1) Mathematics is found in many natural phenomena such as ratios, proportions, geometric shapes, symmetry, and fractals. Concepts like congruence and ratios are important in everyday activities.
2) Foams and bubbles exhibit mathematical properties like Plateau's laws which describe how soap films meet. Fractal patterns are seen in trees, ferns, coastlines, and other natural forms.
3) Symmetry is ubiquitous in nature from bilateral symmetry in animals to radial and rotational symmetries in plants and crystals. Snowflakes have sixfold symmetry while sea stars have fivefold symmetry.
1) Mathematics is present in many natural phenomena such as patterns, symmetry, ratios, and proportions. Concepts like congruence and ratios are important in everyday activities.
2) Geometry describes shapes and spaces and is useful for applications like architecture, engineering, and home projects. Geometric concepts like volume and surface area are important in fields like construction.
3) Many natural structures exhibit self-similarity and fractal properties across different scales, from trees and ferns to coastlines and clouds. Symmetry is also common in nature, seen in structures like snowflakes, sea anemones, and crystals.
This document discusses the mathematical aspects found in natural environmental phenomena. It provides examples of congruence seen in identical flowers or leaves of the same plant. Similarity is discussed with examples like pinecone spirals and same-plant flowers having proportionate dimensions and shape. Geometric shapes in nature are examined like honeycomb, pineapple, and ladies finger. Ratio and proportion are evident in distances like eye to chin and eye to mouth. Symmetry is described and examples given like butterflies, insects, and leaves being exactly alike but for orientation. The document concludes mathematical models can describe the environment and factors affecting it to enable predictions.
This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and the Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes their interior angles sum to 180 degrees. Triangles are classified by angle and side length into right, acute, obtuse, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are shown to be proportional to a ratio of corresponding sides. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
This document discusses trigonometry and its key concepts. It defines trigonometry as a branch of mathematics concerning the study of triangles and the relationship between side lengths and angles. It notes that Hipparchus of Nicaea in the 2nd century BC is considered the father of trigonometry. The document outlines important trigonometric ratios like sine, cosine and tangent, and defines concepts like adjacent, opposite, and hypotenuse sides of a right triangle. It also discusses Pythagoras' theorem relating side lengths in a right triangle.
This document discusses three topics related to triangles: the Basic Proportionality Theorem, areas of similar triangles, and trigonometry. It explains the Basic Proportionality Theorem and its proof. It then discusses that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Finally, it defines trigonometry as the measurement of triangles and describes the ratios of the sides of a right triangle and some real-life applications of trigonometry such as periodic functions and architectural design.
This document is from a geometry textbook lesson on ratios in similar polygons. It begins with examples of identifying similar polygons by comparing corresponding angles and side lengths. A similarity ratio is defined as the ratio of corresponding sides of similar polygons. Examples then show how to determine if polygons are similar, write the similarity ratio, and use proportions to solve problems involving scale factors between similar figures. The lesson concludes with a quiz to assess understanding of identifying and applying properties of similar polygons.
This document discusses three rules for determining if triangles are similar: the AA Similarity Postulate, SAS Similarity Theorem, and SSS Similarity Theorem. It provides examples of applying each rule to determine if triangles are similar, and if so, writing the similarity statement and ratio.
This document provides lesson materials on isosceles and equilateral triangles including:
- Key vocabulary terms like legs, vertex angle, and base of an isosceles triangle.
- The Isosceles Triangle Theorem and its converse.
- Properties and theorems regarding equilateral triangles.
- Examples proving triangles congruent using corresponding parts of congruent triangles (CPCTC).
- A lesson quiz to assess understanding of isosceles triangle properties and angle measures.
1) Mathematics is present in many natural phenomena such as patterns, symmetry, ratios, and proportions. Concepts like congruence and ratios are important in everyday activities.
2) Geometry describes shapes and spaces and is useful for applications like architecture, engineering, and home projects. Geometric concepts like volume and surface area are important in fields like construction.
3) Many natural structures exhibit self-similarity and fractal properties across different scales, from trees and ferns to coastlines and clouds. Symmetry is also common in nature, seen in structures like snowflakes, sea anemones, and crystals.
This document discusses the mathematical aspects found in natural environmental phenomena. It provides examples of congruence seen in identical flowers or leaves of the same plant. Similarity is discussed with examples like pinecone spirals and same-plant flowers having proportionate dimensions and shape. Geometric shapes in nature are examined like honeycomb, pineapple, and ladies finger. Ratio and proportion are evident in distances like eye to chin and eye to mouth. Symmetry is described and examples given like butterflies, insects, and leaves being exactly alike but for orientation. The document concludes mathematical models can describe the environment and factors affecting it to enable predictions.
This document provides an overview of triangles, including their properties, types, similarity, areas of similar triangles, and the Pythagorean theorem. It defines triangles as polygons with three sides and angles, and notes their interior angles sum to 180 degrees. Triangles are classified by angle and side length into right, acute, obtuse, isosceles, equilateral, and scalene varieties. Similarity is discussed, along with criteria like equal angles and proportional sides. Areas of similar triangles are shown to be proportional to a ratio of corresponding sides. Finally, the Pythagorean theorem and its applications to find missing sides of right triangles are covered.
This document discusses trigonometry and its key concepts. It defines trigonometry as a branch of mathematics concerning the study of triangles and the relationship between side lengths and angles. It notes that Hipparchus of Nicaea in the 2nd century BC is considered the father of trigonometry. The document outlines important trigonometric ratios like sine, cosine and tangent, and defines concepts like adjacent, opposite, and hypotenuse sides of a right triangle. It also discusses Pythagoras' theorem relating side lengths in a right triangle.
This document discusses three topics related to triangles: the Basic Proportionality Theorem, areas of similar triangles, and trigonometry. It explains the Basic Proportionality Theorem and its proof. It then discusses that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Finally, it defines trigonometry as the measurement of triangles and describes the ratios of the sides of a right triangle and some real-life applications of trigonometry such as periodic functions and architectural design.
