(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 340592, 6201]*) (*NotebookOutlinePosition[ 341316, 6226]*) (* CellTagsIndexPosition[ 341272, 6222]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ I. Zur Quadratur des Kreises \[Dash] eine Berechnung von \[Pi]\ \>", "Section"], Cell[TextData[{ "Materialien zur Vorlesung ", StyleBox["Elementare Analysis", FontWeight->"Bold"], ", Wintersemester 2003 / 4" }], "SmallText"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1. 1 Einige Presseartikel zu \[Pi] \ \>", "Section"], Cell[TextData[{ StyleBox["Frankfurter Rundschau, ", FontFamily->"Arial", FontWeight->"Bold"], StyleBox["Frankfurt, 5. 8. 95", FontFamily->"Arial"] }], "Text"], Cell[TextData[{ StyleBox["Pi mal Daumen\n\n", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["TOKIO, 4. August (ap). Zwei japanische Mathematiker haben am \ Freitag einen neuen Weltrekord f\[UDoubleDot]r die Berechnung der Zahl \[Pi] \ bekanntgegeben. Wie Yasumasa Kanada und Daisuke Takahashi von der Universit\ \[ADoubleDot]t Tokio erkl\[ADoubleDot]rten, berechneten sie die Zahl, die das \ Verh\[ADoubleDot]ltnis des Kreisumfangs zum Durchmesser angibt, auf 3,22 \ Milliarden Dezimalstellen genau. Wissenschaftler der US-amerikanischen \ Columbia-Universit\[ADoubleDot]t hielten den alten Rekord mit 2,26 Milliarden \ Stellen.\n\nProfessor Kaneda sagte, seine Gruppe habe einen Gro\[SZ]computer \ benutzt, um zwei verschiedene Formeln durchrechnen zu lassen. F\[UDoubleDot]r \ die erste habe der Rechner 36 Stunden und 52 Minuten gebraucht, \ f\[UDoubleDot]r die andere 53 Stunden und 43 Minuten. Mathematiker sind der \ Meinung, da\[SZ] Pi unz\[ADoubleDot]hlig viele Dezimalstellen hat. Die pr\ \[ADoubleDot]zise Berechnung der Kreiszahl hat keinerlei praktischen Wert.\n\n\ ", FormatType->StandardForm, FontFamily->"Arial", FontWeight->"Plain"], StyleBox["\n", FontFamily->"Arial"], StyleBox[ButtonBox["Scientists find even more pi", ButtonData:>{ URL[ "http://www.ananova.com/news/story/sm_722745.html"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Arial"], StyleBox[" ", FontFamily->"Arial"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Arial", FontWeight->"Plain"], StyleBox["\nSiehe auch: ", FormatType->StandardForm, FontFamily->"Arial"], StyleBox[ButtonBox["Mathematik im Internet: \[Pi]", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/MathWWW_Pi.htm"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Arial"], StyleBox[" \n", FormatType->StandardForm, FontFamily->"Arial"] }], "Input", FontWeight->"Plain"], Cell[TextData[{ StyleBox["Extra Tip, ", FontFamily->"Arial", FontWeight->"Bold"], StyleBox["Kassel, 24. 9 1992", FontFamily->"Arial"] }], "Text"], Cell[TextData[{ StyleBox["Zahl Pi falsch\n", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times"], StyleBox["Hermann Wahler ist Rentner. Sein Hobby: Mathematik. Besonders \ angetan hat's ihm der Rauminhalt des Kreises. Der wird \[UDoubleDot]ber den \ Radius und die Zahl Pi berechnet. \"Ich habe das nachgerechnet, \ \[UDoubleDot]ber das Dreieck\", sagt Wahler. Ergebnis: Die Zahl Pi kann so \ nicht stimmen. Hat er sich verrechnet oder die Mathe-Professoren? Lesen Sie \ mehr auf Seite 6.\n\nSeite 6:\n\nZahl Pi neu berechnet!\n\nDr\[UDoubleDot]ckt \ man beim Taschenrechner auf die Taste \"\", erscheint der Wert 3,14159. damit \ errechnet man den Rauminhalt des Kreises, bisher. \"Diese Zahl Pi ist \ falsch\", meint Hermann Wahler aus Kaufungen.* \"Nach meinen Berechnungen ist \ diese Zahl kleiner als 3,14!\" Wie er drauf kommt? Er berechnet Pi nicht \ \[UDoubleDot]ber das Quadrat, sondern \[UDoubleDot]bers Dreieck. \ Normalerweise wird die Zahl Pi \[UDoubleDot]ber die Quadratur des Kreises \ berechnet. Ein Quadrat, dessen Ecken die Innenfl\[ADoubleDot]chen des Kreises \ ber\[UDoubleDot]hren; ein Quadrat, dessen Linien die Au\[SZ]enpunkte des \ Kreises ber\[UDoubleDot]hren.\n\nEine Formel dazu, und raus kommt die Zahl \ Pi: 3,14159.