Euclid of Alexandria the Writer of the Elements
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Applying the Polygon Angle
POLYGONS 8.1.1 – 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more non- intersecting straight line segments connected end-to-end. Using the fact that the sum of the measures of the angles in a triangle is 180°, students learn a method to determine the sum of the measures of the interior angles of any polygon. Next they explore the sum of the measures of the exterior angles of a polygon. Finally they use the information about the angles of polygons along with their Triangle Toolkits to find the areas of regular polygons. See the Math Notes boxes in Lessons 8.1.1, 8.1.5, and 8.3.1. Example 1 4x + 7 3x + 1 x + 1 The figure at right is a hexagon. What is the sum of the measures of the interior angles of a hexagon? Explain how you know. Then write an equation and solve for x. 2x 3x – 5 5x – 4 One way to find the sum of the interior angles of the 9 hexagon is to divide the figure into triangles. There are 11 several different ways to do this, but keep in mind that we 8 are trying to add the interior angles at the vertices. One 6 12 way to divide the hexagon into triangles is to draw in all of 10 the diagonals from a single vertex, as shown at right. 7 Doing this forms four triangles, each with angle measures 5 4 3 1 summing to 180°. -
Geometry Around Us an Introduction to Non-Euclidean Geometry
Geometry Around Us An Introduction to Non-Euclidean Geometry Gaurish Korpal (gaurish4math.wordpress.com) National Institute of Science Education and Research, Bhubaneswar 23 April 2016 Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 1 / 15 Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's Miracle Frank Morley (USA) discovered a theorem about triangle in 1899, approximately 2000 years after first theorems about triangle were published by Euclid. Euclid (Egypt) is referred as father of Plane Geometry, often called Euclidean Geometry. Theorem In any triangle, trisector lines intersect at three points, that are vertices of an equilateral triangle. A. Bogomolny, Morley's Theorem: Proof by R. J. Webster, Interactive Mathematics Miscellany and Puzzle, http://www.cut-the-knot.org/triangle/Morley/Webster.shtml Science Day Celebration Gaurish Korpal(gaurish4math.wordpress.com) (NISER) Geometry Around Us 23 April 2016 2 / 15 Morley's original proof was (published in 1924) stemmed from his results on algebraic curves tangent to a given number of lines. It arose from the consideration of cardioids [(x 2 + y 2 − a2)2 = 4a2((x − a)2 + y 2)]. -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Interior and Exterior Angles of Polygons 2A
Regents Exam Questions Name: ________________________ G.CO.C.11: Interior and Exterior Angles of Polygons 2a www.jmap.org G.CO.C.11: Interior and Exterior Angles of Polygons 2a 1 Which type of figure is shown in the accompanying 5 In the diagram below of regular pentagon ABCDE, diagram? EB is drawn. 1) hexagon 2) octagon 3) pentagon 4) quadrilateral What is the measure of ∠AEB? 2 What is the measure of each interior angle of a 1) 36º regular hexagon? 2) 54º 1) 60° 3) 72º 2) 120° 4) 108º 3) 135° 4) 270° 6 What is the measure, in degrees, of each exterior angle of a regular hexagon? 3 Determine, in degrees, the measure of each interior 1) 45 angle of a regular octagon. 2) 60 3) 120 4) 135 4 Determine and state the measure, in degrees, of an interior angle of a regular decagon. 7 A stop sign in the shape of a regular octagon is resting on a brick wall, as shown in the accompanying diagram. What is the measure of angle x? 1) 45° 2) 60° 3) 120° 4) 135° 1 Regents Exam Questions Name: ________________________ G.CO.C.11: Interior and Exterior Angles of Polygons 2a www.jmap.org 8 One piece of the birdhouse that Natalie is building 12 The measure of an interior angle of a regular is shaped like a regular pentagon, as shown in the polygon is 120°. How many sides does the polygon accompanying diagram. have? 1) 5 2) 6 3) 3 4) 4 13 Melissa is walking around the outside of a building that is in the shape of a regular polygon. -
Water, Air and Fire at Work in Hero's Machines
Water, air and fire at work in Hero’s machines Amelia Carolina Sparavigna Dipartimento di Fisica, Politecnico di Torino Corso Duca degli Abruzzi 24, Torino, Italy Known as the Michanikos, Hero of Alexandria is considered the inventor of the world's first steam engine and of many other sophisticated devices. Here we discuss three of them as described in his book “Pneumatica”. These machines, working with water, air and fire, are clear examples of the deep knowledge of fluid dynamics reached by the Hellenistic scientists. Hero of Alexandria, known as the Mechanicos, lived during the first century in the Roman Egypt [1]. He was probably a Greek mathematician and engineer who resided in the city of Alexandria. We know his work from some of writings and designs that have been arrived nowadays in their Greek original or in Arabic translations. From his own writings, it is