This document is from a geometry textbook lesson on ratios in similar polygons. It begins with examples of identifying similar polygons by comparing corresponding angles and side lengths. A similarity ratio is defined as the ratio of corresponding sides of similar polygons. Examples then show how to determine if polygons are similar, write the similarity ratio, and use proportions to solve problems involving scale factors between similar figures. The lesson concludes with a quiz to assess understanding of identifying and applying properties of similar polygons.
This document discusses three rules for determining if triangles are similar: the AA Similarity Postulate, SAS Similarity Theorem, and SSS Similarity Theorem. It provides examples of applying each rule to determine if triangles are similar, and if so, writing the similarity statement and ratio.
This document provides lesson materials on isosceles and equilateral triangles including:
- Key vocabulary terms like legs, vertex angle, and base of an isosceles triangle.
- The Isosceles Triangle Theorem and its converse.
- Properties and theorems regarding equilateral triangles.
- Examples proving triangles congruent using corresponding parts of congruent triangles (CPCTC).
- A lesson quiz to assess understanding of isosceles triangle properties and angle measures.
This document discusses two postulates for proving triangles congruent: the Side-Side-Side (SSS) postulate and the Side-Angle-Side (SAS) postulate. It provides examples of proofs using each postulate, demonstrating how to prove triangles congruent by showing that three sides or two sides and the included angle are congruent between the two triangles. The document includes practice problems applying these postulate proofs.
Two geometrical figures are similar if one can be obtained from the other by a uniform scaling (enlarging or shrinking) while maintaining the same shape. Specifically, two polygons are similar if they have the same corresponding angles and proportional side lengths, and two triangles are similar if they have the same three angles. The similarity of figures is denoted with a tilde symbol such as ΔABC ~ ΔDEF for similar triangles.
The document contains information about using similarity postulates to determine if triangles are similar and find unknown side lengths. It provides examples of using the AA, SAS, and SSS similarity postulates to show triangles are similar. It also shows using proportional sides to find missing lengths. The objective is for students to use similarity postulates to decide if triangles are similar and find unknowns.
This document outlines steps for teaching trigonometry, including reviewing algebraic and geometric skills, learning about right triangles and trigonometric ratios, applying concepts to non-right triangles using rules like the Sine Rule and Cosine Rule, measuring angles in radians, learning other trigonometric ratios, solving trigonometric equations, and providing tips for instruction. It also discusses common difficulties students face and proposes active learning approaches to help overcome challenges.
This document provides definitions for various math terms starting with letters A through B. Some key terms defined include:
- Absolute Value: Makes a negative number positive. Absolute value uses bars like |-7| = 7.
- Acute Angle: An angle whose measure is less than 90 degrees.
- Arithmetic Mean: The simple average, calculated by adding all values and dividing by the total number.
- Binomial: A polynomial with two terms, like x + y.
The document consists of concise 1-2 sentence definitions for over 50 common math terms used in subjects like algebra, geometry, trigonometry, and statistics. The definitions are arranged alphabetically to serve as a quick
This document discusses angle relationships and congruence. It defines congruent angles as angles that have the same degree of measure. The Vertical Angles Theorem states that vertical angles are congruent. Several theorems are also presented: if two angles are congruent, then their complements and supplements are congruent; if two angles are complementary or supplementary to the same angle, then they are congruent; and if two angles are congruent and supplementary, then each is a right angle. The document provides examples to solve for the measure of unknown angles using these theorems and properties. Homework problems are assigned from pages 126-127, problems 9-21 and 24-28.
1. The document discusses triangle similarity and the different postulates and theorems used to prove triangles are similar, including AA similarity, SSS similarity, and SAS similarity.
2. Examples are provided to demonstrate using these rules to verify similarity between triangles and find missing side lengths in similar triangles using proportional reasoning. Proofs of triangle similarity are also presented.
3. Applications of triangle similarity include finding dimensions in similar right triangles that model roof designs and solving other geometric problems.
4 2 & 4-3 parallel lines and transversalsgwilson8786
This document defines key terms related to parallel lines and transversals, including transversal, interior angles, exterior angles, alternate interior angles, consecutive interior angles, and corresponding angles. It states the theorems that if two parallel lines are cut by a transversal, then each pair of alternate interior angles, consecutive interior angles, alternate exterior angles, and corresponding angles are congruent.
This document discusses several geometry theorems:
- The Basic Proportionality Theorem (also known as Thales' Theorem) states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
- There are four similarity criteria: AAA, SAS, SSS, and ASA, which define conditions for when two triangles are similar.
- The Area Theorem states that the ratio of the areas of similar triangles equals the square of the ratio of their corresponding sides.
- Pythagoras' Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
The document discusses congruence and similarity in geometry. Congruence refers to two objects having the same shape and size. Similarity refers to two objects having the same shape, though not necessarily the same size. Specific examples are provided of congruent and similar line segments, angles, circles, polygons, and triangles. The concepts of uniform scaling, matching corresponding sides and angles, and proportional sides are also covered.
This document provides examples and explanations for proving triangles similar using the Angle-Angle (AA) criterion. It includes examples of showing two triangles are similar by showing they have two pairs of congruent angles. It also includes examples of writing similarity statements and using proportions to find missing side lengths when corresponding angles and one pair of corresponding sides are given. Guided practice problems allow students to practice determining if triangles are similar and writing similarity statements.
This document contains questions about geometric shapes, their properties, and relationships between angles. It asks about definitions of parallelograms, quadrilaterals, Pythagorean theorem, triangle congruency postulates including SAS, SSS, sum of interior angles. It also asks about properties of right triangles, isosceles triangles, HL theorem, congruency of right triangles, transversals, alternate interior angles, corresponding angles, perpendicular bisectors, and angle bisectors.
This module covers triangle congruence and how it can be used to prove that segments and angles are congruent. There are four criteria for triangle congruence: SSS, SAS, ASA, and SAA. There are also criteria for right triangle congruence including LL, LA, HyL, and HyA. Examples are provided of formal proofs using triangle congruence to show that segments and angles are congruent. The document concludes with practice problems for the student to try using triangle congruence to prove statements.
