DOCSLIB.ORG

A Brief History of NON-EUCLIDEAN GEOMETRY DANIEL MARSHALL & PAUL SCOTT

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Brief History of NON-EUCLIDEAN GEOMETRY DANIEL MARSHALL & PAUL SCOTT A brief history of NON-EUCLIDEAN GEOMETRY DANIEL MARSHALL & PAUL SCOTT Euclid It is clear that the fifth postulate is very different to the other four. In fact, in The Around 300 BC, Euclid wrote The Elements, a Elements, the first 28 results are proved major treatise on the geometry of the time, and without it. As a result of this difference, many what would be considered ‘geometry’ for many attempts were made to try to prove the fifth years after. Arguably The Elements is the postulate using the previous four postulates. second most read book of the western world, One earlier attempt at this was made by falling short only to The Bible. In his book, Proclus (410–485). Despite his attempts even- Euclid states five postulates of geometry which tually resulting in failure, Proclus discovered he uses as the foundation for all his proofs. It an equivalent statement for the fifth postulate. is from these postulates we get the term This is now known as Playfair’s Axiom. It says Euclidean geometry, for in these Euclid strove the following: to define what constitutes ‘flat-surface’ geom- etry. These postulates are: Given a line and a point not on the line, it is 1. [It is possible] to draw a straight line possible to draw exactly one line through the from any point to any other. given point parallel to the line. 2. [It is possible] to produce a finite straight line continuously in a straight line. 3. [It is possible] to describe a circle with Saccheri any centre and distance [radius]. 4. That all right angles The attempts to try and prove the fifth postu- are equal to each late in terms of the other four continued. The other. first major breakthrough was due to Girolamo 5. That, if a straight Saccheri in 1697. His technique involves line falling on two assuming the fifth postulate false and straight lines makes attempting to derive a contradiction. What the interior angles Saccheri finds is shown in the diagram on on the same side page 3: the summit angles ADC and BCD are less than two right equal. This gives three cases for him to angles, the two consider: lines, if produced 1. The summit angles are > 90 degrees indefinitely, meet on (hypothesis of the obtuse angle). that side on which 2. The summit angles are < 90 degrees the angles are less (hypothesis of the acute angle). than the two right 3. The summit angles are = 90 degrees Euclid angles. (hypothesis of the right angle). 2 amt 60 (3) Using Euclid’s assumption that a straight necessity of thought. As is often the case in line is infinite, Saccheri manages to derive a mathematics, similar ideas were developed contradiction for the first hypothesis and a independently by Janos Bolyai. His father, hazy contradiction for the second one. Around Wolfgang Bolyai, friend of Gauss, had once 100 years later, Legendre also worked at the told Janos, problem. He gives another equivalent state- ment to the fifth postulate, that is: You ought not to try the road of the parallels; I know the road to its end — I have passed The sum of the angles of a triangle is equal to through this bottomless night, every light two right angles. and every joy of my life has been extin- guished by it — I implore you for God’s sake, Using a similar idea to Saccheri’s, Legendre leave the lesson of the parallels in peace… I showed that the sum of the angles of a triangle had purposed to sacrifice myself to the truth; cannot be greater than two right angles; I would have been prepared to be a martyr if however his proof rests on the assumption of only I could have delivered to the human race infinite lines. Legendre also provided a proof a geometry cleansed of this blot. I have on the sum not being less than two right performed dreadful, enormous labours; I angles, but again there was a flaw, in that he have accomplished far more than was makes an assumption equivalent to the fifth accomplished up until now; but never have I postulate. found complete satisfaction… When I discov- ered that the bottom of this night cannot be reached from the earth, I turned back Gauss and Bolyai without solace, pitying myself and the entire human race. The first person to understand the problem of the fifth postulate was Gauss. In 1817, after Janos ignored his father’s impassioned looking at the problem for many years, he had plea, however, and worked on the problem become convinced it was independent of the himself. Like Gauss, he looked at the conse- other four. Gauss then began to look at the quences of the fifth postulate not being consequences of a geometry