as foundations for geometry. He then used deductive reasoning to prove hundreds of theorems, establishing geometry as a logical science. While little is known about Euclid's life, his work The Elements has had an immense impact and remains highly influential to this day.

Maths Exploration

1. The document discusses a new technique called the "amplituhedron" developed by physicist Nima Arkani-Hamed and his research team to more easily calculate scattering amplitudes in quantum field theory.
2. The amplituhedron represents scattering processes geometrically rather than through thousands of Feynman diagrams. It allows the calculation of scattering amplitudes as the volume of the geometric shape.
3. The document provides an example of how the amplituhedron could represent the scattering of gluons, with each part of the shape corresponding to one gluon collision and the overall volume giving the scattering amplitude.

Geometry, Nature and Architecture

The document discusses the relationships between nature, geometry, and architecture. It provides examples of geometric patterns found in nature, such as hexagonal honeycomb structures and symmetrical flower patterns. Geometry has historically been used in architecture for spatial layout, decoration, and meeting structural requirements. Examples discussed include ancient Egyptian pyramids, Greek temples, Islamic tiling patterns, Chinese tulou structures, and Gothic cathedrals. Modern architects like Gaudi incorporated complex geometries in works like the Sagrada Familia, drawing inspiration from nature.

The Platonic Solids

The document discusses the five Platonic solids - geometric shapes that are the only possible convex regular polyhedra. These five shapes are the tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces). The Platonic solids have been known since antiquity and are notable for their symmetry, with all faces being the same regular polygon and all vertices having the same configuration. They can all be inscribed in a sphere and satisfy Euler's formula relating the number of vertices, edges, and faces.

Maths in culturar life new

This document provides an overview of the role of mathematics in cultural and social life. It discusses how mathematics has been used in fields like architecture, art, music, literature, and movies. Some key points:
- Ancient civilizations like the Babylonians, Egyptians, and Greeks used basic math principles in areas like construction, astronomy, calendars, and measurement.
- Many artists through history have been inspired by mathematical themes and patterns like the golden ratio, fractals, and platonic solids.
- Music incorporates mathematical concepts like ratios, patterns, and geometry. Some composers incorporated things like the Fibonacci sequence.
- Literature and movies sometimes feature or are inspired by mathematical ideas, theories, and un

3 d chapter 3 3d design elements

This document discusses various design elements of 3D form including depth, viewing angles, mass and space interaction, line, plane, surface qualities like texture and color, light, and time/motion. It provides examples and definitions for each element and explores how they contribute to the expression and perception of 3D forms in art and design.

Futura Type Specimen_LeighCavanaugh

This document is a type specimen book created by Leigh Cavanaugh in 2014 entirely in the Futura typeface. It contains samples of Futura in different styles and weights showing the glyphs and characters, as well as body copy text providing an overview of classical Greek geometry. The book includes sections on early Greek mathematicians such as Thales and Pythagoras, as well as Plato, Euclid, and Archimedes and their contributions to the field of geometry.

Mathematical Models and Modern Art

Paper from the Bridges 2010 conference proceedings. In the late 1800’s and early 1900’s mathematicians were creating models of mathematical surfaces out of plaster, wire, and other materials. These models were used to illustrate research and for university instruction. Gradually, mathematical interest in these models faded, but the models themselves were still on display in universities and museums. Here they were found by several artists from the Constructivist and Surrealist movements, two movements of abstract art that were active in the early 20th century. Artists from each of these movements drew some inspiration from these models of surfaces. We trace the paths of this inﬂuence, concluding with some locations in which models can still be found as well as some current artistic interest.

Thesis Book

This document summarizes an investigation into cymatics, the study of visible sound vibrations. It describes various cymatics experiments using different materials like salt, sand, and water. It also discusses images by photographer Harald Finster that resemble cymatic patterns. The document then outlines the development of a design project exploring cymatics and its relationship to concepts in origami, architecture, and transportation in Kyoto, Japan. Models were created to represent proposed metro station designs applying principles of movement derived from cymatics research.

