ASIA-EUROPE CLASSROOM NETWORK (AEC-NET) Title: “Famous Mathematicians in Greece”
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Extending Euclidean Constructions with Dynamic Geometry Software
Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) Extending Euclidean constructions with dynamic geometry software Alasdair McAndrew [email protected] College of Engineering and Science Victoria University PO Box 18821, Melbourne 8001 Australia Abstract In order to solve cubic equations by Euclidean means, the standard ruler and compass construction tools are insufficient, as was demonstrated by Pierre Wantzel in the 19th century. However, the ancient Greek mathematicians also used another construction method, the neusis, which was a straightedge with two marked points. We show in this article how a neusis construction can be implemented using dynamic geometry software, and give some examples of its use. 1 Introduction Standard Euclidean geometry, as codified by Euclid, permits of two constructions: drawing a straight line between two given points, and constructing a circle with center at one given point, and passing through another. It can be shown that the set of points constructible by these methods form the quadratic closure of the rationals: that is, the set of all points obtainable by any finite sequence of arithmetic operations and the taking of square roots. With the rise of Galois theory, and of field theory generally in the 19th century, it is now known that irreducible cubic equations cannot be solved by these Euclidean methods: so that the \doubling of the cube", and the \trisection of the angle" problems would need further constructions. Doubling the cube requires us to be able to solve the equation x3 − 2 = 0 and trisecting the angle, if it were possible, would enable us to trisect 60◦ (which is con- structible), to obtain 20◦. -
Es, and (3) Toprovide -Specific Suggestions for Teaching Such Topics
DOCUMENT RESUME ED 026 236 SE 004 576 Guidelines for Mathematics in the Secondary School South Carolina State Dept. of Education, Columbia. Pub Date 65 Note- I36p. EDRS Price MF-$0.7511C-$6.90 Deseriptors- Advanced Programs, Algebra, Analytic Geometry, Coucse Content, Curriculum,*Curriculum Guides, GeoMetry,Instruction,InstructionalMaterials," *Mathematics, *Number ConCepts,NumberSystems,- *Secondar.. School" Mathematies Identifiers-ISouth Carcilina- This guide containsan outline of topics to be included in individual subject areas in secondary school mathematics andsome specific. suggestions for teachin§ them.. Areas covered inclUde--(1) fundamentals of mathematicsincluded in seventh and eighth grades and general mathematicsin the high school, (2) algebra concepts for COurset one and two, (3) geometry, and (4) advancedmathematics. The guide was written With the following purposes jn mind--(1) to assist local .grOupsto have a basis on which to plan a rykathematics 'course of study,. (2) to give individual teachers an overview of a. particular course Or several cOur:-:es, and (3) toprovide -specific sUggestions for teaching such topics. (RP) Ilia alb 1 fa...4...w. M".7 ,noo d.1.1,64 III.1ai.s3X,i Ala k JS& # Aso sA1.6. It tilatt,41.,,,k a.. -----.-----:--.-:-:-:-:-:-:-:-:-.-. faidel1ae,4 icii MATHEMATICSIN THE SECONDARYSCHOOL Published by STATE DEPARTMENT OF EDUCATION JESSE T. ANDERSON,State Superintendent Columbia, S. C. 1965 Permission to Reprint Permission to reprint A Guide, Mathematics in Florida Second- ary Schools has been granted by the State Department of Edu- cation, Tallahassee, Flmida, Thomas D. Bailey, Superintendent. The South Carolina State Department of Education is in- debted to the Florida State DepartMent of Education and the aahors of A Guide, Mathematics in Florida Secondary Schools. -
ANCIENT PROBLEMS VS. MODERN TECHNOLOGY 1. Geometric Constructions Some Problems, Such As the Search for a Construction That Woul