This document discusses ways to prove that two lines are parallel using postulates and theorems about angles formed when lines are cut by a transversal. Specifically, it states that two lines are parallel if:
- A pair of corresponding angles is congruent
- A pair of alternate interior angles is congruent
- A pair of alternate exterior angles is congruent
- A pair of consecutive interior angles is supplementary
- The two lines are perpendicular to a third line
Examples are provided to demonstrate applying these rules to identify parallel lines and find values that prove lines are parallel.
This document discusses properties of isosceles and equilateral triangles. It begins with examples calculating angle measures using properties of isosceles triangles, such as the fact that the two base angles of an isosceles triangle are congruent. It then proves theorems about isosceles and equilateral triangles, including that an equilateral triangle is also equiangular. Coordinate proofs are presented to show properties such as the midpoint of the two sides of an isosceles triangle bisecting the base. The document concludes with a lesson quiz reviewing the material.
The document discusses the topics of algebra and its history. It defines algebra as the study of mathematical symbols and the rules for manipulating symbols. Algebra originated from Arabic and Persian mathematicians in the 9th century who developed methods for solving equations. The basics of algebra taught in elementary courses involve using variables, expressions, and terms. More advanced areas of algebra study abstract algebraic structures and their properties.
In this first volume of our main scientific research We showed that the «Armenian Theory of Special Relativity» is full of fine and difficult ideas to understand, which in many cases seems to conflict with our everyday experiences and legacy conceptions. This new crash course book is the simplified version for broad audiences. This book is not just generalizing transformation equations and all relativistic formulas; It is also without limitations and uses a pure mathematical approach to bring forth new revolutionary ideas in the theory of relativity. It also paves the way to build general theory of relativity and finally for the construction of the unified field theory – the ultimate dream of every truth seeking physicist.
Armenian Theory of Relativity is such a mathematically solid and perfect theory that it cannot be wrong. Therefore, our derived transformation equations and all relativistic formulas have the potential to not just replace legacy relativity formulas, but also rewrite all modern physics. Lorentz transformation equations and other relativistic formulas is a very special case of the Armenian Theory of Relativity when we put s = 0 and g = -1 .
The proofs in this book are very brief, therefore with just a little effort, the readers themselves can prove all the provided formulas in detail. You can find the more detailed proofs of the formulas in our main research book «Armenian Theory of Special Relativity», published in Armenia of June 2013.
In this visual book, you will set your eyes on many new and beautiful formulas which the world has never seen before, especially the crown jewel of the Armenian Theory of Relativity - Armenian energy and Armenian momentum formulas, which can change the future of the human species and bring forth the new golden age.
The time has come to reincarnate the ether as a universal reference medium which does not contradict relativity theory. Our new theory explains all these facts and peacefully brings together followers of absolute ether theory, relativistic ether theory and dark energy theory. We just need to mention that the absolute ether medium has a very complex geometric character, which has never been seen before.
Hyperbolic geometry was developed in the 19th century as a non-Euclidean geometry that discards one of Euclid's parallel postulate. It assumes that through a point not on a given line there are multiple parallel lines. This led to discoveries like triangles having interior angles summing to less than 180 degrees. Key figures who developed it include Gauss, Bolyai, Lobachevsky, and models include the Klein model, Poincaré disk model, and hyperboloid model.
This document provides an overview of non-Euclidean geometry, which studies shapes and constructions that do not follow Euclidean geometry. It contrasts Euclidean geometry, where parallel lines do not intersect, with hyperbolic geometry, where parallel lines diverge, and elliptic geometry, where there are no parallels. The development of non-Euclidean geometries began with ancient mathematicians attempting to prove Euclid's parallel postulate, and it was fully established in the 19th century by Bolyai, Lobachevsky, Gauss, and Riemann. Non-Euclidean geometry is important in physics, appearing in Einstein's theory of relativity.
This document provides a history of non-Euclidean geometry, beginning with Euclid's fifth postulate and early attempts to prove it from the other four postulates. In the early 19th century, Bolyai, Lobachevsky, and Gauss independently developed hyperbolic geometry by replacing the fifth postulate. However, their work was initially rejected by the mathematical community. Later, Riemann generalized the concept of geometry and Beltrami provided a model showing the consistency of non-Euclidean geometry. Klein classified the three types of geometry as hyperbolic, elliptic and Euclidean. Non-Euclidean geometry has since found applications in Einstein's theory of relativity and GPS systems.
Mathematics is evident everywhere in nature and is an integral part of our lives. It is the science of patterns, quantities and relationships. The document discusses several examples of patterns in nature like geometric shapes, symmetry, the Fibonacci sequence and golden ratio that are all deeply rooted in mathematics. It also elaborates on the importance and applications of mathematics in fields like science, technology, medicine and more, establishing it as an indispensable and universal language.
- Geometric shapes like triangles, squares, pentagons and hexagons are commonly found throughout nature in structures like mountain formations, plant leaves and flowers, animal skins, crystal structures, and more. Their repeated precise geometric patterns suggest an underlying mathematical order or structure inherent in nature.
- Triangles in particular are ubiquitous in nature, seen in mountain ranges, volcanoes, plant structures like clover and trillium flowers, seed pods, and more. Their presence may be partly due to erosion processes but their precise repetition over large areas is remarkable.
- Many types of polygons from triangles to dodecahedrons can be observed throughout the natural world, demonstrating the pervasive role of mathematics and geometry in shaping the forms and
This document discusses two postulates for proving triangles congruent: the Side-Side-Side (SSS) postulate and the Side-Angle-Side (SAS) postulate. It provides examples of proofs using each postulate, demonstrating how to prove triangles congruent by showing that three sides or two sides and the included angle are congruent between the two triangles. The document includes practice problems applying these postulate proofs.
Two geometrical figures are similar if one can be obtained from the other by a uniform scaling (enlarging or shrinking) while maintaining the same shape. Specifically, two polygons are similar if they have the same corresponding angles and proportional side lengths, and two triangles are similar if they have the same three angles. The similarity of figures is denoted with a tilde symbol such as ΔABC ~ ΔDEF for similar triangles.