Integralsm symbol

Integralsm symbol

BARBARAKERWINART_45pp

BARBARAKERWINART_45pp

Add Maths Project Work

Add Maths Project Work

Mathematics and Art

Mathematics and Art

Mathematics in the Modern World

Mathematics in the Modern World

Forca Barca Math's quiz finals

Forca Barca Math's quiz finals

HISTORY: Greek Architecture Codes (Part 1)

HISTORY: Greek Architecture Codes (Part 1)

Trondheim small

Trondheim small

Geometry

Geometry

Golden ratio and golden rectangle

Golden ratio and golden rectangle

Introduction to Crystallography

Introduction to Crystallography

Presentation on the Euclid

Presentation on the Euclid

Maths Exploration

Maths Exploration

Geometry, Nature and Architecture

Geometry, Nature and Architecture

The Platonic Solids

The Platonic Solids

Maths in culturar life new

Maths in culturar life new

3 d chapter 3 3d design elements

3 d chapter 3 3d design elements

Futura Type Specimen_LeighCavanaugh

Futura Type Specimen_LeighCavanaugh

Mathematical Models and Modern Art

Mathematical Models and Modern Art

Thesis Book

Thesis Book

Generative Art: A Systemic Approach to Manufacturing Inspirado - Nate Moser

From providing that "happy accident" that spurs organic creativity, to filling in the background with intricate detail, to expressing a whole piece at the intersection of art and math, generative art techniques have been used for centuries by visual and musical creators. In this talk, we'll explore a few techniques and tools that anyone can use on even modest computer hardware to inspire or guide their own work.
This talk was presented at Convox 2015.

From Stretching to Striving: 3 Life Lessons from Yoga

My recent yoga teacher training not only upped my knowledge of poses, but also taught me some pretty deep life lessons. In this talk, I share three such lessons: nothing is difficult, a daily practice teaches you to strive, and striving can give life meaning. We also do a little posing by way of illustration.
This talk was presented at Convox 2015.

Gaining perspective on Your Perspective - Olga Bergstrom

Have you ever drilled down on your own perspective? Each of us has a view of the world that is as unique as a fingerprint. This talk discusses how gaining perspective on your own perspective can help you cultivate relationships with others and the person you are around everyday - you! Pondering your thoughts and influences can lead you to a new normal - that you are special and unique, just like everyone else!

Language Contact and Its Outcomes - Kyle Shiells

Languages have been coexisting and influencing each other since long before history was recorded. What are the situations in which contact can arise, how are the languages and communities changed in the process, and how can we learn about histories of contact from the languages themselves?

Sticker Shock - What is Life Really Worth - Pravin Mahajan

Establishing the value of life is a problematic exercise at best. A mystery billionaire recently insured himself at $200 million as distinct from his net worth, prompting one to reflect on the monetary value of a life. In this talk, we'll explore various methods used today to put a number to life. But are they correct? In this age of limitless data are we using the right metrics? Are there other ways to measure life, and not just the human one? Should we even try?

Cultivating Equanimity - Chris LuVogt

Being calm and composed in difficult situations is not only an immensely useful life skill, but a learnable one, "in just 15 minutes a day!" In this talk, you'll not only learn about the what, why, and how of equanimity, but also get an interactive lesson on just how easy it is to cultivate it in your daily life.

IfCon0.1: Life Lessons from Backgammon - Marco Zagha

This document discusses life lessons that can be learned from playing the game of backgammon. It touches on finding an opponent to play with, either in person or using artificial intelligence software, learning that winning isn't everything and dealing with variance in outcomes, thinking strategically about important decisions, and maintaining a positive outlook. The document suggests backgammon can provide lessons about connections between events and accepting imperfect outcomes.

IfCon0.1: An iPhone fit for a Queen - Alex Dow

The document discusses Andy Warhol's view that rich and poor consumers buy essentially the same products, as illustrated by examples of a Coke or hot dog. It then argues this also applies to smartphones like the iPhone, as the Queen could have no better iPhone than a regular consumer, despite her wealth. It notes iPhones are expensive but most Americans have cell phones, with over half having smartphones. The document suggests Warhol's view extends to other goods like watches, truffles, and proposes iPhone-inspired truffle fries.