where this fifth necessary. His major breakthrough, was not postulate was not necessarily true. He never his work, which had already been done by published his work due to pressure of time, Gauss, but the fact that he believed that this perhaps illustrating Kant’s statement that ‘other’ geometry actually existed. Despite the Euclidean geometry requires the inevitable revolutionary new ideas that were being put forward, there was little public recognition to be had. D C Lobachevsky Another mathematician, Lobachevsky, worked on the same problems as Gauss and Bolyai but again, despite working at the same time, he knew nothing of their work. Lobachevsky A B also assumed the fifth postulate was not necessary and from this formed a new geom- ∆ABD is congruent to ∆BAC (two sides and etry. In 1840, he explained how this new included angle). Hence AC = BD so ∆ADC is geometry would work (see diagram on page 4): congruent to ∆BCD (three sides). Therefore ∠ADC = ∠BCD. All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two Saccheri’s quadrilateral classes — into cutting and non-cutting. The amt 60 (3) 3 E G H C Riemann and Klein F The next example of what we could now call a ‘non-euclidean’ geometry was given by D D Riemann. A lecture he gave which was A published in 1868, two years after his death, speaks of a ‘spherical’ geometry in which every F line through a point P not on a line AB meets Acquiring the essential life skills of managing the line AB. Here, no parallels are possible. H G E B Earning, money doesn’t come easily to any of us. Also, in 1868, Eugenio Beltrami wrote a paper But for many Australian students, it’s AD is the perpendicular from A to BC. in which he puts forward a model called a becoming a lot easier. AE is perpendicular to AD. ‘pseudo-sphere’. The importance of this model Within the angle EAD, some lines (such as AF) will is that it gave an example of the first four spending, Building on our School Banking program meet BC. Assume that AE is not the only line which postulates holding but not the fifth. From this, which has helped thousands of young people does not meet BC, so let AG be another such line. it can be seen that non-euclidean geometry is to save, the Commonwealth Bank actively AF is a cutting line and AG is a non-cutting line. just as consistent as euclidean geometry. supports financial literacy in Australian youth. There must be a boundary between cutting and non- In 1871, Klein completed the ideas of non- stashing, In consultation with State and Territory education cutting lines and we may take AH as this boundary. euclidean geometry and gave the solid departments, the Commonwealth Bank has underpinnings to the subject. He shows that developed www.DollarsandSense.com.au there are essentially three types of geometry: Part of Lobachevsky’s calculation. – a money management and life skills web • that proposed by Bolyai and growing, site for teenagers between 14 and 17 years. Lobachevsky, where straight lines have two infinitely distant points, Enhancing the curriculum. boundary lines of the one and the other class • the Riemann ‘spherical’ geometry, where protecting The content of www.DollarsandSense.com.au of those lines will be called parallel to the lines have no infinitely distant points, has been mapped to complement Australian upper given line. and secondary Mathematics and Business curricula. • Euclidean geometry, where for each line From this, Lobachevsky’s geometry has a there are two coincident infinitely and Site features include practical information about new fifth postulate, that is: distant points. managing money; budgeting for goals such as a car or going to uni; financial skill tests and There exist two lines parallel to a given line tips; and forums with experts such as Telstra’s through a given point not on the line. losing it. Business Woman of the Year Di Yerbury, Commonwealth Bank Chief Economist Michael Clearly, this is not equivalent to Euclid’s Blythe and young entrepreneur Ainsley Gilkes. geometry. Lobachevsky went on to develop many trigonometric identities for triangles The web site is designed to be used which held in this geometry, showing that as independently by students and as a teaching tool the triangle becomes small the identities tend for teachers. Features of the teachers’ section to the usual trigonometric identities. include a guide for delivering learning outcomes; Pseudosphere a curriculum library; as well as quizzes and research questions to set for students. Reference Now financial common sense is just a click away. Eves, H. (1972). A Survey of Geometry. Allyn and Bacon.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
    BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein.
