41. TRISEK Translate this page Bis ins Jahr 1837 mußte man warten, bis pierre Laurent wantzel (1814-1848) dieUnmöglichkeit der Trisektion (=Dreiteilung) eines Winkels mit Lineal und http://did.mat.uni-bayreuth.de/studium/seminar/antike/kraus/trisek.html

Seminar: Klassische Probleme in der Antike SS 1997 Peter Kraus Die Trisektierer Die Gruppe der Trisektierer erkennt man an folgenden Charakeristiken: auseinandersetzen. 3. Im Allgemeinem besitzt man nur Mathematikkenntnisse aus der Schulzeit. Die Dreiteilung verstehen. Konstruktionsbeschreibungen. einfachste Weise dreiteilen kann, beachtet man Platos Spielregeln nicht. Konstruktionsbeschreibung: Ein beliebiger Winkel Der entstandene Winkel b Konstruktion: Berechnung des Winkels b D ACX Es gilt: Es gilt: AF = AB + BF ; aus (2) und (3) folgt: In (1): Fehler des Winkels b Winkel Fehler Fehlerkurve: = 120° liegt der Fehler bei schon fast bei 5°). Konstruktionsbeschreibung Ein beliebiger Winkel Durch S zieht man eine Parallele zu SF. Der Schnittpunkt mit dem entfernten Schenkel sei F . Ein Kreis um A mit Radius AF schneidet die Parallele g in H. Der Mittelpunkt der Strecke S H sei X . Es entsteht ein Winkel b X Die Parallele g schneidet einen Schenkel in P, die Parallele h schneidet den anderen Schenkel in P’. Das Lot durch S auf den entfernteren Schenkel schneidet diesen in B. Das Lot durch S auf den entfernteren Schenkel schneidet diesen in E.

42. Exkurs: Die Klassischen Probleme Der Antike Translate this page Der Hinweis auf pierre Laurent wantzel (1814 - 1848), der 1837 einen Beweis überdie Unmöglichkeit der Winkeldreiteilung und der Würfelverdoppelung mit http://did.mat.uni-bayreuth.de/~matthias/geometrieids/pythagoras/html/node6.html

Next: Hippokrates von Chios Up: Previous: Beginning of the document: Pythagoras und kein Ende Exkurs: Die klassischen Probleme der Antike Der Erfolg bei den Mond-Quadraturen mag der Grund sein, warum die antiken Wissenschaftler (und nicht nur diese) glaubten, auch die Quadratur des Kreises Dreiteilung eines bliebigen Winkels und der unter dem Namen klassische Probleme der Antike APPOS AUSS (1777 - 1855) und Evariste G ALOIS ANTZEL Auch der Transzendenzbeweis von durch Ferdinand L INDEMANN The Lady's Diary or Woman's Almanach Next: Hippokrates von Chios Up: Previous: Beginning of the document: Pythagoras und kein Ende Matthias Ehmann

43. Trissecção Do Ângulo E Duplicação Do Cubo - Resumo Translate this page XIX pelo matemático francês pierre Laurent wantzel e depende de conceitosalgébricos que foram sendo desenvolvidos ao longo de vários séculos. http://www.prof2000.pt/users/miguel/tese/resumo.htm

Trissecção do Ângulo e Duplicação do Cubo: as Soluções na Antiga Grécia José Miguel Rodrigues de Sousa Tese submetida à Faculdade de Ciências da Universidade do Porto para obtenção do grau de Mestre em Matemática - Fundamentos e Aplicações RESUMO O presente trabalho aborda dois problemas clássicos da geometria antiga: a trissecção do ângulo (capítulo 1) e a duplicação do cubo (capítulo 2). Neste se analisam as contribuições de vários matemáticos para a resolução dos dois problemas, ao longo do período helénico (compreendido entre o século VI a.C. e o séc. V d.C.). Foi neste período que se iniciou o estudo destes problemas geométricos que desafiaram o poder inventivo de inúmeros matemáticos e intelectuais, durante mais de dois mil anos. Durante séculos, diversas soluções foram propostas para a sua resolução mas não estavam de acordo com as regras do jogo, presumivelmente, colocadas na Academia de Platão. As soluções aqui estudadas envolvem construções geométricas que, embora não sejam da matemática elementar, fazem apelo a métodos geométricos simples. A impossibilidade de resolução com régua não graduada e compasso (abordada no epílogo da dissertação), destes problemas clássicos da geometria grega, só foi demonstrada no séc. XIX pelo matemático francês Pierre Laurent Wantzel e depende de conceitos algébricos que foram sendo desenvolvidos ao longo de vários séculos.