IfCon0.1: Longevity, Health, and Happiness - Chris LuVogt

The document discusses factors that can contribute to longevity and happiness. It suggests striving for a productive and socially rich life by engaging in physical activities you enjoy, maintaining mental focus, nurturing relationships, helping your community, and finding fulfillment in your work. Staying on healthy pathways physically, mentally, and socially through persistence and balance can lead to both longevity and happiness. It proposes measuring daily behaviors related to conscientiousness, social connection, and physical activity in order to maintain overall health and well-being.

IfCon0.1: Tomorrow is a new day...what if you really believed it? - Jesse Bri...

The document discusses strategies for starting fresh and living with self-compassion. It suggests thinking of past mistakes as actions of a previous version of yourself rather than your current self. This allows you to feel compassion for past failures while focusing on the present. The document also advocates embracing each new day as an opportunity to leave past shortcomings behind and work towards your current goals.

IfCon0.1: Journeys in Cambodia - Chad Carson

The document discusses Cambodia, including its history of being dominated by various outside powers, the atrocities of the Khmer Rouge regime in the 1970s, and current socioeconomic conditions. It then outlines several nonprofit organizations working in Cambodia today, focusing on education initiatives, community development projects, conservation efforts in the Tonle Sap region, and the importance of sustainability over temporary aid.

Generative Art: A Systemic Approach to Manufacturing Inspirado - Nate Moser

Generative Art: A Systemic Approach to Manufacturing Inspirado - Nate Moser

From Stretching to Striving: 3 Life Lessons from Yoga

From Stretching to Striving: 3 Life Lessons from Yoga

Gaining perspective on Your Perspective - Olga Bergstrom

Gaining perspective on Your Perspective - Olga Bergstrom

Language Contact and Its Outcomes - Kyle Shiells

Language Contact and Its Outcomes - Kyle Shiells

Sticker Shock - What is Life Really Worth - Pravin Mahajan

Sticker Shock - What is Life Really Worth - Pravin Mahajan

Cultivating Equanimity - Chris LuVogt

Cultivating Equanimity - Chris LuVogt

IfCon0.1: Life Lessons from Backgammon - Marco Zagha

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IfCon0.1: An iPhone fit for a Queen - Alex Dow

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IfCon0.1: Longevity, Health, and Happiness - Chris LuVogt

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IfCon0.1: Tomorrow is a new day...what if you really believed it? - Jesse Bri...

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IfCon0.1: Journeys in Cambodia - Chad Carson

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BANANA BUNCHY TOP K R.pptx

BANANA BUNCHY TOP K R.pptx

Explainable Deepfake Image/Video Detection

Presentation of our paper, "Towards Quantitative Evaluation of Explainable AI Methods for Deepfake Detection", by K. Tsigos, E. Apostolidis, S. Baxevanakis, S. Papadopoulos, V. Mezaris. Presented at the ACM Int. Workshop on Multimedia AI against Disinformation (MAD’24) of the ACM Int. Conf. on Multimedia Retrieval (ICMR’24), Thailand, June 2024. https://doi.org/10.1145/3643491.3660292 https://arxiv.org/abs/2404.18649
Software available at https://github.com/IDT-ITI/XAI-Deepfakes

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

CSIR UGC NET CONTENT

MICROBIAL INTERACTION PPT/ MICROBIAL INTERACTION AND THEIR TYPES // PLANT MIC...