The document contains information about using similarity postulates to determine if triangles are similar and find unknown side lengths. It provides examples of using the AA, SAS, and SSS similarity postulates to show triangles are similar. It also shows using proportional sides to find missing lengths. The objective is for students to use similarity postulates to decide if triangles are similar and find unknowns.
This document outlines steps for teaching trigonometry, including reviewing algebraic and geometric skills, learning about right triangles and trigonometric ratios, applying concepts to non-right triangles using rules like the Sine Rule and Cosine Rule, measuring angles in radians, learning other trigonometric ratios, solving trigonometric equations, and providing tips for instruction. It also discusses common difficulties students face and proposes active learning approaches to help overcome challenges.
This document provides definitions for various math terms starting with letters A through B. Some key terms defined include:
- Absolute Value: Makes a negative number positive. Absolute value uses bars like |-7| = 7.
- Acute Angle: An angle whose measure is less than 90 degrees.
- Arithmetic Mean: The simple average, calculated by adding all values and dividing by the total number.
- Binomial: A polynomial with two terms, like x + y.
The document consists of concise 1-2 sentence definitions for over 50 common math terms used in subjects like algebra, geometry, trigonometry, and statistics. The definitions are arranged alphabetically to serve as a quick
This document discusses angle relationships and congruence. It defines congruent angles as angles that have the same degree of measure. The Vertical Angles Theorem states that vertical angles are congruent. Several theorems are also presented: if two angles are congruent, then their complements and supplements are congruent; if two angles are complementary or supplementary to the same angle, then they are congruent; and if two angles are congruent and supplementary, then each is a right angle. The document provides examples to solve for the measure of unknown angles using these theorems and properties. Homework problems are assigned from pages 126-127, problems 9-21 and 24-28.
1. The document discusses triangle similarity and the different postulates and theorems used to prove triangles are similar, including AA similarity, SSS similarity, and SAS similarity.
2. Examples are provided to demonstrate using these rules to verify similarity between triangles and find missing side lengths in similar triangles using proportional reasoning. Proofs of triangle similarity are also presented.
3. Applications of triangle similarity include finding dimensions in similar right triangles that model roof designs and solving other geometric problems.
4 2 & 4-3 parallel lines and transversalsgwilson8786
This document defines key terms related to parallel lines and transversals, including transversal, interior angles, exterior angles, alternate interior angles, consecutive interior angles, and corresponding angles. It states the theorems that if two parallel lines are cut by a transversal, then each pair of alternate interior angles, consecutive interior angles, alternate exterior angles, and corresponding angles are congruent.
This document discusses several geometry theorems:
- The Basic Proportionality Theorem (also known as Thales' Theorem) states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
- There are four similarity criteria: AAA, SAS, SSS, and ASA, which define conditions for when two triangles are similar.
- The Area Theorem states that the ratio of the areas of similar triangles equals the square of the ratio of their corresponding sides.
- Pythagoras' Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
The document discusses congruence and similarity in geometry. Congruence refers to two objects having the same shape and size. Similarity refers to two objects having the same shape, though not necessarily the same size. Specific examples are provided of congruent and similar line segments, angles, circles, polygons, and triangles. The concepts of uniform scaling, matching corresponding sides and angles, and proportional sides are also covered.
This document provides examples and explanations for proving triangles similar using the Angle-Angle (AA) criterion. It includes examples of showing two triangles are similar by showing they have two pairs of congruent angles. It also includes examples of writing similarity statements and using proportions to find missing side lengths when corresponding angles and one pair of corresponding sides are given. Guided practice problems allow students to practice determining if triangles are similar and writing similarity statements.
This document contains questions about geometric shapes, their properties, and relationships between angles. It asks about definitions of parallelograms, quadrilaterals, Pythagorean theorem, triangle congruency postulates including SAS, SSS, sum of interior angles. It also asks about properties of right triangles, isosceles triangles, HL theorem, congruency of right triangles, transversals, alternate interior angles, corresponding angles, perpendicular bisectors, and angle bisectors.
This module covers triangle congruence and how it can be used to prove that segments and angles are congruent. There are four criteria for triangle congruence: SSS, SAS, ASA, and SAA. There are also criteria for right triangle congruence including LL, LA, HyL, and HyA. Examples are provided of formal proofs using triangle congruence to show that segments and angles are congruent. The document concludes with practice problems for the student to try using triangle congruence to prove statements.
This document discusses ways to prove that two lines are parallel using postulates and theorems about angles formed when lines are cut by a transversal. Specifically, it states that two lines are parallel if:
- A pair of corresponding angles is congruent
- A pair of alternate interior angles is congruent
- A pair of alternate exterior angles is congruent
- A pair of consecutive interior angles is supplementary
- The two lines are perpendicular to a third line
Examples are provided to demonstrate applying these rules to identify parallel lines and find values that prove lines are parallel.
This document discusses properties of isosceles and equilateral triangles. It begins with examples calculating angle measures using properties of isosceles triangles, such as the fact that the two base angles of an isosceles triangle are congruent. It then proves theorems about isosceles and equilateral triangles, including that an equilateral triangle is also equiangular. Coordinate proofs are presented to show properties such as the midpoint of the two sides of an isosceles triangle bisecting the base. The document concludes with a lesson quiz reviewing the material.
The document discusses the topics of algebra and its history. It defines algebra as the study of mathematical symbols and the rules for manipulating symbols. Algebra originated from Arabic and Persian mathematicians in the 9th century who developed methods for solving equations. The basics of algebra taught in elementary courses involve using variables, expressions, and terms. More advanced areas of algebra study abstract algebraic structures and their properties.
In this first volume of our main scientific research We showed that the «Armenian Theory of Special Relativity» is full of fine and difficult ideas to understand, which in many cases seems to conflict with our everyday experiences and legacy conceptions. This new crash course book is the simplified version for broad audiences. This book is not just generalizing transformation equations and all relativistic formulas; It is also without limitations and uses a pure mathematical approach to bring forth new revolutionary ideas in the theory of relativity. It also paves the way to build general theory of relativity and finally for the construction of the unified field theory – the ultimate dream of every truth seeking physicist.