\n\nDoch Wahler gibt sich damit nicht zufrieden. Sein Trick: Er \ zieht ein Vieleck innerhalb des Kreises. Je mehr Ecken den Kreis ber\ \[UDoubleDot]hren, desto genauer spiegelt dieses Vieleck, den Rauminhalt des \ Kreises wieder.\n\nDas Vieleck unterteilt den Kreis in viele Dreiecke plus \ die Summe der Kreisabschnitte. Wahlers Ergebnis: Die Zahl Pi ist kleiner als \ 3,14. denn beim 600-Eck liegt sie bei 3.139999998, beim 6000-Eck bei \ 3,139999976 und beim 60.000-Eck bei 3,13999981.\nHermann Wahler: \"Je mehr \ Ecken ich berechne, desto genauer wird die endg\[UDoubleDot]ltige Zahl Pi. \ Sicher schon jetzt: Die Zahl, die sich bei den bisherigen Berechnungen \ ergibt, kann so nicht stimmen.\"\n\nStimmt seine Rechnung? \"Ich glaube \ schon, doch ich hoffe, da\[SZ] das jemand auf einem Gro\[SZ]rechner \ nachrechnet\", meint Wahler. Denn er selber hat nur einen Taschenrechner zur \ Verf\[UDoubleDot]gung. Es k\[ODoubleDot]nnte sein, da\[SZ] ein Rundungsfehler \ innerhalb seiner komplizierten Berechnung die Zahl Pi ver\[ADoubleDot]ndert.\n\ \"Die ganze Geschichte der Zahl habe ich verfolgt, es wurde immer um den \ Rauminhalt des Kreises gestritten\", meint Wahler. Bis eben jemand die Formel \ mit Quadratur des Kreises herausfand, doch der Kaufunger gibt sich damit \ nicht zufrieden.\n\n*Name und Ort ge\[ADoubleDot]ndert\n\n\n", FormatType->StandardForm] }], "Input", FontFamily->"Arial", FontWeight->"Plain"], Cell[TextData[StyleBox["Kurhessische Landeszeitung, Kassel, 7. 3. 1934\n", FontFamily->"Arial", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["Eine sensationelle Ver\[ODoubleDot]ffentlichung des K.f.d.K., \ Landesleitung Kurhessen:\n\nQuadratur des Kreises entdeckt\n", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times"], StyleBox["Ein epochales Forschungsergebnis von unsch\[ADoubleDot]tzbarer \ Bedeutung - Der Musikforscher Willi Oberle, Kassel, l\[ODoubleDot]st ein \ Jahrhundert altes Problem. Der Konstruktion des Ton-Prismas entgegen.\nDurch \ die \"Kurhessische Landeszeitung\" gibt der Landesleiter des Kampfbundes f\ \[UDoubleDot]r deutsche Kultur, Max K\[ODoubleDot]hler, der \ \[CapitalODoubleDot]ffentlichkeit folgendes Forschungsergebnis von Willi \ Oberle, Kassel-Niederzwehren, bekannt, das besagt, da\[SZ] dieser durch die \ Musikgeometrie zur L\[ODoubleDot]sung der Quadratur des Kreises gelangt ist:\n\ \n\"Dem Musikforscher Willi Oberle in Kassel-Niederzwehren, \ fr\[UDoubleDot]her in Baden-Baden, Mitarbeiter des Kampfbundes \ f\[UDoubleDot]r deutsche Kultur, Landesleitung Kurhessen, ist es gelungen, \ aus den Ergebnissen seiner Forschungen auf dem Gebiete der Musikgeometrie die \ Quadratur des Kreises aufzudecken. [ ... ]\n\nDer Weg zur L\[ODoubleDot]sung\n\ \nDa\[SZ] in seinen Konstruktionen die Quadratur des Kreises verborgen sein k\ \[ODoubleDot]nne, vermutete Oberle schon seit 1927. Die vielen anderen \ Probleme, wie das Kristallisationsph\[ADoubleDot]nomen, das \ Schwingungsproblem (Urschwingungen), die Architektonik der Zukunft, das \ Rassenproblem (Chromosomen = Ariogermanischer Urtyp), das Urwort (Runen, \ \[ADoubleDot]yptische und griechische Inschriften), in erster Linie das \ Tonprisma erschienen Oberle aber weit wichtiger als die Quadratur des \ Kreises, denn er erkannte, da\[SZ] die L\[ODoubleDot]sung der genannten \ Probleme aus seiner Musikgeometrie so gut wie gewi\[SZ] ist. Erst seit Oberle \ Mitarbeiter des Kurhessischen Kampfbundes f\[UDoubleDot]r deutsche Kultur \ geworden war und durch ihn neuen Schaffensimpuls erhielt, wurde sein \ Interesse f\[UDoubleDot]r das Problem der Quadratur des Kreises wach.