possible to gather that he knew the works of Archimedes and of Philo the Byzantian, who was a contemporary of Ctesibius [2]. It is almost certain that Heron taught at the Museum, a college for combined philosophy and literary studies and a religious place of cult of Muses, that included the famous Library. For this reason, Hero claimed himself a pupil of Ctesibius, who was probably the first head of the Museum of Alexandria. Most of Hero’s writings appear as lecture notes for courses in mathematics, mechanics, physics and pneumatics [2]. In optics, Hero formulated the Principle of the Shortest Path of Light, principle telling that if a ray of light propagates from a point to another one within the same medium, the followed path is the shortest possible. -
Plato As "Architectof Science"
Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries. -
15 Famous Greek Mathematicians and Their Contributions 1. Euclid
15 Famous Greek Mathematicians and Their Contributions 1. Euclid He was also known as Euclid of Alexandria and referred as the father of geometry deduced the Euclidean geometry. The name has it all, which in Greek means “renowned, glorious”. He worked his entire life in the field of mathematics and made revolutionary contributions to geometry. 2. Pythagoras The famous ‘Pythagoras theorem’, yes the same one we have struggled through in our childhood during our challenging math classes. This genius achieved in his contributions in mathematics and become the father of the theorem of Pythagoras. Born is Samos, Greece and fled off to Egypt and maybe India. This great mathematician is most prominently known for, what else but, for his Pythagoras theorem. 3. Archimedes Archimedes is yet another great talent from the land of the Greek. He thrived for gaining knowledge in mathematical education and made various contributions. He is best known for antiquity and the invention of compound pulleys and screw pump. 4. Thales of Miletus He was the first individual to whom a mathematical discovery was attributed. He’s best known for his work in calculating the heights of pyramids and the distance of the ships from the shore using geometry. 5. Aristotle Aristotle had a diverse knowledge over various areas including mathematics, geology, physics, metaphysics, biology, medicine and psychology. He was a pupil of Plato therefore it’s not a surprise that he had a vast knowledge and made contributions towards Platonism. Tutored Alexander the Great and established a library which aided in the production of hundreds of books. -
Analytic Geometry
STATISTIC ANALYTIC GEOMETRY SESSION 3 STATISTIC SESSION 3 Session 3 Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). 2D Shapes Activity: Sorting Shapes Triangles Right Angled Triangles Interactive Triangles Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite Interactive Quadrilaterals Shapes Freeplay Perimeter Area Area of Plane Shapes Area Calculation Tool Area of Polygon by Drawing Activity: Garden Area General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more: Pentagon Pentagra m Hexagon Properties of Regular Polygons Diagonals of Polygons Interactive Polygons The Circle Circle Pi Circle Sector and Segment Circle Area by Sectors Annulus Activity: Dropping a Coin onto a Grid Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Here is a short reference for you: -
Theon of Alexandria and Hypatia
CREATIVE MATH. 12 (2003), 111 - 115 Theon of Alexandria and Hypatia Michael Lambrou Abstract. In this paper we present the story of the most famous ancient female math- ematician, Hypatia, and her father Theon of Alexandria. The mathematician and philosopher Hypatia flourished in Alexandria from the second part of the 4th century until her violent death incurred by a mob in 415. She was the daughter of Theon of Alexandria, a math- ematician and astronomer, who flourished in Alexandria during the second part of the fourth century. Information on Theon’s life is only brief, coming mainly from a note in the Suda (Suida’s Lexicon, written about 1000 AD) stating that he lived in Alexandria in the times of Theodosius I (who reigned AD 379-395) and taught at the Museum. He is, in fact, the Museum’s last attested member. Descriptions of two eclipses he observed in Alexandria included in his commentary to Ptolemy’s Mathematical Syntaxis (Almagest) and elsewhere have been dated as the eclipses that occurred in AD 364, which is consistent with Suda. Although originality in Theon’s works cannot be claimed, he was certainly immensely influential in the preservation, dissemination and editing of clas- sic texts of previous generations. Indeed, with the exception of Vaticanus Graecus 190 all surviving Greek manuscripts of Euclid’s Elements stem from Theon’s edition. A comparison to Vaticanus Graecus 190 reveals that Theon did not actually change the mathematical content of the Elements except in minor points, but rather re-wrote it in Koini and in a form more suitable for the students he taught (some manuscripts refer to Theon’s sinousiai). -