Armenian Theory of Relativity is such a mathematically solid and perfect theory that it cannot be wrong. Therefore, our derived transformation equations and all relativistic formulas have the potential to not just replace legacy relativity formulas, but also rewrite all modern physics. Lorentz transformation equations and other relativistic formulas is a very special case of the Armenian Theory of Relativity when we put s = 0 and g = -1 .
The proofs in this book are very brief, therefore with just a little effort, the readers themselves can prove all the provided formulas in detail. You can find the more detailed proofs of the formulas in our main research book «Armenian Theory of Special Relativity», published in Armenia of June 2013.
In this visual book, you will set your eyes on many new and beautiful formulas which the world has never seen before, especially the crown jewel of the Armenian Theory of Relativity - Armenian energy and Armenian momentum formulas, which can change the future of the human species and bring forth the new golden age.
The time has come to reincarnate the ether as a universal reference medium which does not contradict relativity theory. Our new theory explains all these facts and peacefully brings together followers of absolute ether theory, relativistic ether theory and dark energy theory. We just need to mention that the absolute ether medium has a very complex geometric character, which has never been seen before.
Hyperbolic geometry was developed in the 19th century as a non-Euclidean geometry that discards one of Euclid's parallel postulate. It assumes that through a point not on a given line there are multiple parallel lines. This led to discoveries like triangles having interior angles summing to less than 180 degrees. Key figures who developed it include Gauss, Bolyai, Lobachevsky, and models include the Klein model, Poincaré disk model, and hyperboloid model.
This document provides an overview of non-Euclidean geometry, which studies shapes and constructions that do not follow Euclidean geometry. It contrasts Euclidean geometry, where parallel lines do not intersect, with hyperbolic geometry, where parallel lines diverge, and elliptic geometry, where there are no parallels. The development of non-Euclidean geometries began with ancient mathematicians attempting to prove Euclid's parallel postulate, and it was fully established in the 19th century by Bolyai, Lobachevsky, Gauss, and Riemann. Non-Euclidean geometry is important in physics, appearing in Einstein's theory of relativity.
This document provides a history of non-Euclidean geometry, beginning with Euclid's fifth postulate and early attempts to prove it from the other four postulates. In the early 19th century, Bolyai, Lobachevsky, and Gauss independently developed hyperbolic geometry by replacing the fifth postulate. However, their work was initially rejected by the mathematical community. Later, Riemann generalized the concept of geometry and Beltrami provided a model showing the consistency of non-Euclidean geometry. Klein classified the three types of geometry as hyperbolic, elliptic and Euclidean. Non-Euclidean geometry has since found applications in Einstein's theory of relativity and GPS systems.
Mathematics is evident everywhere in nature and is an integral part of our lives. It is the science of patterns, quantities and relationships. The document discusses several examples of patterns in nature like geometric shapes, symmetry, the Fibonacci sequence and golden ratio that are all deeply rooted in mathematics. It also elaborates on the importance and applications of mathematics in fields like science, technology, medicine and more, establishing it as an indispensable and universal language.
- Geometric shapes like triangles, squares, pentagons and hexagons are commonly found throughout nature in structures like mountain formations, plant leaves and flowers, animal skins, crystal structures, and more. Their repeated precise geometric patterns suggest an underlying mathematical order or structure inherent in nature.
- Triangles in particular are ubiquitous in nature, seen in mountain ranges, volcanoes, plant structures like clover and trillium flowers, seed pods, and more. Their presence may be partly due to erosion processes but their precise repetition over large areas is remarkable.
- Many types of polygons from triangles to dodecahedrons can be observed throughout the natural world, demonstrating the pervasive role of mathematics and geometry in shaping the forms and
Mathematics is evident throughout nature in intricate and complex ways. Symmetry, shapes like spheres and hexagons, patterns like fractals and the Fibonacci sequence, and mathematical structures like tessellations are all demonstrated in the natural world. What was once seen as randomness is now understood as the application of mathematical principles that illustrate the complexity of the natural order.
1. The document discusses the concepts of natural resources, congruence, and similarity in geometry. It defines congruence as two figures having the same shape and size, and similarity as two figures having the same shape, whether normally or as mirrors of each other.
2. Methods for determining congruence of polygons are described, including matching and labeling corresponding vertices and rotating/reflecting one figure to match the other.
3. Two figures are similar if they have the same shape, normally or as mirrors, and corresponding sides are in proportion while corresponding angles have the same measure.
1. The document discusses the concepts of natural resources, congruence, and similarity in geometry. It defines congruence as two figures having the same shape and size, and similarity as two figures having the same shape, whether in their normal or mirrored form.
2. Methods for determining congruence of polygons are described, including matching and labeling corresponding vertices and rotating/reflecting one figure to match the other.
3. Two figures are similar if they have the same shape, either normally or as mirrors. Corresponding sides of similar polygons are in proportion and corresponding angles have the same measure.
This document discusses the mathematical aspects found in natural environmental phenomena. It provides examples of congruence seen in identical flowers or leaves of the same plant. Similarity is seen in pinecones and same-plant flowers that have proportionate dimensions and shapes. Geometric shapes like honeycombs, pineapples, and ladies fingers are found in nature. Ratios explain proportions like the distance between eyes and mouth. Symmetry is visible in butterflies, insects, and leaves, and spirals describe hurricane structures. The document concludes that mathematical models can describe the environment and factors affecting it to enable predictions.
This document provides an overview of a workshop on permaculture and sacred geometry. The workshop will cover basic concepts of permaculture like working with nature, ethics of care for the earth, people, and fair share. It will then discuss sacred geometry, exploring shapes like the platonic solids, golden ratio, mandala, and how they relate to nature. The workshop will involve group work using intuition to understand intention and the alchemical process of transforming a current reality into a vision using sacred geometry as a tool.