MICROBIAL INTERACTION PPT/ MICROBIAL INTERACTION AND THEIR TYPES // PLANT MIC...ABHISHEK SONI NIMT INSTITUTE OF MEDICAL AND PARAMEDCIAL SCIENCES , GOVT PG COLLEGE NOIDA

Microbial interaction
Microorganisms interacts with each other and can be physically associated with another organisms in a variety of ways.
One organism can be located on the surface of another organism as an ectobiont or located within another organism as endobiont.
Microbial interaction may be positive such as mutualism, proto-cooperation, commensalism or may be negative such as parasitism, predation or competition
Types of microbial interaction
Positive interaction: mutualism, proto-cooperation, commensalism
Negative interaction: Ammensalism (antagonism), parasitism, predation, competition
I. Mutualism:
It is defined as the relationship in which each organism in interaction gets benefits from association. It is an obligatory relationship in which mutualist and host are metabolically dependent on each other.
Mutualistic relationship is very specific where one member of association cannot be replaced by another species.
Mutualism require close physical contact between interacting organisms.
Relationship of mutualism allows organisms to exist in habitat that could not occupied by either species alone.
Mutualistic relationship between organisms allows them to act as a single organism.
Examples of mutualism:
i. Lichens:
Lichens are excellent example of mutualism.
They are the association of specific fungi and certain genus of algae. In lichen, fungal partner is called mycobiont and algal partner is called
II. Syntrophism:
It is an association in which the growth of one organism either depends on or improved by the substrate provided by another organism.
In syntrophism both organism in association gets benefits.
Compound A
Utilized by population 1
Compound B
Utilized by population 2
Compound C
utilized by both Population 1+2
Products
In this theoretical example of syntrophism, population 1 is able to utilize and metabolize compound A, forming compound B but cannot metabolize beyond compound B without co-operation of population 2. Population 2is unable to utilize compound A but it can metabolize compound B forming compound C. Then both population 1 and 2 are able to carry out metabolic reaction which leads to formation of end product that neither population could produce alone.
Examples of syntrophism:
i. Methanogenic ecosystem in sludge digester
Methane produced by methanogenic bacteria depends upon interspecies hydrogen transfer by other fermentative bacteria.
Anaerobic fermentative bacteria generate CO2 and H2 utilizing carbohydrates which is then utilized by methanogenic bacteria (Methanobacter) to produce methane.
ii. Lactobacillus arobinosus and Enterococcus faecalis:
In the minimal media, Lactobacillus arobinosus and Enterococcus faecalis are able to grow together but not alone.
The synergistic relationship between E. faecalis and L. arobinosus occurs in which E. faecalis require folic acid
Gadgets for management of stored product pests_Dr.UPR.pdf

Insectsplayamajorroleinthedeteriorationoffoodgrainscausingbothquantitativeandqualitativelosses
Wellprovedthatnogranariescanbefilledwithgrainswithoutinsectsastheharvestedproducecontainegg(or)larvae(or)pupae(or)adultinsectinthembecauseoffieldcarryoverinfestationwhichcannotbeavoidedindevelopingcountrieslikeIndia
Simpletechnologiesfortimelydetectionofinsectsinthestoredproduceandtherebyplantimelycontrolmeasures

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.

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Rodents, Birds and locust_Pests of crops.pdf

Mole rat or Lesser bandicoot rat, Bandicotabengalensis
•Head -round and broad muzzle
•Tail -shorter than head, body
•Prefers damp areas
•Burrows with scooped soil before entrance
•Potential rat, one pair can produce more than 800 offspringsin one year

一比一原版美国佩斯大学毕业证如何办理

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Discovery of Merging Twin Quasars at z=6.05

We report the discovery of two quasars at a redshift of z = 6.05 in the process of merging. They were
serendipitously discovered from the deep multiband imaging data collected by the Hyper Suprime-Cam (HSC)
Subaru Strategic Program survey. The quasars, HSC J121503.42−014858.7 (C1) and HSC J121503.55−014859.3
(C2), both have luminous (>1043 erg s−1
) Lyα emission with a clear broad component (full width at half
maximum >1000 km s−1
). The rest-frame ultraviolet (UV) absolute magnitudes are M1450 = − 23.106 ± 0.017
(C1) and −22.662 ± 0.024 (C2). Our crude estimates of the black hole masses provide log 8.1 0. ( ) M M BH = 3
in both sources. The two quasars are separated by 12 kpc in projected proper distance, bridged by a structure in the
rest-UV light suggesting that they are undergoing a merger. This pair is one of the most distant merging quasars
reported to date, providing crucial insight into galaxy and black hole build-up in the hierarchical structure
formation scenario. A companion paper will present the gas and dust properties captured by Atacama Large
Millimeter/submillimeter Array observations, which provide additional evidence for and detailed measurements of
the merger, and also demonstrate that the two sources are not gravitationally lensed images of a single quasar.
Unified Astronomy Thesaurus concepts: Double quasars (406); Quasars (1319); Reionization (1383); High-redshift
galaxies (734); Active galactic nuclei (16); Galaxy mergers (608); Supermassive black holes (1663)