ANCIENT PROBLEMS VS. MODERN TECHNOLOGY SˇARKA´ GERGELITSOVA´ AND TOMA´ Sˇ HOLAN Abstract. Geometric constructions using a ruler and a compass have been known for more than two thousand years. It has also been known for a long time that some problems cannot be solved using the ruler-and-compass method (squaring the circle, angle trisection); on the other hand, there are other prob- lems that are yet to be solved. Nowadays, the focus of researchers’ interest is different: the search for new geometric constructions has shifted to the field of recreational mathematics. In this article, we present the solutions of several construction problems which were discovered with the help of a computer. The aim of this article is to point out that computer availability and perfor- mance have increased to such an extent that, today, anyone can solve problems that have remained unsolved for centuries. 1. Geometric constructions Some problems, such as the search for a construction that would divide a given angle into three equal parts, the construction of a square having an area equal to the area of the given circle or doubling the cube, troubled mathematicians already hundreds and thousands of years ago. Today, we not only know that these problems have never been solved, but we are even able to prove that such constructions cannot exist at all [8], [10]. On the other hand, there is, for example, the problem of finding the center of a given circle with a compass alone. This is a problem that was admired by Napoleon Bonaparte [11] and one of the problems that we are able to solve today (Mascheroni found the answer long ago [9]). -
Cambridge University Press 978-1-108-48147-2 — Scale, Space and Canon in Ancient Literary Culture Reviel Netz Index More Information
Cambridge University Press 978-1-108-48147-2 — Scale, Space and Canon in Ancient Literary Culture Reviel Netz Index More Information Index Aaker, Jennifer, 110, 111 competition, 173 Abdera, 242, 310, 314, 315, 317 longevity, 179 Abel, N. H., 185 Oresteia, 197, 200, 201 Academos, 189, 323, 324, 325, 337 papyri, 15 Academy, 322, 325, 326, 329, 337, 343, 385, 391, Persians, 183 399, 404, 427, 434, 448, 476, 477–8, 512 portraits, 64 Achilles Tatius, 53, 116, 137, 551 Ptolemaic era, 39 papyri, 16, 23 Aeschylus (astronomer), 249 Acta Alexandrinorum, 87, 604 Aesop, 52, 68, 100, 116, 165 adespota, 55, 79, 81–5, 86, 88, 91, 99, 125, 192, 194, in education, 42 196, 206, 411, 413, 542, 574 papyri, 16, 23 Adkin, Neil, 782 Aethiopia, 354 Adrastus, 483 Aetia, 277 Adrastus (mathematician), 249 Africa, 266 Adrianople, 798 Agatharchides, 471 Aedesius (martyr), 734, 736 Agathocles (historian), 243 Aegae, 479, 520 Agathocles (peripatetic), 483 Aegean, 338–43 Agathon, 280 Aegina, 265 Agias (historian), 373 Aelianus (Platonist), 484 agrimensores, 675 Aelius Aristides, 133, 657, 709 Ai Khanoum, 411 papyri, 16 Akhmatova, Anna, 186 Aelius Herodian (grammarian), 713 Albertus Magnus, 407 Aelius Promotus, 583 Albinus, 484 Aenesidemus, 478–9, 519, 520 Alcaeus, 49, 59, 61–2, 70, 116, 150, 162, 214, 246, Aeolia, 479 see also Aeolian Aeolian, 246 papyri, 15, 23 Aeschines, 39, 59, 60, 64, 93, 94, 123, 161, 166, 174, portraits, 65, 67 184, 211, 213, 216, 230, 232, 331 Alcidamas, 549 commentaries, 75 papyri, 16 Ctesiphon, 21 Alcinous, 484 False Legation, 22 Alcmaeon, 310 -
Construction of Regular Polygons a Constructible Regular Polygon Is One That Can Be Constructed with Compass and (Unmarked) Straightedge
DynamicsOfPolygons.org Construction of regular polygons A constructible regular polygon is one that can be constructed with compass and (unmarked) straightedge. For example the construction on the right below consists of two circles of equal radii. The center of the second circle at B is chosen to lie anywhere on the first circle, so the triangle ABC is equilateral – and hence equiangular. Compass and straightedge constructions date back to Euclid of Alexandria who was born in about 300 B.C. The Greeks developed methods for constructing the regular triangle, square and pentagon, but these were the only „prime‟ regular polygons that they could construct. They also knew how to double the sides of a given polygon or combine two polygons together – as long as the sides were relatively prime, so a regular pentagon could be drawn together with a regular triangle to get a regular 15-gon. Therefore the polygons they could construct were of the form N = 2m3k5j where m is a nonnegative integer and j and k are either 0 or 1. The constructible regular polygons were 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ... but the only odd polygons in this list are 3,5 and 15. The triangle, pentagon and 15-gon are the only regular polygons with odd sides which the Greeks could construct. If n = p1p2 …pk where the pi are odd primes then n is constructible iff each pi is constructible, so a regular 21-gon can be constructed iff both the triangle and regular 7-gon can be constructed. -