The document discusses natural resources, geometry, geometric shapes, and symmetry. It defines natural resources as materials found on Earth that satisfy human needs. Geometry is a branch of mathematics that studies geometric shapes. A geometric shape is defined by its properties that remain after transformations like moving, enlarging, rotating or reflecting the shape. Symmetry refers to an object being unchanged after transformations like flipping, sliding or turning, with reflection symmetry occurring when one half of a shape is a mirror image of the other half.
This document discusses concepts of congruence and similarity found in nature and everyday life. It provides definitions of congruence as shapes being the same size and shape, while similarity means shapes have the same proportions. Examples of congruence and similarity discussed include leaves being similar in shape on a tree, buildings having congruent structures, insects having congruent wing designs, fruits displaying similarity through seed alignment, and the human body exhibiting congruence through structures like hands and DNA. The conclusion states mathematics can be found all around us and understanding concepts like congruence enables better handling of everyday situations and resources.
This document provides an overview of Module 1 of a mathematics course titled "Mathematics in the Modern World". The module introduces patterns and sequences found in nature and discusses different types of patterns, symmetries, and sequences. It explains patterns seen in nature through Fibonacci sequences, spirals in shells and flowers, packing problems in honeycombs. It also discusses different types of sequences like arithmetic, geometric, and harmonic sequences. The module emphasizes how mathematics is important and applicable in understanding the world, for organization, prediction, and control of systems. It provides learning outcomes and examples of patterns, symmetries and sequences.
1) Mathematics is used extensively in science for measurements, calculations, and showing relationships between scientific properties. Arithmetic, algebra, and advanced mathematics like calculus are applied in areas like physics, chemistry, and astronomy.
2) Specific mathematical concepts are used for different scientific domains. For example, fractions and decimals are used in chemistry for compositions, graphs are made in meteorology, and trigonometry is applied to measure tree heights.
3) Advanced mathematics is necessary for more complex theories and equations, like relativity or quantum mechanics. Overall, mathematics provides the means to make quantitive analyses and predictions in scientific disciplines.
This document discusses various topics in mensuration including the formulas for calculating the area and perimeter of basic shapes like rectangles, parallelograms, trapezoids, triangles, cubes, cuboids, and cylinders. It also covers the concepts of direct and inverse proportion and provides examples. Finally, it discusses the construction of a quadrilateral given specific length measurements of its sides.
Crop circles first appeared over 300 years ago in England. Over time, they grew in size and complexity, evolving from simple circles to intricate geometric patterns. Mathematicians have studied crop circles and discovered that many follow precise geometric rules and relationships. Some mathematicians even proposed new theorems based on analyses of crop circle patterns. While the origins of crop circles remain mysterious, they have inspired new mathematical discoveries and interest in geometry.
This document describes four different projects related to geometry:
1) Investigating the efficiency of packing cylindrical objects in a cuboid container using square and hexagonal packing arrangements. Hexagonal packing was found to be slightly more efficient (80.3% vs 78.5%).
2) Using properties of similar triangles to solve problems like determining the height of a mirror or tree, or the width of a pathway.
3) Comparing the probability of birthdays calculated for each month to experimental probabilities found by surveying birthdays in classrooms.
4) Studying how the distances between points and the ratio of the centroid remain unchanged when a geometric figure like a triangle is displaced or rotated using coordinate geometry.
This document discusses how mathematical concepts like symmetry, shapes, patterns and sequences are evident in nature. It provides examples of bilateral symmetry (like in humans and butterflies), radial symmetry (like in flowers and starfish), parallel lines (like in leaf veins and zebra stripes), basic shapes (like hexagons in bee hives and spheres in planets), the Fibonacci sequence (seen in rabbit birth rates), and patterns (like leaf arrangements, zebra fur and sand ripples). The document emphasizes that mathematics can be observed throughout nature from atoms to galaxies.
The document thanks the teacher, Mrs. Leena Saji, for giving the opportunity to do a presentation. It also thanks parents and friends for their help in finishing the presentation on time. Finally, it thanks God for everything and bringing the presentation to fruition.
The document provides information on surface area and volume formulas and calculations for basic 3D shapes including prisms, cubes, cylinders, cones, and spheres. It defines key terms like surface area and volume and provides example calculations and formulas for finding the surface area and volume of cubes, rectangular prisms, cylinders, cones, and spheres. Diagrams and examples are included to illustrate the different shapes and how to set up the surface area and volume calculations.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
Three-Dimensional Geometry discusses spatial relations and three-dimensional figures. It explains that three-dimensional figures have faces, edges, and vertices. The document provides examples and formulas for calculating the volumes of prisms, cylinders, cones, pyramids and cubes. It also discusses surface area and provides examples and formulas for calculating surface areas of prisms and cylinders.
The document discusses the prevalence and applications of the golden ratio, also known as phi, in mathematics, nature, art, architecture, music, and the human body. Some key points include:
- The golden ratio is approximately 1.618 and can be seen in the proportions of flowers, shells, galaxies, DNA, and the human face/body.
- It has been used intentionally in architecture for centuries, appearing in structures like the Parthenon and pyramids of Giza.
- The Fibonacci sequence is related to the golden ratio and can be observed in patterns in nature as well as the piano keyboard.
- Artists, architects and designers continue to find inspiration from the golden ratio's
Similar to 4. aswathi. m.s online assignment natural resourses (20)
Anti-Universe And Emergent Gravity and the Dark UniverseSérgio Sacani
Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and provide evidence for the fact that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Compositions of iron-meteorite parent bodies constrainthe structure of the pr...Sérgio Sacani
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.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
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.