Evaluation and Identification of J'BaFofi the Giant Spider of Congo and Moke...

ABSTRACT
The J'BaFofi, or "Giant Spider," is a mainly legendary arachnid by reportedly inhabiting the dense rain forests of
the Congo. As despite numerous anecdotal accounts and cultural references, the scientific validation remains more elusive.
My study aims to proper evaluate the existence of the J'BaFofi through the analysis of historical reports,indigenous
testimonies and modern exploration efforts.

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.

GBSN - Microbiology (Unit 2) Antimicrobial agents

Antimicrobial Agents in Therapy

Lattice Defects in ionic solid compound.pptx

lattice of ionic solid

22PH503 - Astronomy and Astrophysics - Unit 2 - Spectral Classification of Stars

Undergrad Physics - BSc Physics

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gastroretentive drug delivery system-PPT.pptx

PPT of gastro retentive drug delivery system

JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDS

The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.

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

WEB PROGRAMMING bharathiar university bca unitII

web programming unitII

BANANA BUNCHY TOP K R.pptx

BANANA BUNCHY TOP K R.pptx

Explainable Deepfake Image/Video Detection

Explainable Deepfake Image/Video Detection

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

MICROBIAL INTERACTION PPT/ MICROBIAL INTERACTION AND THEIR TYPES // PLANT MIC...

MICROBIAL INTERACTION PPT/ MICROBIAL INTERACTION AND THEIR TYPES // PLANT MIC...

Gadgets for management of stored product pests_Dr.UPR.pdf

Gadgets for management of stored product pests_Dr.UPR.pdf

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

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Rodents, Birds and locust_Pests of crops.pdf

Rodents, Birds and locust_Pests of crops.pdf

一比一原版美国佩斯大学毕业证如何办理

一比一原版美国佩斯大学毕业证如何办理

Discovery of Merging Twin Quasars at z=6.05

Discovery of Merging Twin Quasars at z=6.05

Evaluation and Identification of J'BaFofi the Giant Spider of Congo and Moke...

Evaluation and Identification of J'BaFofi the Giant Spider of Congo and Moke...

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

GBSN - Microbiology (Unit 2) Antimicrobial agents

GBSN - Microbiology (Unit 2) Antimicrobial agents

Lattice Defects in ionic solid compound.pptx

Lattice Defects in ionic solid compound.pptx

22PH503 - Astronomy and Astrophysics - Unit 2 - Spectral Classification of Stars

22PH503 - Astronomy and Astrophysics - Unit 2 - Spectral Classification of Stars

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gastroretentive drug delivery system-PPT.pptx