Trisect Angle
HOW TO TRISECT AN ANGLE (Using P-Geometry) (DRAFT: Liable to change) Aaron Sloman School of Computer Science, University of Birmingham (Philosopher in a Computer Science department) NOTE Added 30 Jan 2020 Remarks on angle-trisection without the neusis construction can be found in Freksa et al. (2019) NOTE Added 1 Mar 2015 The discussion of alternative geometries here contrasts with the discussion of the nature of descriptive metaphysics in "Meta-Descriptive Metaphysics: Extending P.F. Strawson’s ’Descriptive Metaphysics’" http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-descriptive-metaphysics.html This document makes connections with the discussion of perception of affordances of various kinds, generalising Gibson’s ideas, in http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#gibson Talk 93: What’s vision for, and how does it work? From Marr (and earlier) to Gibson and Beyond Some of the ideas are related to perception of impossible objects. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html JUMP TO CONTENTS Installed: 26 Feb 2015 Last updated: A very nice geogebra applet demonstrates the method described below: http://www.cut-the-knot.org/pythagoras/archi.shtml. Feb 2017: Added note about my 1962 DPhil thesis 25 Apr 2016: Fixed typo: ODB had been mistyped as ODE (Thanks to Michael Fourman) 29 Oct 2015: Added reference to discussion of perception of impossible objects. 4 Oct 2015: Added reference to article by O’Connor and Robertson. 25 Mar 2015: added (low quality) ’movie’ gif showing arrow rotating. 2 Mar 2015 Formatting problem fixed. 1 Mar 2015 Added draft Table of Contents. -
A Concise History of the Philosophy of Mathematics
A Thumbnail History of the Philosophy of Mathematics "It is beyond a doubt that all our knowledge begins with experience." - Imannuel Kant ( 1724 – 1804 ) ( However naïve realism is no substitute for truth [1] ) [1] " ... concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to comprehend the problem of the á priori as posed by Kant", from "The Problem of Space, Ether, and the Field in Physics" ( "Das Raum-, Äether- und Feld-Problem der Physik." ), by Albert Einstein, 1934 - source: "Beyond Geometry: Classic Papers from Riemann to Einstein", Dover Publications, by Peter Pesic, St. John's College, Sante Fe, New Mexico Mathematics does not have a universally accepted definition during any period of its development throughout the history of human thought. However for the last 2,500 years, beginning first with the pre - Hellenic Egyptians and Babylonians, mathematics encompasses possible deductive relationships concerned solely with logical truths derived by accepted philosophic methods of logic which the classical Greek thinkers of antiquity pioneered. Although it is normally associated with formulaic algorithms ( i.e., mechanical methods ), mathematics somehow arises in the human mind by the correspondence of observation and inductive experiential thinking together with its practical predictive powers in interpreting future as well as "seemingly" ephemeral phenomena. Why all of this is true in human progress, no one can answer faithfully. In other words, human experiences and intuitive thinking first suggest to the human mind the abstract symbols for which the economy of human thinking mathematics is well known; but it is those parts of mathematics most disconnected from observation and experience and therefore relying almost wholly upon its internal, self - consistent, deductive logics giving mathematics an independent reified, almost ontological, reality, that mathematics most powerfully interprets the ultimate hidden mysteries of nature. -
Chapter Three