44. From The Time Of Euclid, Geometric Constructions Were Done Solely With A Straigh pierre wantzel was the mathematician that came up with the proof by turningthe geometric problem into an algebraic problem (Greenburg, 16). http://www.cs.gsu.edu/~matdnv/Math6371/JASD/project1/TrisectingAngles.html

HOMEWORK 6 WEDNESDAY, FEBRUARY 10, 1999 TRISECTING ANGLES AND SQUARING CIRCLES USING GEOMETRIC TOOLS JASD SANDRA CARTER DAWN DANIELS ALYSON ELLIOTT JILL PRUITT th Century, when p was shown to be a transcendental number by Lindemann, that the impossibility of squaring a circle was proven It is not possible to construct with a ruler and compass a line whose length is the numerical value of a root of a cubic equation with rational coefficients having no rational roots (Davis, 227). The proof of this theorem establishes that "a line whose length is the numerical value of a root of a cubic equation with rational coefficients having no rational root cannot be constructed by performing a finite number of rational operations and the extractions of real square roots (Davis, 227)." This idea leads to the angle of trisection proof. 4 cos A – 3 cos A = cos 3A we can multiply each term by two to obtain 8 cos A – 6 cos A = 2 cos 3 A. Let x = 2 cos A and substitute into the above equation. The resulting equation is x – 3x – 2 cos 3 A = 0. If 3A = 60 o , then the cos 3A = ½. If 3A = 60

45. Euclid Challenge - Squaring A Circle By Straightedge And Compass - Page 10 pierre wantzel proved during the 19 th century that it was impossible to squarea circle by straightedge and compass, when following the “Traditional http://www.euclidchallenge.org/pg_10.htm

EUCLID CHALLENGE Successful Response by Milton Mintz May 10, 2002 Page 10: Squaring A Circle By Straightedge And Compass Introduction Converting the area of the circle to a Rectangle. Converting the area of the Rectangle to a Square. This Method Is Successful Because Of The Following: Points are moved at a uniform rate; The method goes beyond traditional euclidian methods The Method was accomplished by using an unmarked straightedge and compass only. Uniform Rate ”: This Method uses the SAME Uniform Rate by BOTH hands. (Hippias used a “Uniform Rate” in his “Non-Euclidian device” over two thousand years ago. His use of a “Uniform Rate” has been accepted over the years, but he used tools other than “straightedge and compass only”.) Traditional Euclidian Methods Pierre Wantzel proved during the 19 th century that it was impossible to square a circle by straightedge and compass, when following the “Traditional Euclidian Methods”. That is why to “successfully square a circle”, unmarked straightedge and compass only, it was necessary for the attached Method to go “Beyond the Traditional Euclidian Methods”. Previous Page Top of Page Next Page

46. Euclid Challenge - Trisection Of Any Angle By Straightedge And Compass Traditional Euclidian Methods pierre wantzel proved during the 19 th century thatit was impossible to trisect all angles by straightedge and compass, when http://www.euclidchallenge.org/pg_03.htm

EUCLID CHALLENGE Successful Response by Milton Mintz May 10, 2002 Page 3: Trisection of Any Angle by Straightedge and Compass Introduction This method received the following review by an eminent American professor of mathematics: Construction of an angle trisector is successful Points are moved at a uniform rate The method goes beyond traditional Euclidian methods Angle trisector: Accomplished by using straightedge and compass only. No scratches on the straightedge that were used by Archimedes. “Uniform Rate”: This Method includes PROOF of the accuracy of its use of the SAME Uniform Rate by BOTH hands. (Hippias used a “Uniform Rate” in his “Non-Euclidian device” over two thousand years ago. His use of a “Uniform Rate” has been accepted over the years, but he used tools other than “straightedge and compass only”). Traditional Euclidian Methods: Pierre Wantzel proved during the 19 th century that it was impossible to trisect all angles by straightedge and compass, when following the “Traditional Euclidian Methods”. That is why to “successfully construct a Trisector”, using only a “straightedge and compass only”, it was necessary for the attached Method to go “Beyond the Traditional Euclidian Methods”.