gastroretentive drug delivery system-PPT.pptx

JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDS

JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDS

Methods of grain storage Structures in India.pdf

Methods of grain storage Structures in India.pdf

WEB PROGRAMMING bharathiar university bca unitII

WEB PROGRAMMING bharathiar university bca unitII

- 1. Tilings in Math, Art and Science Bob Culley
- 2. Tiling: universal, fundamental pattern making
- 3. Islamic Tiling Patterns Perhaps the most extensive use and development of tiling patterns is found in Islamic Art. These patterns spread with the rise of Islamic societies from Spain to China: the Umayyads, Abbasids, Fatimids, Seljuqs, Ilkhanids, Timurids, Safavids, Ottomans, Mughals, 7th century on Darb-i Imam Shrine, Isfahan, Iran 1453 Tash Hauli Palace, Khiva, Uzbekistan, 19th century
- 4. “Know, oh brother...that the study of sensible geometry leads to skill in all the practical arts, while the study of intelligible geometry leads to skill in the intellectual arts because this science is one of the gates through which we move to the knowledge of the essence of the soul, and that is the root of all knowledge... “ from the Rasa’il of the Brethren of Purity, 10th century C.E., translated by S.H.Nasr, in “Islamic Patterns”, Keith Critchlow “The artist and the mathematician in Arab civilization have become one. And I mean quite literally.” - Jacob Bronowski, quoted in “Symmetries of Islamic Geometrical Patterns”, Syed Jan Abas and Amer Shaker Salman
- 5. Connecting Art and Mathematics Historical documents from the House of Wisdom: On the Geometric Constructions Necessary for the Artisan, by Abu’l-Wafa Buzjani (ca. 940– 998), anonymous work, On Interlocks of Similar or Corresponding Figures (ca. 1300)
- 6. The Wallpaper (a.k.a. Plane Crystallographic) Groups are 17 symmetries composed of translations, rotations, reflections and glide reflections Starting with Edith Müller’s thesis in 1944, who found 12 of the groups, mathematicians have debated whether all 17 occur in the Alhambra. Branko Grünbaum in 2006 questioned the definition of the problem: does color count in the symmetries or just shape? If colors count, there are 17. He also wrote: “Groups of symmetry had no significance to the artists and artisans who decorated the Alhambra”
- 7. Modern Mathematical View of Tiling Some single (monohedral) shapes don’t tile. Which do? What’s a good prototile? Not a packing, which can have gaps Not a covering, which can have overlaps A plane tiling T is a countable family of closed sets which cover the (Euclidean) plane without gaps or overlaps. Each tile T is a closed topological disk, i.e. the tile boundaries are simple closed curves. The union of the tiles is the plane, and the interiors of the tiles are pairwise disjoint. (from Tilings and Patterns, Grünbaum and Shephard)
- 8. Single Regular Polygon Tiling: 3, 4, 6, not 5
- 9. Many other kinds of tilings Regular polygons, but not edge-to-edge Voderberg’s non-convex 9-gon (ennagon) spiral tiling One of Kepler’s tilings including star polygons
- 10. Aperiodic Tilings Periodic Tiling: lines up with itself after a plane translation. Aperiodic: can’t 1961: Hao Wang conjectures: If a finite set of tiles will tile the plane, it can do so periodically. and that there should exist an algorithm to determine whether any given set of tiles will do this. 1966: Robert Berger showed (using an equivalence to the Halting Problem!) that no such algorithm exists, and aperiodic sets of tiles exist. Berger came up with a set of 20,426 such tiles. Then reduced them to 104. Don Knuth reduced them to 92. Karel Culik came up with these 13:
- 11. Aperiodic 2: Robinson to Penrose 1971: Raphael M. Robinson found an aperiodic set of 6 modified rectangular tiles; used projections and dents rather than coloring matching 1974: Roger Penrose produces the dart and kite aperiodic prototile set. John H. Conway suggests a colored line based matching rule. Can a single prototile set be aperiodic? It is still an open question.
- 12. “Everywhere there is found...a silent swerving from accuracy by an inch that is the uncanny element in everything… a sort of secret treason in the universe.” G.K. Chesterton as quoted by Martin Gardner (in Penrose Tiles and Trapdoor Ciphers) Aperiodic patterns with Penrose tiles
- 13. X-Ray Crystallography ~1912 Max Von Laue developed X-Ray Crystallography (Physics Nobel 1914)