2008032. Markovic. 03_Chapter3. Proef 4. 7-5-2008:11.24, page 83. chapter three ARGUMENTS AND ORNAMENTS In the introduction we made a case for Epicurean argumentation as an essentially rhetorical phenomenon. Before we enter rhetorical analysis of the arguments advanced in DRN, let us corroborate this starting point with some further explanation. The word ‘argument’ denotes a kind of proof. The differentia specifica of this kind of proof is best understood vis-à-vis demonstration. Demon- stration is a formal logical proof; argument is a rhetorical proof. Unlike demonstrations, which may use artificial language (in logic or mathe- matics for example) to present objectively true conclusions, regardless of an audience, arguments always use natural language, present relative truths, and are addressed to an audience. The distinction between the two types of proof goes back to Aristotle’s distinction between “ana- lytical” (demonstrative) reasoning, discussed in the Prior and Posterior Analytics, and “dialectical” (argumentative) reasoning, discussed in the Topics, Rhetoric,andSophistical Refutations.1 Within this division Epicurus’ scientific discourse falls under the heading of argumentation. While Epicurus did claim that he dealt with absolute truths, his explanations were entirely audience-oriented, and he used reasoning to explain notoriously controversial phenomena, such as the nature of gods. Although he somewhat confusingly aspired to use natural language in the way in which artificial languages are used, i.e., with one-to-one correspondence between significans and signi- ficatum,2 and although he maintained certain standards of proof, Epicu- rus rejected formal logic and mathematics. Ignoring the development which these disciplines had undergone in Aristotle’s philosophical sys- tem, Epicurus introduced a radically empirical scientific methodology, based entirely on sensations as the fundamental means of acquiring correct knowledge.3 The main principle of Epicurus’ methodology was that an explanation must be in accordance with the data provided by 1 Ar. -
Trisecting an Angle and Doubling the Cube Using Origami Method
広 島 経 済 大 学 研 究 論 集 第38巻第 4 号 2016年 3 月 Note Trisecting an Angle and Doubling the Cube Using Origami Method Kenji Hiraoka* and Laura Kokot** meaning to fold, and kami meaning paper, refers 1. Introduction to the traditional art of making various attractive Laura Kokot, one of the authors, Mathematics and decorative figures using only one piece of and Computer Science teacher in the High school square sheet of paper. This art is very popular, of Mate Blažine Labin, Croatia came to Nagasaki, not only in Japan, but also in other countries all Japan in October 2014, for the teacher training over the world and everyone knows about the program at the Nagasaki University as a MEXT paper crane which became the international (Ministry of education, culture, sports, science symbol of peace. Origami as a form is continu- and technology) scholar. Her training program ously evolving and nowadays a lot of other was conducted at the Faculty of Education, possibilities and benefits of origami are being Nagasaki University in the field of Mathematics recognized. For example in education and other. Education under Professor Hiraoka Kenji. The goal of this paper is to research and For every teacher it is important that pupils learn more about geometric constructions by in his class understand and learn the material as using origami method and its properties as an easily as possible and he will try to find the best alternative approach to learning and teaching pedagogical approaches in his teaching. It is not high school geometry. -
A Acerbi, F., 45 Adam, C., 166–169, 171, 175, 176, 178–181, 183–185
Index A 80, 81, 84, 86, 87, 93, 97, 98, 107, 112, Acerbi, F., 45 137, 198, 201, 212, 233, 235, 236 Adam, C., 166–169, 171, 175, 176, 178–181, Arius Didymus, 35 183–185, 187, 188, 200, 201 Arnauld, 217, 227, 233, 239–242, 245 Adams, R.M., 234, 245, 306 Arnzen, R., 55 Adorno, T.W., 147 Arriaga, R.de, 6 Adrastos, 77–81, 89, 96 Arthur, R, 254 Aetius, 35 Athenaeus, 39 Agapius, 56 Atherton, M., 170, 174 Aglietta, M., 151 Aujac, G., 145 Aichelin, J., 232 Autolycus, 15, 20, 21, 23, 24 Aiken, J.A., 148 Ayers, M., 229 Aime, M., 145 Aksamija, N., 152 Alberti, L.B., 9, 148, 150 B Alembert, J.d', 70 Bacon, R., 165, 166 Alexander of Aphrodisias, 16, 35, 43, 54, 56, Banu Musa, 53 58, 61, 77, 78, 81 Barnes, J., 38, 212, 235 Algra, K., 34, 35, 38 