47. Doğum Tarihlerine Göre Gelmiş Geçmiş Tüm Matematikçiler Translate this page 1813-1854) Laurent, pierre. (1777-1859) Poinsot, (1797-1886) Saint-Venant, (1814-1894)Catalan. (1778-1853) Wronski, (1798-1840) Bobillier, (1814-1894) wantzel. http://www.sanalhoca.com/matematik/matematikci3.htm

sanal hoca Ana Sayfa Kimya Matematik Fizik ... E-Posta (1759-1789) Bernoulli, Jac(II) (1786-1837) Horner (1802-1829) Abel (1759-1803) Arbogast (1786-1853) Arago (1802-1860) Bolyai, János (1760-1826) Kramp (1788-1827) Fresnel (1803-1853) Doppler (1761-1840) Budan de BL (1788-1856) Hamilton W (1803-1855) Sturm, Charles (1765-1822) Ruffini (1788-1867) Poncelet (1803-1869) Libri (1765-1825) Pfaff (1789-1854) Ohm (1803-1880) Bellavitis (1765-1832) Osipovsky (1789-1857) Cauchy (1804-1849) Verhulst (1765-1836) Girard, Pierre (1790-1868) Möbius (1804-1851) Jacobi (1765-1842) Ivory (1791-1820) Petit (1804-1863) Jerrard (1765-1843) Lacroix (1791-1841) Savart (1804-1889) Bunyakovsky (1766-1832) Leslie (1791-1858) Peacock (1804-1891) Weber (1768-1810) Francais, F

48. Trisecting An Angle The proof of the impossibility had to await the mathematics of the 19th century.The final pieces of the argument were put together by pierre wantzel. http://www2.ittu.edu.tm/math/bosna.net/Geometry/papers/Trisecting an angle/Trise

49. FRACTALES.ORG Ciencia. Los 3 Problemas De Los Clásicos Griegos. Translate this page Fue el matemático francés pierre L. wantzel (1814-1848) a quien debemos la demostraciónde la imposibilidad de la trisección del ángulo y la duplicación http://www.fractales.org/ciencia/3clasicos.shtml

50. Costruzioni Geometriche Translate this page Gauss provò solo la sufficienza della condizione (all'età di 19 anni!!), mentrela necessità fu provata successivamente da pierre-Laurent wantzel nel 1836. http://digilander.libero.it/lucianobattaia/matematica/a_costruz/ciclotomia/ciclo

Il problema generale della ciclotomia Costruzioni geometriche Indice Si tratta del problema di dividere una circonferenza in n parti uguali, cioè di costruire un poligono regolare di n lati inscritto nella circonferenza, o, equivalentemente, di costruire un poligono regolare di lato assegnato. Il problema di sapere quali sono le divisioni possibili con riga e compasso, già affrontato nell'antichità e diffusamente studiato dai Greci, fu definitivamente risolto da Gauss, nel 1796, che giunse alla seguente conclusione: Poligoni costruibili con riga e compasso Un poligono regolare di n lati può essere costruito con riga e compasso se e solo se n è un numero del tipo dove k è un intero e p p s sono numeri di Fermat primi. La formula comprende anche il caso in cui n=2 k . In realtà Gauss provò solo la sufficienza della condizione (all'età di 19 anni!!), mentre la necessità fu provata successivamente da Pierre-Laurent Wantzel nel 1836. I greci conoscevano la costruzione dei poligoni di 3, 4, 5 e 15 lati (oltre a quelli ovviamente che si ottengono raddoppiando questi numeri), ma non riuscirono a trarre conclusioni generali. La costruzione dei poligoni di tre lati ( triangolo equilatero ) e di quattro lati ( quadrato ) è quasi banale. Molto più interessante la costruzione del

51. History Of Mathematics: Chronology Of Mathematicians A list of all of the important mathematicians working in a given century.Category Science Math Mathematicians Directories...... pierre Laurent wantzel (18141848); Eugène Charles Catalan (1814-1894); LudwigSchläfli (1814-1895) *MT; James Joseph Sylvester (1814-1897) *MT 1840. http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan Table of Contents 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below List of Mathematicians 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *MT 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) 500 B.C.E. Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB

52. Knoten Translate this page Erst 1837 hat pierre wantzel bewiesen, dass das Problem allein mit Hilfe von Linealund Zirkel nicht zu lösen ist. Nur wenn man Kurven 2. oder 3. Grades bzw. http://www.uni-siegen.de/~ifan/koblenz/bobzin.htm