Physics Investigatory Project on transformers. Class 12thpihuart12
Physics investigatory project on transformers with required details for 12thes. with index, theory, types of transformers (with relevant images), procedure, sources of error, aim n apparatus along with bibliography🗃️📜. Please try to add your own imagination rather than just copy paste... Hope you all guys friends n juniors' like it. peace out✌🏻✌🏻
Rodents, Birds and locust_Pests of crops.pdfPirithiRaju
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
Embracing Deep Variability For Reproducibility and Replicability
Abstract: Reproducibility (aka determinism in some cases) constitutes a fundamental aspect in various fields of computer science, such as floating-point computations in numerical analysis and simulation, concurrency models in parallelism, reproducible builds for third parties integration and packaging, and containerization for execution environments. These concepts, while pervasive across diverse concerns, often exhibit intricate inter-dependencies, making it challenging to achieve a comprehensive understanding. In this short and vision paper we delve into the application of software engineering techniques, specifically variability management, to systematically identify and explicit points of variability that may give rise to reproducibility issues (eg language, libraries, compiler, virtual machine, OS, environment variables, etc). The primary objectives are: i) gaining insights into the variability layers and their possible interactions, ii) capturing and documenting configurations for the sake of reproducibility, and iii) exploring diverse configurations to replicate, and hence validate and ensure the robustness of results. By adopting these methodologies, we aim to address the complexities associated with reproducibility and replicability in modern software systems and environments, facilitating a more comprehensive and nuanced perspective on these critical aspects.
https://hal.science/hal-04582287
Order : Trombidiformes (Acarina) Class : Arachnida
Mites normally feed on the undersurface of the leaves but the symptoms are more easily seen on the uppersurface.
Tetranychids produce blotching (Spots) on the leaf-surface.
Tarsonemids and Eriophyids produce distortion (twist), puckering (Folds) or stunting (Short) of leaves.
Eriophyids produce distinct galls or blisters (fluid-filled sac in the outer layer)
2. Natural resources- Mathematical
aspects
found in Environmental phenomena
congruence,
similarity, ratio and proportion,
geometric shapes, symmetric property
etc
3. Natural resources
INTRODUCTION
Math is all around us, even in the kitchen. When cooking a meal, we
sometimes checks the recipe to measure the portion or double-check the ratio of
ingredients. Choosing between a cup of sugar or a half cup is a mathematical
decision. Even setting and checking the timer to make sure the dish cooks for the
appropriate amount of time requires math skills. When at the mall looking for an
outfit to wear, we use math. Patterns in nature are visible regularities of form found
in the natural world. These patterns recur in different contexts and can sometimes
be modelled mathematically.
congruence
The concept of congruence is not restricted to the study of geometry. it plays
an important role in everyday living. We may be able to buy a refill for our pen when
the ink runs dry. We use congruence to replace a worn out part of the car, looking for
the 'same part number'. In construction, they have to use the blocks of a 's tandard
size'. When using screws, we look for the kind of screw that we need and also
screwdriver that fits. In each of these examples, the idea of sameness of shape and
sizes applies.
Ratio and proportion
Ratios: Relationships between quantities: That ingredients have relationships to
each other in a recipe is an important concept in cooking. It's also an important math
concept. In math, this relationship between 2 quantities is called a ratio. If a recipe
calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2.
In mathematical language, that relationship can be written in two ways:
4. 1/2 or 1:2
Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter
that ratio, the results may not be edible.
Working with proportion
All recipes are written to serve a certain number of people or yield a certain amount
of food. You might come across a cookie recipe that makes 2 dozen cookies, for
example. What if you only want 1 dozen cookies? What if you want 4 dozen
cookies? Understanding how to increase or decrease the yield without spoiling the
ratio of ingredients is a valuable skill for any cook.
Let's say you have a mouth-watering cookie recipe:
1 cup flour
1/2 tsp. baking soda
1/2 tsp. salt
1/2 cup butter
1/3 cup brown sugar
1/3 cup sugar
1 egg
1/2 tsp. vanilla
1 cup chocolate chips
This recipe will yield 3 dozen cookies. If you want to make 9 dozen cookies, you'll
have to increase the amount of each ingredient listed in the recipe. You'll also need to
make sure that the relationship between the ingredients stays the same. To do this,
you'll need to understand proportion. A proportion exists when you have 2 equal
ratios, such as 2:4 and 4:8. Two unequal ratios, such as 3:16 and 1:3, don't result in a
proportion. The ratios must be equal
Eg. Golden ratio
5. A golden rectangle with longer sidea and shorter side b, when placed adjacent to a
square with sides of length a, will produce a similar golden rectangle with longer
side a + b and shorter side a. This illustrates the relationship .
In mathematics, two quantities are in the golden ratio if their ratio is the same as the
ratio of their sum to the larger of the two quantities. The figure on the right illustrates
the geometric relationship. Expressed algebraically, for
quantities a and b with a > b > 0,
where the Greek letter phi (φ) represents the golden ratio. Its value is:
The golden ratio is also called the golden section (Latin: sectio aurea)
or golden mean. Other names include extreme and mean ratio, medial
section, divine proportion, divine section (Latin: sectio divina), golden
proportion, golden cut, and golden number.
GEOMETRY
Geometry is one of the oldest sciences and is concerned with questions of shape, size
and relative position of figures and with properties of space.Geometry is considered
an important field of study because of its applications in daily life.Geometry is
mainly divided in two ;
Plane geometry - It is about all kinds of two dimensional shapes such as lines,circles
and triangles.Solid geometry - It is about all kinds of three dimensional shapes like
polygons,prisms,pyramids,sphere and cylinder.
Role of geometry in daily life:- Role of geometry in the daily life is the foundation
of physical mathematics. A room, a car, a ball anything with physical things is
geometrically formed. Geometry applies us to accurately calculate physical spaces.n
the world , Anything made use of geometrical constraints this is important
application in daily life of geometry.
Example: Architecture of a thing, design, engineering, building etc.
Geometry is particularly useful in home building or improvement projects. If you
need to find the floor area of a house, you need to use geometry. If you want to
replace a piece of furniture, you need to calculate the amount of fabric you want, by
6. calculating the surface area of the furniture. Geometry has applications in hobbies.
The goldfish tank water needs to have a certain volume as well as surface area in
order for the fish to thrive. We can calculate the volume and surface area using
geometry
Bubbles, foam
A soap bubble forms a sphere, a surface with minimal area — the smallest
possible surface area for the volume enclosed. Two bubbles together form a more
complex shape: the outer surfaces of both bubbles are spherical; these surfaces are
joined by a third spherical surface as the smaller bubble bulges slightly into the larger
one.