Barozzi, F., 115, 119, 120 Al-Haytham, 164–167 Barrow, I., 6, 220 Alhazen, see Ibn Al-Haytham Basileides of Tyre, 37 Al-Khwārizmī, 92 Baudrillard, J., 144 Al-Kindī, 161, 162, 164, 165 Bayle, P., 237 Al-Nayrizi, 55–57, 59 Beauchamp, T.L., 274 Al-Sijzī, 110 Bechtle, G., 105 Allison, H.E., 283 Belting, H., 145, 148 Andronicus of Rhodes, 102 Benatouïl, T., 114 Apollodorus, 41, 42 Benjamin, W., 144, 145, 148 Apollonios of Perge (Apollonius), 15, 19–21, Bentley, R., 223–226 23–27, 37, 38, 52, 53, 63, 81, 106, 123 Berggren, J.L., 94 Apostle Thomas, 147 Berkeley, G., 11, 160, 161, 166, 170, 186 Aratus, 24 Bernadete, J., 212, 213 Arbini, R., 170 Bernanos, G., 137 Archedemus, 41 Bernoulli, J., 233 Archimedes, 15, 19–21, 23, 24, 27, 29, 38, 39, Bessel, F.W., 72 44, 45, 47, 68, 106, 123 Bioesmat-Martagon, L., 72 Ariew, R., 110, 212, 214 Blumenberg, H., 143 Aristaeus, 27 Boer, E., 93 Ariston, 39 Bolyai, J., 11 Aristotle, 3, 5, 6, 15, 16, 18, 19, 24–34, 36, Bonola, R., 70 40–45, 47–50, 55, 58, 60, 61, 68, 75, Borelli, G.A., 6 © Springer International Publishing Switzerland 2015 311 V. -
Quadratrix of Hippias -- from Wolfram Mathworld
12/3/13 Quadratrix of Hippias -- from Wolfram MathWorld Search MathWorld Algebra Applied Mathematics Geometry > Curves > Plane Curves > Polar Curves > Geometry > Geometric Construction > Calculus and Analysis Interactive Entries > Interactive Demonstrations > Discrete Mathematics THINGS TO TRY: Quadratrix of Hippias quadratrix of hippias Foundations of Mathematics 12-w heel graph Geometry d^4/dt^4(Ai(t)) History and Terminology Number Theory Probability and Statistics Recreational Mathematics Hippias Quadratrix Bruno Autin Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Contribute to MathWorld Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal Send a Message to the Team parts, and circle squaring. It has polar equation MathWorld Book (1) Wolfram Web Resources » 13,191 entries with corresponding parametric equation Last updated: Wed Nov 6 2013 (2) Created, developed, and nurtured by Eric Weisstein at Wolfram Research (3) and Cartesian equation (4) Using the parametric representation, the curvature and tangential angle are given by (5) (6) for . SEE ALSO: Angle trisection, Cochleoid REFERENCES: Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 223, 1987. Law rence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972. Loomis, E. S. "The Quadratrix." §2.1 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. -
Mathematical Theory of Big Game Hunting
DOCUMEWT RESUME ED 121 522 SE 020 770 AUTHOR Schaaf, Willie' L. TITLE A Bibliography of Recreational Mathematics, Volume 1. Fourth, Edition. INSTITUTION National Council of Teachers of Mathematics, Inc., Reston, Va. PUB DATE 70 NOTE 160p.; For related documents see Ed 040 874 and ED 087 631 AVAILABLE FROMNational Council of Teachers of Mathematics, Inc., 1906 Association Drive, Reston, Virginia 22091 EDRS PRICE MP-$0.83 Plus Postage. HC Not Available from EDRS. DESCRIPTORS *Annotated Bibliographies; *Games; Geometric Concepts; History; *Literature Guides; *Mathematical Enrichment; Mathematics; *Mathematics Education; Number Concepts; Reference Books ABSTRACT This book is a partially annotated bibliography of books, articles, and periodicals concerned with mathematical games, puzzles, and amusements. It is a reprinting of Volume 1 of a three-volume series. This volume, originally published in 1955, treats problems and recreations which have been important in the history of mathematics as well as some of more modern invention. The book is intended for use by both professional and amateur mathematicians. Works on recreational mathematics are listed in eight broad categories: general works, arithmetic and algebraic recreations, geometric recreations, assorted recreations, magic squares, the Pythagorean relationship, famous problems of antiquity, and aathematical 'miscellanies. (SD) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC sakes every effort * * to obtain the best copy available. Nevertheless, items of marginal * * reproducibility are often encountered and this affects the quality* * of the microfiche and hardcopy reproductions ERIC aakes available * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document.