Institut für Angewandtes Nichtwissen e.V. Knoten Hagen Bobzin Institut für Angewandtes Nichtwissen Jul 11, 2000 Der gordische Knoten Als Alexander der Große im Jahr 333 v. Chr. auf seinem Marsch durch Anatolien die phrygische Hauptstadt Gordium erreichte, wurde er zum Tempel des Zeus geführt. Dort stand der Ochsenkarren des Stadtgründers Gordius, der das Joch und die Deichsel des Wagens durch einen kunstvollen Knoten verbunden hatte. Da sich der Knoten lange Zeit von niemandem entwirren ließ, erhielt er für die Umwohner religiöse Bedeutung und schließlich entstand die Sage, dass derjenige, der den Knoten zu lösen vermag, zum Herrscher der gesamten Welt - also ganz Asiens - aufsteigen würde. Mit seinem Schwerthieb präsentiert Alexander der Große eine erstaunlich einfache wie sprichwörtlich grobe Methode, das Problem zu lösen. Die Sage um den gordischen Knoten enthält einige Aspekte des Angewandten Nichtwissens , während andere Details eher irreführend sind. So beschäftigt sich das Institut eher selten mit Fragen der Religion oder des Glaubens, was nicht heißt, dass etwa die Frage nach einem Gottesbeweis hin und wieder die Gemüter erhitzt. Auch der Irrtum des Orakels bezüglich der Herrschaft über den gesamten Orient und Okzident spielt für uns eine untergeordnete Rolle. Außerdem haben unsere Fragestellungen häufig keine exakte Lösung, wie sie Alexander der Große durchaus gefunden hat. Wie aber geht man mit Problemen um, die sich nicht so einfach lösen lassen? Existiert überhaupt eine Lösung, wenn man beispielsweise die Frage nach Gerechtigkeit aufwirft? Und wie geht man mit Vermutungen um, die über lange Zeit nicht widerlegt worden sind?

53. THE BRAVES SOLVE A PROBLEM OF ANTIQUITY? Plato's rules. pierre wantzel (1814 1848) proved the impossibilityof trisecting any given angle by Plato's restrictions. The method http://jwilson.coe.uga.edu/emt669/Student.Folders/Perkins.Catherine/Tomahawk.Tri

THE BRAVES SOLVE A PROBLEM OF ANTIQUITY? BY CATHY PERKINS Can this be true? Are the World Champion Atlanta Braves so outstanding that now they are even solving mathematics problems that have stumped mathematicians for over 2000 years? The problem that I am referring to is one of the three construction problems of antiquity that Plato supposedly insisted be performed with a collapsible compass and unmarked straightedge. The 3 problems are as follows: ** Construct a square whose area is equal to the area of a given circle. (Squaring of the Circle, Quadrature of the Circle) ** Find the edge of a cube having a volume twice that of a given cube. ** Trisect a arbitrary angle. The problem that the Braves have "solved" is the trisection of an arbitrary angle. Actually, the problem has not been solved because the solution was accomplished mechanically, not according to Plato's rules. Pierre Wantzel (1814 - 1848) proved the impossibility of trisecting any given angle by Plato's restrictions. The method of "solving" the trisection problem is to construct a

54. Tangram Two thousand years later, in 1837, pierre Laurent wantzel showed, by an algebraicprocess, that there are angles that can't be trisected with rule and compasses http://www.univ.trieste.it/~nirtv/tanweb/texten.html

Video (Intranet) Video (Intranet) You need: Windows Media Player 6.2 References Pictures Home Links Play Tangram on line Tangram for Mac Tangram for Windows World Mathematical Year 2000 ... [Conclusion] Text of the video: WHAT ARE WE PLAYING: TANGRAM OR MATH? C.Pellegrino - L.Zuccheri 1. What is Tangram? Tangram is an antique game that originally comes from China. It is formed by dividing a square into seven parts that are called "tan": a square, a parallelogram and five isosceles right-angled triangles, two big ones, a medium and two little ones. The traditional rules of the game are simple: you have to lay the seven tans on a plane, without overlapping them, trying to form a figure that reproduces, maintaining the proportions, the figures that you have seen earlier in the instruction book. It may appear very easy to play the game Tangram, especially if you see the pieces already assembled in a square, but normally a beginner has already difficulties to reform the square, after having taken the pieces out of the box. But Tangram isn't a puzzle as many others. After having played a little bit, you begin to enjoy the subtle elegance with which the square has been divided.