The Ptolemies: an Unloved and Unknown Dynasty. Contributions to a Different Perspective and Approach
THE PTOLEMIES: AN UNLOVED AND UNKNOWN DYNASTY. CONTRIBUTIONS TO A DIFFERENT PERSPECTIVE AND APPROACH JOSÉ DAS CANDEIAS SALES Universidade Aberta. Centro de História (University of Lisbon). Abstract: The fifteen Ptolemies that sat on the throne of Egypt between 305 B.C. (the date of assumption of basileia by Ptolemy I) and 30 B.C. (death of Cleopatra VII) are in most cases little known and, even in its most recognised bibliography, their work has been somewhat overlooked, unappreciated. Although boisterous and sometimes unloved, with the tumultuous and dissolute lives, their unbridled and unrepressed ambitions, the intrigues, the betrayals, the fratricides and the crimes that the members of this dynasty encouraged and practiced, the Ptolemies changed the Egyptian life in some aspects and were responsible for the last Pharaonic monuments which were left us, some of them still considered true masterpieces of Egyptian greatness. The Ptolemaic Period was indeed a paradoxical moment in the History of ancient Egypt, as it was with a genetically foreign dynasty (traditions, language, religion and culture) that the country, with its capital in Alexandria, met a considerable economic prosperity, a significant political and military power and an intense intellectual activity, and finally became part of the world and Mediterranean culture. The fifteen Ptolemies that succeeded to the throne of Egypt between 305 B.C. (date of assumption of basileia by Ptolemy I) and 30 B.C. (death of Cleopatra VII), after Alexander’s death and the division of his empire, are, in most cases, very poorly understood by the public and even in the literature on the topic. -
From Ancient Greece to Byzantium
Proceedings of the European Control Conference 2007 TuA07.4 Kos, Greece, July 2-5, 2007 Technology and Autonomous Mechanisms in the Mediterranean: From Ancient Greece to Byzantium K. P. Valavanis, G. J. Vachtsevanos, P. J. Antsaklis Abstract – The paper aims at presenting each period are then provided followed by technology and automation advances in the accomplishments in automatic control and the ancient Greek World, offering evidence that transition from the ancient Greek world to the Greco- feedback control as a discipline dates back more Roman era and the Byzantium. than twenty five centuries. II. CHRONOLOGICAL MAP OF SCIENCE & TECHNOLOGY I. INTRODUCTION It is worth noting that there was an initial phase of The paper objective is to present historical evidence imported influences in the development of ancient of achievements in science, technology and the Greek technology that reached the Greek states from making of automation in the ancient Greek world until the East (Persia, Babylon and Mesopotamia) and th the era of Byzantium and that the main driving force practiced by the Greeks up until the 6 century B.C. It behind Greek science [16] - [18] has been curiosity and was at the time of Thales of Miletus (circa 585 B.C.), desire for knowledge followed by the study of nature. when a very significant change occurred. A new and When focusing on the discipline of feedback control, exclusively Greek activity began to dominate any James Watt’s Flyball Governor (1769) may be inherited technology, called science. In subsequent considered as one of the earliest feedback control centuries, technology itself became more productive, devices of the modern era. -
Constructive Solid Geometry Concepts
3-1 Chapter 3 Constructive Solid Geometry Concepts Understand Constructive Solid Geometry Concepts Create a Binary Tree Understand the Basic Boolean Operations Use the SOLIDWORKS CommandManager User Interface Setup GRID and SNAP Intervals Understand the Importance of Order of Features Use the Different Extrusion Options 3-2 Parametric Modeling with SOLIDWORKS Certified SOLIDWORKS Associate Exam Objectives Coverage Sketch Entities – Lines, Rectangles, Circles, Arcs, Ellipses, Centerlines Objectives: Creating Sketch Entities. Rectangle Command ................................................3-10 Boss and Cut Features – Extrudes, Revolves, Sweeps, Lofts Objectives: Creating Basic Swept Features. Base Feature .............................................................3-9 Reverse Direction Option ........................................3-16 Hole Wizard ..............................................................3-20 Dimensions Objectives: Applying and Editing Smart Dimensions. Reposition Smart Dimension ..................................3-11 Feature Conditions – Start and End Objectives: Controlling Feature Start and End Conditions. Reference Guide Reference Extruded Cut, Up to Next .......................................3-25 Associate Certified Constructive Solid Geometry Concepts 3-3 Introduction In the 1980s, one of the main advancements in solid modeling was the development of the Constructive Solid Geometry (CSG) method. CSG describes the solid model as combinations of basic three-dimensional shapes (primitive solids). The