- 14. This becomes a standard crystallography method: over 400,000 solids were characterized in 70 years. All fit this “3,4,6 and not 5” model. Three-fold, four-fold, and six-fold symmetries. Five-fold and higher than six-fold symmetries are “proven” impossible.
- 15. Connecting Aperiodicity to Crystals 1982: Alan Lindsay Mackay, a crystallographer, puts circles at the intersections of a Penrose tiling, computes the diffraction pattern: the result is a 10-fold symmetry April 8 1982: Dr. Dan Shechtman, from the Technion in Israel, doing metallurgy experiments at NBS (now NIST) on AlMg sees 10-fold symmetry in his electron beam crystal diffraction patterns. He tries unsuccessfully to publish his results, has other crystallographers review and check his results.
- 16. 10 Fold ???
- 17. "There's no such thing as quasicrystals, only quasi- scientists." - Linus Pauling 1984: Physicists Paul Steinhardt and Dov Levine connect the work of Mackay and Shechtman, coin the term quasicrystal in an article weeks after Shechtman’s work is finally published. Significant resistance to quasicrystals continued, but so did experimental results. In 1993, the International Union of Crystallographers changed the definition of “crystal” to include quasiperiodic crystals and many other structures. 2010: Mackay, Steinhardt and Levine get the Buckley Prize in Physics 2011: Dan Shechtman receives the Nobel Prize in Chemistry Hundreds of quasicrystals have now been found, including the natural quasicrystal icosahedrite.
- 18. The Girih Tiles In 2007, Peter Lu, a student of Paul Steinhardt, came up with the Girih tiles They match many Islamic tiling patterns and they match the Penrose dart/kites. Later Lu found the Girih tiles match the Topkapi scroll (~1500 c.e.), a Timurid set of instructions for artisans
- 19. The Girih Tiles as a Penrose Tiling The Girih Tiles fit the tiling of the Darb-i Imam Shrine (and many other tilings)
- 20. References and Resources: The Wikipedia pages on tessellation, Wallpaper Group, Crystallographic Restriction Theorem, aperiodic tiling, quasicrystal and each of the individuals named in these slides are pretty good. YouTube has presentations of their work by Dan Shechtman, Paul Steinhardt, and Peter Yu. Books: Introduction to Tessellations by Dale Seymour and Jill Britton: very simple, graphic explanations, art oriented, but includes the algebra shown in this talk Tilings and Patterns by Branko Grünbaum and G.C. Sheppard: encyclopedic, definitive work on mathematics of tilings Penrose Tilings to Trapdoor Ciphers and the Return of Dr. Matrix by Martin Gardner: the first two chapters give the history of aperiodic tilings up to the discovery of quasicrystals Symmetries of Islamic Geometric Patterns by Syed Jan Abas and Amer Shaker Salman: discusses some of the history of Islamic patterns (not just tilings) then catalogs many patterns according to Wallpaper group symmetry
- 21. Appendix: Archimedean tilings Archimedes (ca. 287-212 B.C.E) gave us the Stomachion (see the Archimedes Codex) - the oldest known geometric puzzle - but not the Archimedean tilings
- 22. Semi-regular polygonal tilings The general formula for the interior angle of an n-gon: So for three regular polygons with sides n1, n2, n3 to fit together: This simplifies to: Extending this analysis to combinations of regular polygons, there are 21 combinations possible. 17 are distinct combinations: 4 are just different orderings of the same sets of polygons. 11 of these fit together to tile the plane, called the uniform or Archimedean tilings
- 23. Semi-regular Polygonal Tilings Similarly for four, five or six polygons: But six is a maximum, since 60 degrees is the smallest regular polygon angle Each solution corresponds to a tiling pattern, e. g.

- Found all over the world history; Bricks: mass manufacturing, std. parts; natural question of possible patterns; pictures: modern mud brick palestine, road paving,Uruk (Sumerian ~3400 BC), Alhambra, Escher Lizards, graphene
- Elaborate on Abu’l-Wafa Buzjani (Astronomy, Spherical Trigonometry) or more generally House of Wisdom guys: Algebra, Algorithm Or Archimedes and Stomachicon.
- 3, 4 or 6 not 5 Do diagrams as click animation Show algebraic solution
- Confirmation bias congeals into dogma
- Shectman announced running for President in Israel, Jan. 2014! (explain Twining)
- Paradigm shift: A spectrum of organization from crystal to glass. Icosahedrite found
- 2007 Top 100 Science Stories Show Penrose dart/kite inside Girih tiles
- Actually, there are 21 solutions, but some are the same polygon combinations in different orders