55. Tangram Translate this page Solo dopo più di duemila anni, nel 1837, pierre Laurent wantzel dimostrò, con unprocedimento algebrico, che esistono angoli che non possono essere trisecati http://www.univ.trieste.it/~nirtv/tanweb/textit.html

Filmato (Intranet) Filmato (Intranet) E' necessario alla visione: Windows Media Player 6.2 Bibliografia Immagini Home Altri Siti per giocare al Tangram in linea per scaricare una versione del Tangram per Mac per scaricare una versione Window del Tangram per informazioni sull'Anno Internazionale della Matematica ... [Conclusione] Testo del video: A CHE GIOCO GIOCHIAMO: TANGRAM O MATEMATICA? C.Pellegrino - L.Zuccheri Il tangram è un antico gioco di origine cinese, ottenuto scomponendo un quadrato in sette parti dette tan: un quadrato, un romboide, e cinque triangoli rettangoli isosceli, di cui due grandi, uno medio e due piccoli . Le regole tradizionali del gioco sono semplici: si tratta di disporre sul piano, evitando sovrapposizioni, tutti i sette tan in modo da formare figure che riproducano, rispettando le proporzioni, quelle riportate in formato ridotto sui libretti che accompagnano il gioco. Giocare con il tangram può sembrare facile, troppo facile, soprattutto quando lo si vede già assemblato sotto forma di quadrato: normalmente però un principiante trova già difficoltà a comporre il quadrato, una volta tolti i pezzi dalla scatola.

56. Les Mathématiciens Français Translate this page Halphen Hamilton Herigone Hudde Georges Humbert pierre Humbert Jonquieres Julia LaHire Servois Tannery Jules Tannery Paul Tinseau Varignon Viète wantzel Weil. http://www.mathsnut.com/languages/french_mathem.html

Appell Arbogast Bachet Binet ... page d'accueil Jamie Coventry 24th Jan 2001

57. Terminkalender Translate this page 05.06.2003 Donnerstag, Geburtstag Sterbetag wantzel (1814), Adams (1819), Keynes(1883 Maxwell (1831), Steinitz (1871), Gosset (1876), pierre Humbert (1891 http://www.fmi.uni-leipzig.de/aktuell/termine.html

Mathematisches Institut Terminkalender gekennzeichnet! Dienstag Geburtstag: Sterbetag: Clavius Gudermann Shatunovsky Wessel Mittwoch Geburtstag: Sterbetag: Bowditch Andreev Hurwitz Max Abraham ... Wheeler Donnerstag Geburtstag: Sterbetag: Pearson Hartree Carl Neumann Shatunovsky ... Escher Freitag Geburtstag: Sterbetag: Herstein Grothendieck Dechales Lhuilier ... Hellinger Samstag Geburtstag: Sterbetag: Levi-Civita Ackermann Condorcet Cochran ... Maurice Kendall Sonntag Geburtstag: Sterbetag: Leshniewski Banach Ries Boys ... George Batchelor Beginn der Sommerzeit Montag Geburtstag: Sterbetag: Descartes Louis Richard Kirkman Korteweg ... Arthur Walker Dienstag Geburtstag: Sterbetag: Mohr Germain Kulik Moriarty ... Lev Landau Mittwoch Geburtstag: Sterbetag: Iyanaga Cohen 10 Uhr s.t. Arbeitsgemeinschaft NUMERIK Eugene Tyrtyshnikov (Russian Academy of Science, Moscow) Hierarchical Kronecker Tensor-Product Approximations Donnerstag Geburtstag: Sterbetag: Amringe Rademacher Ingham Ulam ... Cartwright Freitag Geburtstag: Sterbetag: Benjamin Peirce Lucas Eberhard Hopf Yau ... Siegel Samstag Geburtstag: Sterbetag: Hobbes Fabri Viviani Chaplygin ... Bertrand Sonntag Geburtstag: Sterbetag: Abel Burkill Montag Geburtstag: Sterbetag: Francois Francais Fredholm du Bois-Reymond Paley ... Kantorovich Vorlesungsbeginn - Sommersemester 2003 Dienstag Geburtstag: Sterbetag: Stone Wintner Peurbach Mittwoch Geburtstag: Sterbetag: Peacock Anstice Delaunay Laguerre ... Matsushima Donnerstag Geburtstag: Sterbetag: Tschirnhaus West Dudeney Lagrange ... Loyd Freitag Geburtstag: Sterbetag: Finsler Hall Kuttner Wiles ... Robinson Samstag Geburtstag: Sterbetag: Dandelin Zolotarev Lindemann Youden ... David Crighton Sonntag

