Comparison of Euclidean and Non-Euclidean Geometry
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Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. -
Five-Dimensional Design
PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 50, NO. 1, PP. 35–41 (2006) FIVE-DIMENSIONAL DESIGN Elek TÓTH Department of Building Constructions Budapest University of Technology and Economics H–1521 Budapest, POB. 91. Hungary Received: April 3 2006 Abstract The method architectural and engineering design and construction apply for two-dimensional rep- resentation is geometry, the axioms of which were first outlined by Euclid. The proposition of his 5th postulate started on its course a problem of geometry that provoked perhaps the most mistaken demonstrations and that remained unresolved for two thousand years, the question if the axiom of parallels can be proved. The quest for the solution of this problem led Bolyai János to revolutionary conclusions in the wake of which he began to lay the foundations of a new (absolute) geometry in the system of which planes bend and become hyperbolic. It was Einstein’s relativity theory that eventually created the possibility of the recognition of new dimensions by discussing space as well as a curved, non-linear entity. Architectural creations exist in what Kant termed the dual form of intuition, that is in time and space, thus in four dimensions. In this context the fourth dimension is to be interpreted as the current time that may be actively experienced. On the other hand there has been a dichotomy in the concept of time since ancient times. The introduction of a time-concept into the activity of the architectural designer that comprises duration, the passage of time and cyclic time opens up a so-far unknown new direction, a fifth dimension, the perfection of which may be achieved through the comprehension and the synthesis of the coded messages of building diagnostics and building reconstruction. -
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)]. -
Using Non-Euclidean Geometry to Teach Euclidean Geometry to K–12 Teachers
Using non-Euclidean Geometry to teach Euclidean Geometry to K–12 teachers David Damcke Department of Mathematics, University of Portland, Portland, OR 97203 [email protected] Tevian Dray Department of Mathematics, Oregon State University, Corvallis, OR 97331 [email protected] Maria Fung Mathematics Department, Western Oregon University, Monmouth, OR 97361 [email protected] Dianne Hart Department of Mathematics, Oregon State University, Corvallis, OR 97331 [email protected] Lyn Riverstone Department of Mathematics, Oregon State University, Corvallis, OR 97331 [email protected] January 22, 2008 Abstract The Oregon Mathematics Leadership Institute (OMLI) is an NSF-funded partner- ship aimed at increasing mathematics achievement of students in partner K–12 schools through the creation of sustainable leadership capacity. OMLI’s 3-week summer in- stitute offers content and leadership courses for in-service teachers. We report here on one of the content courses, entitled Comparing Different Geometries, which en- hances teachers’ understanding of the (Euclidean) geometry in the K–12 curriculum by studying two non-Euclidean geometries: taxicab geometry and spherical geometry. By confronting teachers from mixed grade levels with unfamiliar material, while mod- eling protocol-based pedagogy intended to emphasize a cooperative, risk-free learning environment, teachers gain both content knowledge and insight into the teaching of mathematical thinking. Keywords: Cooperative groups, Euclidean geometry, K–12 teachers, Norms and protocols, Rich mathematical tasks, Spherical geometry, Taxicab geometry 1 1 Introduction The Oregon Mathematics Leadership Institute (OMLI) is a Mathematics/Science Partner- ship aimed at increasing mathematics achievement of K–12 students by providing professional development opportunities for in-service teachers. -
Squaring the Circle in Elliptic Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1]. -
Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence
Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence • Ax1. For every pair of antipodal point P and P’ and for every pair of antipodal point Q and Q’ such that P≠Q and P’≠Q’, there exists a unique circle incident with both pairs of points. • Ax2. For every great circle c, there exist at least two distinct pairs of antipodal points incident with c. • Ax3. There exist three distinct pairs of antipodal points with the property that no great circle is incident with all three of them. Betweenness Axioms Betweenness fails on circles What is the relation among points? • (A,B/C,D)= points A and B separates C and D 2. Axioms of Separation • Ax4. If (A,B/C,D), then points A, B, C, and D are collinear and distinct. In other words, non-collinear points cannot separate one another. • Ax5. If (A,B/C,D), then (B,A/C,D) and (C,D/A,B) • Ax6. If (A,B/C,D), then not (A,C/B,D) • Ax7. If points A, B, C, and D are collinear and distinct then (A,B/C,D) , (A,C/B,D) or (A,D/B,C). • Ax8. If points A, B, and C are collinear and distinct then there exists a point D such that(A,B/C,D) • Ax9. For any five distinct collinear points, A, B, C, D, and E, if (A,B/E,D), then either (A,B/C,D) or (A,B/C,E) Definition • Let l and m be any two lines and let O be a point not on either of them. -
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: -
And Are Lines on Sphere B That Contain Point Q