    [Show full text]
  • Canada Archives Canada Published Heritage Direction Du Branch Patrimoine De I'edition
    Rhetoric more geometrico in Proclus' Elements of Theology and Boethius' De Hebdomadibus A Thesis submitted in Candidacy for the Degree of Master of Arts in Philosophy Institute for Christian Studies Toronto, Ontario By Carlos R. Bovell November 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-43117-7 Our file Notre reference ISBN: 978-0-494-43117-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
    [Show full text]
  • The Birth of Hyperbolic Geometry Carl Friederich Gauss
    The Birth of Hyperbolic Geometry Carl Friederich Gauss • There is some evidence to suggest that Gauss began studying the problem of the fifth postulate as early as 1789, when he was 12. We know in letters that he had done substantial work of the course of many years: Carl Friederich Gauss • “On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case, the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure.“ – Letter to Gerling, 1816 Carl Friederich Gauss • "I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. Perhaps in another world, we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics.“ –Letter to Olbers, 1817 Carl Friederich Gauss • " There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180°.
    [Show full text]
  • P. 1 Math 490 Notes 7 Zero Dimensional Spaces for (SΩ,Τo)
    p. 1 Math 490 Notes 7 Zero Dimensional Spaces For (SΩ, τo), discussed in our last set of notes, we can describe a basis B for τo as follows: B = {[λ, λ] ¯ λ is a non-limit ordinal } ∪ {[µ + 1, λ] ¯ λ is a limit ordinal and µ < λ}. ¯ ¯ The sets in B are τo-open, since they form a basis for the order topology, but they are also closed by the previous Prop N7.1 from our last set of notes. Sets which are simultaneously open and closed relative to the same topology are called clopen sets. A topology with a basis of clopen sets is defined to be zero-dimensional. As we have just discussed, (SΩ, τ0) is zero-dimensional, as are the discrete and indiscrete topologies on any set. It can also be shown that the Sorgenfrey line (R, τs) is zero-dimensional. Recall that a basis for τs is B = {[a, b) ¯ a, b ∈R and a < b}. It is easy to show that each set [a, b) is clopen relative to τs: ¯ each [a, b) itself is τs-open by definition of τs, and the complement of [a, b)is(−∞,a)∪[b, ∞), which can be written as [ ¡[a − n, a) ∪ [b, b + n)¢, and is therefore open. n∈N Closures and Interiors of Sets As you may know from studying analysis, subsets are frequently neither open nor closed. However, for any subset A in a topological space, there is a certain closed set A and a certain open set Ao associated with A in a natural way: Clτ A = A = \{B ¯ B is closed and A ⊆ B} (Closure of A) ¯ o Iτ A = A = [{U ¯ U is open and U ⊆ A}.
    [Show full text]
  • Molecular Symmetry
    Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry .
    [Show full text]
  • 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.
    [Show full text]
  • Circle Theorems
    Circle theorems A LEVEL LINKS Scheme of work: 2b. Circles – equation of a circle, geometric problems on a grid Key points • A chord is a straight line joining two points on the circumference of a circle. So AB is a chord. • A tangent is a straight line that touches the circumference of a circle at only one point. The angle between a tangent and the radius is 90°. • Two tangents on a circle that meet at a point outside the circle are equal in length. So AC = BC. • The angle in a semicircle is a right angle. So angle ABC = 90°. • When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference. So angle AOB = 2 × angle ACB. • Angles subtended by the same arc at the circumference are equal. This means that angles in the same segment are equal. So angle ACB = angle ADB and angle CAD = angle CBD. • A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. Opposite angles in a cyclic quadrilateral total 180°. So x + y = 180° and p + q = 180°. • The angle between a tangent and chord is equal to the angle in the alternate segment, this is known as the alternate segment theorem. So angle BAT = angle ACB. Examples Example 1 Work out the size of each angle marked with a letter. Give reasons for your answers. Angle a = 360° − 92° 1 The angles in a full turn total 360°. = 268° as the angles in a full turn total 360°.