A foam is a mass of bubbles; foams of different materials occur in nature.
Foams composed of soap films obey Plateau's laws, which require three soap films to
meet at each edge at 120° and four soap edges to meet at each vertex at the
tetrahedral angle of about 109.5°. Plateau's laws further require films to be smooth
and continuous, and to have a constant average at every point. For example, a film
may remain nearly flat on average by being curved up in one direction (say, left to
right) while being curved downwards in another direction (say, front to back).
Structures with minimal surfaces can be used as tents. Lord Kelvin identified the
problem of the most efficient way to pack cells of equal volume as a foam in 1887;
his solution uses just one solid, the truncated cubic honeycomb with very slightly
curved faces to meet Plateau's laws.
At the scale of living cells, foam patterns are common; radiolarians, sponge
spicules, silicoflagellate exoskeletons and the calcite skeleton of a sea urchin,
Cidaris rugosa, all resemble mineral casts of Plateau foam boundaries. The skeleton
of the Radiolarian, Aulonia hexagona, a beautiful marine form drawn by Haeckel,
looks as if it is a sphere composed wholly of hexagons, but this is mathematically
impossible. The Euler characteristic states that for any convex polyhedron, the
number of faces plus the number of vertices (corners) equals the number of edges
plus two. A result of this formula is that any closed polyhedron of hexagons has to
include exactly 12 pentagons, like a soccer ball, Buckminster Fuller geodesic dome,
or fullerene molecule. This can be visualised by noting that a mesh of hexagons is
flat like a sheet of chicken wire, but each pentagon that is added forces the mesh to
bend (there are fewer corners, so the mesh is pulled in).
7.
Foam of soap bubbles: 4 edges meet at each vertex, at angles close to 109.5°, as in
two C-H bonds in methane.
Beautiful examples of nature’s self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a
part of itself (i.e. the whole has the same shape as one or more of the parts). Many
objects in the real world, such as coastlines, are statistically self-similar: parts of
them show the same statistical properties at many scales. Self-similarity is a typical
. property of fractals
Cloud cotton
Trees, fractals:- Fractals are infinitely self-similar, iterated mathematical constructs
having fractal dimensions. Infinite iteration is not possible in nature so all 'fractal'
patterns are only approximate. For example, the leaves
of ferns and umbellifers(Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels.
Fern-like growth patterns occur in plants and in animals
including bryozoa, corals, hydrozoa like the air fern, Sertularia argentea, and in non-living
things, notably electrical discharges. Lindenmayer system fractals can model
different patterns of tree growth by varying a small number of parameters including
8. branching angle, distance between nodes or branch points (internode length), and
number of branches per branch point.
Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river
networks, geologic fault lines,mountains, coastlines, animal coloration, snow
flakes, crystals, blood vessel branching, and ocean waves.
Leaf of Cow Parsley,Anthriscus sylvestris, is 2- or 3-pinnate, not infinite
Fractal spirals:Romanesco broccolishowing self-similar form
Angelica flowerhead, a sphere made of spheres (self-similar)
Trees: Lichtenberg figure: high voltage dielectric breakdown in an acrylic polymer
block
9.
Trees: dendritic Copper crystals (in microscope)
Symmetry Symmetry in biology, Floral symmetry and crystal symmetry
Symmetry is pervasive in living things. Animals mainly have bilateral or mirror
symmetry, as do the leaves of plants and some flowers such as orchids. Plants often
have radial or rotational symmetry, as do many flowers and some groups of animals
such. as sea anemones Fivefold symmetry is found in the echinoderms, the group that
includes starfish, sea urchins, and sea lilies.
Among non-living things, snowflakes have striking sixfold symmetry: each flake is
unique, its structure forming a record of the varying conditions during its
crystallisation, with nearly the same pattern of growth on each of its six
arms. Crystals in general have a variety of symmetries and crystal habits; they can be
cubic or octahedral, but true crystals cannot have fivefold symmetry
(unlike quasicrystals). Rotational symmetry is found at different scales among non-living
things including the crown-shaped splash pattern formed when a drop falls into
a pond,[28] and both the spheroidal shape and rings of a planet like Saturn.
Symmetry has a variety of causes. Radial symmetry suits organisms like sea
anemones whose adults do not move: food and threats may arrive from any direction.
But animals that move in one direction necessarily have upper and lower sides, head
and tail ends, and therefore a left and a right. The head becomes specialised with a
mouth and sense organs (cephalisation), and the body becomes bilaterally symmetric
(though internal organs need not be). More puzzling is the reason for the fivefold
(pentaradiate) symmetry of the echinoderms. Early echinoderms were bilaterally
symmetrical, as their larvae still are. Sumrall and Wray argue that the loss of the old
symmetry had both developmental and ecological causes.
Animals often show mirror or bilateral symmetry, like this tiger.
10.
Echinoderms like thisstarfish have fivefold symmetry.
Fivefold symmetry can be seen in many flowers and some fruits like thismedlar.
Snowflakes have sixfold symmetry.
Each snowflake is unique but symmetrical.
Fluorite showing cubiccrystal habit
12. conclusion
Mathematics plays a key role in environmental studies, modeling, etc. Basic
mathematics - calculus, percents, ratios, graphs and charts, sequences, sampling,
averages, a population growth model, variability and probability - all relate to
current, critical issues such as pollution, the availability of resources, environmental
clean-up, recycling, CFC's, and population growth.
Mathematics plays a central role in our scientific picture of the world.
How the connection between mathematics and the world is to be accounted for
remains one of the most challenging problems in philosophy of science, philosophy
of mathematics, and general philosophy. A very important aspect of this problem is
that of accounting for the explanatory role mathematics seems to play in the account
of physical phenomena.
REFERENCES:
Stevens, Peter. Patterns in Nature, 1974.
Balaguer, Mark (12 May 2004, revised 7 April 2009). "Stanford Encyclopedia of
Philosophy"
WWW.WIKIPEDIA.COM .