11-5 Spherical Geometry Name each of the following on sphere B. 3. a triangle SOLUTION: are examples of triangles on sphere B. 1. two lines containing point Q SOLUTION: and are lines on sphere B that contain point Q. ANSWER: 4. two segments on the same great circle SOLUTION: are segments on the same great circle. ANSWER: and 2. a segment containing point L SOLUTION: is a segment on sphere B that contains point L. ANSWER: SPORTS Determine whether figure X on each of the spheres shown is a line in spherical geometry. 5. Refer to the image on Page 829. SOLUTION: Notice that figure X does not go through the pole of ANSWER: the sphere. Therefore, figure X is not a great circle and so not a line in spherical geometry. ANSWER: no eSolutions Manual - Powered by Cognero Page 1 11-5 Spherical Geometry 6. Refer to the image on Page 829. 8. Perpendicular lines intersect at one point. SOLUTION: SOLUTION: Notice that the figure X passes through the center of Perpendicular great circles intersect at two points. the ball and is a great circle, so it is a line in spherical geometry. ANSWER: yes ANSWER: PERSEVERANC Determine whether the Perpendicular great circles intersect at two points. following postulate or property of plane Euclidean geometry has a corresponding Name two lines containing point M, a segment statement in spherical geometry. If so, write the containing point S, and a triangle in each of the corresponding statement. If not, explain your following spheres. reasoning. 7. The points on any line or line segment can be put into one-to-one correspondence with real numbers. -
+ 2R - P = I + 2R - P + 2, Where P, I, P Are the Invariants P and Those of Zeuthen-Segre for F
232 THE FOURTH DIMENSION. [Feb., N - 4p - (m - 2) + 2r - p = I + 2r - p + 2, where p, I, p are the invariants p and those of Zeuthen-Segre for F. If I is the same invariant for F, since I — p = I — p, we have I = 2V — 4p — m = 2s — 4p — m. Hence 2s is equal to the "equivalence" in nodes of the point (a, b, c) in the evaluation of the invariant I for F by means of the pencil Cy. This property can be shown directly for the following cases: 1°. F has only ordinary nodes. 2°. In the vicinity of any of these nodes there lie other nodes, or ordinary infinitesimal multiple curves. In these cases it is easy to show that in the vicinity of the nodes all the numbers such as h are equal to 2. It would be interesting to know if such is always the case, but the preceding investigation shows that for the applications this does not matter. UNIVERSITY OF KANSAS, October 17, 1914. THE FOURTH DIMENSION. Geometry of Four Dimensions. By HENRY PARKER MANNING, Ph.D. New York, The Macmillan Company, 1914. 8vo. 348 pp. EVERY professional mathematician must hold himself at all times in readiness to answer certain standard questions of perennial interest to the layman. "What is, or what are quaternions ?" "What are least squares?" but especially, "Well, have you discovered the fourth dimension yet?" This last query is the most difficult of the three, suggesting forcibly the sophists' riddle "Have you ceased beating your mother?" The fact is that there is no common locus standi for questioner and questioned. -
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 -
Lecture 3: Geometry
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-2015 Lecture 3: Geometry Geometry is the science of shape, size and symmetry. While arithmetic dealt with numerical structures, geometry deals with metric structures. Geometry is one of the oldest mathemati- cal disciplines and early geometry has relations with arithmetics: we have seen that that the implementation of a commutative multiplication on the natural numbers is rooted from an inter- pretation of n × m as an area of a shape that is invariant under rotational symmetry. Number systems built upon the natural numbers inherit this. Identities like the Pythagorean triples 32 +42 = 52 were interpreted geometrically. The right angle is the most "symmetric" angle apart from 0. Symmetry manifests itself in quantities which are invariant. Invariants are one the most central aspects of geometry. Felix Klein's Erlanger program uses symmetry to classify geome- tries depending on how large the symmetries of the shapes are. In this lecture, we look at a few results which can all be stated in terms of invariants. In the presentation as well as the worksheet part of this lecture, we will work us through smaller miracles like special points in triangles as well as a couple of gems: Pythagoras, Thales,Hippocrates, Feuerbach, Pappus, Morley, Butterfly which illustrate the importance of symmetry. Much of geometry is based on our ability to measure length, the distance between two points. A modern way to measure distance is to determine how long light needs to get from one point to the other. This geodesic distance generalizes to curved spaces like the sphere and is also a practical way to measure distances, for example with lasers. -
Feb 23 Notes: Definition: Two Lines L and M Are Parallel If They Lie in The
Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal. We define alternate interior angles, corresponding angles, alternate exterior angles, and interior angles on the same side of the transversal using various betweeness and half-plane notions. Suppose line l intersects lines m and n at points B and E, respectively, with points A and C on line m and points D and F on line n such that A-B-C and D-E-F, with A and D on the same side of l. Suppose also that G and H are points such that H-E-B- G. Then pABE and pBEF are alternate interior angles, as are pCBE and pDEB. pABG and pFEH are alternate exterior angles, as are pCBG and pDEH. pGBC and pBEF are a pair of corresponding angles, as are pGBA & pBED, pCBE & pFEH, and pABE & pDEH. pCBE and pFEB are interior angles on the same side of the transversal, as are pABE and pDEB. Our Last Theorem in Absolute Geometry: If two lines in the same plane are cut by a transversal so that a pair of alternate interior angles are congruent, the lines are parallel. Proof: Let l intersect lines m and n at points A and B respectively. Let p1 p2. Suppose m and n meet at point C. Then either p1 is exterior to ªABC, or p2 is exterior to ªABC. In the first case, the exterior angle inequality gives p1 > p2; in the second, it gives p2 > p1.