    [Show full text]
  • Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD
    10 Descriptive Geometry Section 10.1 Basic Descriptive Geometry and Board Drafting Section 10.2 Solving Descriptive Geometry Problems with CAD Chapter Objectives • Locate points in three-dimensional (3D) space. • Identify and describe the three basic types of lines. • Identify and describe the three basic types of planes. • Solve descriptive geometry problems using board-drafting techniques. • Create points, lines, planes, and solids in 3D space using CAD. • Solve descriptive geometry problems using CAD. Plane Spoken Rutan’s unconventional 202 Boomerang aircraft has an asymmetrical design, with one engine on the fuselage and another mounted on a pod. What special allowances would need to be made for such a design? 328 Drafting Career Burt Rutan, Aeronautical Engineer Effi cient travel through space has become an ambi- tion of aeronautical engineer, Burt Rutan. “I want to go high,” he says, “because that’s where the view is.” His unconventional designs have included every- thing from crafts that can enter space twice within a two week period, to planes than can circle the Earth without stopping to refuel. Designed by Rutan and built at his company, Scaled Composites LLC, the 202 Boomerang aircraft is named for its forward-swept asymmetrical wing. The design allows the Boomerang to fl y faster and farther than conventional twin-engine aircraft, hav- ing corrected aerodynamic mistakes made previously in twin-engine design. It is hailed as one of the most beautiful aircraft ever built. Academic Skills and Abilities • Algebra, geometry, calculus • Biology, chemistry, physics • English • Social studies • Humanities • Computer use Career Pathways Engineers should be creative, inquisitive, ana- lytical, detail oriented, and able to work as part of a team and to communicate well.
    [Show full text]
  • 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.
    [Show full text]
  • 1-1 Understanding Points, Lines, and Planes Lines, and Planes
    Understanding Points, 1-11-1 Understanding Points, Lines, and Planes Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Objectives Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Holt Geometry 1-1 Understanding Points, Lines, and Planes Vocabulary undefined term point line plane collinear coplanar segment endpoint ray opposite rays postulate Holt Geometry 1-1 Understanding Points, Lines, and Planes The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms point, line, and plane are the building blocks of geometry. Holt Geometry 1-1 Understanding Points, Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear. Points that lie on the same plane are coplanar. Otherwise they are noncoplanar. K L M N Holt Geometry 1-1 Understanding Points, Lines, and Planes Example 1: Naming Points, Lines, and Planes A. Name four coplanar points. A, B, C, D B. Name three lines. Possible answer: AE, BE, CE Holt Geometry 1-1 Understanding Points, Lines, and Planes Holt Geometry 1-1 Understanding Points, Lines, and Planes Example 2: Drawing Segments and Rays Draw and label each of the following. A. a segment with endpoints M and N. N M B. opposite rays with a common endpoint T. T Holt Geometry 1-1 Understanding Points, Lines, and Planes Check It Out! Example 2 Draw and label a ray with endpoint M that contains N.
    [Show full text]
  • Can One Design a Geometry Engine? on the (Un) Decidability of Affine
    Noname manuscript No. (will be inserted by the editor) Can one design a geometry engine? On the (un)decidability of certain affine Euclidean geometries Johann A. Makowsky Received: June 4, 2018/ Accepted: date Abstract We survey the status of decidabilty of the consequence relation in various ax- iomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu’s orthogonal and metric geometries (Wen- Ts¨un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991) are undecidable. It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geom- etry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable. The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving. Keywords Euclidean Geometry · Automated Theorem Proving · Undecidability arXiv:1712.07474v3 [cs.SC] 1 Jun 2018 J.A. Makowsky Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel E-mail: [email protected] 2 J.A.
    [Show full text]