\nMit gr\ \[ODoubleDot]\[SZ]ter Intensit\[ADoubleDot]t widmete sich dann Oberle der \ Aufgabe, das Problem aus seinen Konstruktionen aufzudecken. Da\[SZ] es sich \ nur um seine Aufdeckung handelt, ist ausdr\[UDoubleDot]cklich zu betonen, \ weil die Quadratur des Kreises, wie gesagt, seit 1927 in den Konstruktionen \ Oberles bis 1933 einen Dornr\[ODoubleDot]schenschlaf hielt. [ ... ]\n\nWir \ beschr\[ADoubleDot]nken uns hier zun\[ADoubleDot]chst auf die Verk\ \[UDoubleDot]ndigung der gefundenen Grundresultate:\n\n\"Der absolute Inhalt \ eines jeden Kreises ist gleich dem Quadrat \[UDoubleDot]ber der Hypothenuse \ des rechtwinckligen Dreiecks, dessen eine Kathede die gr\[ODoubleDot]\[SZ]te \ Seite des Oberleschen Trapezes darstellt, und dessen kleine Kathede gleich \ der Entfernung vom Kreismittelpunkt bis zum Schnittpunkte der beiden \ Diagonalen des das Trapez umschlie\[SZ]enden Kreises ist. Die beiden \ genannten Diagonalen schneiden sich im goldenen Schnitt und zugleich im \ rechten Winkel. Die kleinste Seite des Trapezes ist gleich dem Radius des \ Kreises (Pythagor\[ADoubleDot]ischer Lehrsatz!)\".\n\nForschungserfolg von \ unaussprechlicher Bedeutung\nDie Bedeutung der Entdeckung der Quadratur des \ Kreises l\[ADoubleDot]\[SZ]t sich erst daran ermessen, da\[SZ] sich die \ Menschheit seit Jahrhunderten mit diesem Problem besch\[ADoubleDot]ftigt \ hatte, ohne zu einer L\[ODoubleDot]sung zu kommen. [ ... ] ", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["\:f35f", FormatType->StandardForm, FontFamily->"Times"] }], "Input", FontWeight->"Plain"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1.2 N\[ADoubleDot]herungsweise Berechnung von \[Pi] (-287 (?) - -212 )\ \>", "Section"], Cell[TextData[{ ButtonBox["Archimedes' Approximation of Pi", ButtonData:>{ URL[ "http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.\ html"], None}, ButtonStyle->"Hyperlink"], " " }], "Text"], Cell[TextData[{ StyleBox["Approximation des Kreises durch regul\[ADoubleDot]re Polygone \ nach ", FontFamily->"Arial", FontWeight->"Bold"], ButtonBox["Archimedes", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.\ html"], None}, ButtonStyle->"Hyperlink"], StyleBox[" (-287 (?) - -212)", FontFamily->"Arial", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox[" ARCHIMEDES approximierte einen Kreis von innen und au\[SZ]en \ erst durch ein 6-Eck, dann durch je ein 12-Eck, 24-, 48- und 96-Eck.\n\nDer \ Umfang des einbeschriebenen 6-Ecks ist beim Einheitskreis (also dem Kreis mit \ Radius eins) 6. Damit ist 3 die erste N\[ADoubleDot]herung von (wie auch in \ der ", FontFamily->"Arial", FontSize->14], StyleBox[ButtonBox["Bibel: 1. K\[ODoubleDot]nige, 7. 23", ButtonData:>{ URL[ "http://www.bibel-online.net/buch/11.1-koenige/7.html#7,23"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Arial", FontSize->14], StyleBox[").\n\nARCHIMEDES verdoppelte bei jedem Schritt die Anzahl der \ Ecken der Polygone, bzw. halbierte die Winkel. Wenn Sie eine \ N\[ADoubleDot]herung kennen, k\[ODoubleDot]nnen Sie verstehen, wie die n\ \[ADoubleDot]chste berechnet wird.", FontFamily->"Arial", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .86603 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Times %%IncludeFont: Times /Times findfont 12 scalefont setfont % Scaling calculations 0.5 0.408852 0.433013 0.408852 [ [ 0 0 0 0 ] [ 1 .86603 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .86603 L 0 .86603 L closepath clip newpath 0 1 0 r 1.5 Mabswid [ ] 0 setdash .90476 .43301 m .70238 .78355 L .29762 .78355 L .09524 .43301 L .29762 .08248 L .70238 .08248 L .90476 .43301 L s 1 0 0 r .97619 .43301 m .7381 .84541 L .2619 .84541 L .02381 .43301 L .2619 .02062 L .7381 .02062 L .97619 .43301 L s 0 0 1 r 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Schreiben Sie \ alle von Ihnen berechneten Zwischenwerte in eine Tabelle.", FormatType->StandardForm, FontFamily->"Arial", FontSize->14, FontSlant->"Italic"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["Aufgabe: Rechnen Sie mit Ihrem Taschenrechner bis zum 13. Schritt \ ", FormatType->StandardForm, FontFamily->"Arial", FontSize->14], StyleBox["E", FormatType->StandardForm, FontFamily->"Arial", FontSize->14, FontSlant->"Italic"], StyleBox["n", FormatType->StandardForm, FontFamily->"Arial", FontSize->14, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" aus und zwar nach den beiden Formeln (1) und (2). 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