58. Construções Com Régua E Compasso. Translate this page Isto foi feito em 1837 por pierre wantzel (1814-1848) que, além disso, mostroua impossibilidade da duplicação do cubo e da trisseção do ângulo. http://socrates.if.usp.br/~pellicer/x/algebra/algebraIII/trab1/node2.html

59. Barbara Jakimuszko Dopiero w roku 1837 matematyk francuski pierre wantzel udowodnil, ze dla wielukatów trysekcja jest niewykonalna – np. dla kata o mierze 60°. http://www.gimkorycin.com/Konspekt.htm

Barbara Jakimuszko KONSPEKT LEKCJI MATEMATYKI W KLASIE VI TEMAT: Rozwi¹zywanie zadañ z zastosowaniem poznanych konstrukcji. Po lekcji uczeñ: poprawnie wykreœla symetraln¹ odcinka, poprawnie wykreœla dwusieczn¹ k¹ta, ustala sposób rozwi¹zywania zadañ konstrukcyjnych, uzasadnia poprawnoœæ konstrukcji, formu³uje wnioski na podstawie wykonanej konstrukcji, wykonuje konstrukcje starannie i dok³adnie. METODA: dzia³ania praktyczne, dyskusja, problemowa FORMA PRACY: indywidualna, zespo³owa POMOCE : cyrkiel, linia, karty pracy (mapka Polski, narysowany okr¹g bez œrodka) PRZEBIEG LEKCJI I. CZÊŒÆ WSTÊPNA sprawdzenie obecnoœci, sprawdzenie pracy domowej, zapoznanie z celem lekcji ( zastosowanie poznanych konstrukcji w zadaniach problemowych), zapoznanie ze sposobem oceniania uczniów na lekcji. II. CZÊŒÆ G£ÓWNA nauczyciel pokazuje narysowany k¹t, a uczniowie wypowiadaj¹ zdania kojarz¹ce siê im z k¹tem – nauczyciel s³ownie ocenia aktywnoœæ i wiedzê uczniów, na tablicy narysowany jest schemat sto³u bilardowego i uczniowie odpowiadaj¹ za pomoc¹ konstrukcji, czy bia³a bila uderzy w czarn¹ ( uczniowie przedstawiaj¹ wczeœniej odszukane przez siebie w literaturze prawo odbicia i stosuj¹ w konstrukcji) – wybrany uczeñ wykonuje konstrukcjê na tablicy ( nauczyciel ocenia sposób wykonania konstrukcji), wszyscy wykonuj¹ konstrukcjê na kartach pracy i wklejaj¹ j¹ do zeszytu, uczniowie otrzymuj¹ karty pracy z mapk¹ Polski i rozwi¹zuj¹ zadanie: Zadanie.

60. Revista Comunicación Translate this page un ángulo arbitrario. pierre wantzel demostró en 1837 que es imposiblelograr esa construcción geométrica. Lo anterior no significa http://www.itcr.ac.cr/revistacomunicacion/2_2001/reflexiones_punto_crucial.htm

Reflexiones a partir del libro “El punto crucial” de Fritjof Capra M.B.A. Luis Gerardo Meza Cascante Profesor Catedrático Escuela de Matemática Instituto Tecnológico de Costa Rica VOLVER PRINCIPAL NÚMEROS ANTERIORES I. Introducción La lectura del libro “ El punto crucial ” de Fritjof Capra me acercó a planteamientos sumamente agudos, tres de los cuales deseo rescatar en este trabajo. Primero, el autor plantea que vivimos en un mundo que enfrenta una profunda crisis que afecta la salud, amenaza el sustento, la calidad del medio, la relación con los semejantes, la economía, la política y la tecnología. Una crisis que se caracteriza por tener dimensiones tanto políticas, como intelectuales, morales y espirituales. Además de la amenaza de una guerra nuclear, los alimentos y el agua resultan contaminados por la emisión de elementos radiactivos generados en las plantas productoras de energía. El exceso de población y la tecnología industrial degradan el entorno lo que afecta a las personas, a las plantas y a los animales. A estos efectos visibles del uso creciente de la tecnología tenemos que agregar los invisibles, que son, tal vez, más graves. Vivimos en un mundo, dice el autor, en el que aparecen

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