(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 5266, 131]*) (*NotebookOutlinePosition[ 34979, 1187]*) (* CellTagsIndexPosition[ 34935, 1183]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["A Brief History of Quaternions", "Subtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell["\<\ Complex numbers were a hot subject for research in the early eighteen \ hundreds. An obvious question was that if a rule for multiplying two numbers \ together was known, what about multiplying three numbers? For over a decade, \ this simple question had bothered Hamilton, the big mathematician of his day. \ The pressure to find a solution was not merely from within. Hamilton wrote \ to his son:\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ \"Every morning in the early part of the above-cited month [Oct. 1843] on my \ coming down to breakfast, your brother William Edwin and yourself used to ask \ me, 'Well, Papa, can you multiply triplets?' Whereto I was always obliged to \ reply, with a sad shake of the head, 'No, I can only add and subtract \ them.'\"\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ We can guess how Hollywood would handle the Brougham Bridge scene in Dublin. \ Strolling along the Royal Canal with Mrs. H-, he realizes the solution to the \ problem, jots it down in a notebook. So excited, he took out a knife and \ carved the answer in the stone of the bridge.\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ Hamilton had found a long sought-after solution, but it was weird, very \ weird, it was 4D. One of the first things Hamilton did was get rid of the \ fourth dimension, setting it equal to zero, and calling the result a \"proper \ quaternion.\" He spent the rest of his life trying to find a use for \ quaternions. By the end of the nineteenth century, quaternions were viewed \ as an oversold novelty.\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ In the early years of this century, Prof. Gibbs of Yale found a use for \ proper quaternions by reducing the extra fluid surrounding Hamilton's work \ and adding key ingredients from Rodrigues concerning the application to the \ rotation of spheres. He ended up with the vector dot product and cross \ product we know today. This was a useful and potent brew. Our investment in \ vectors is enormous, eclipsing their place of birth (Harvard had >1000 \ references under \"vector\", about 20 under \"quaternions\", most of those \ written before the turn of the century).\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ In the early years of this century, Albert Einstein found a use for four \ dimensions. In order to make the speed of light constant for all inertial \ observers, space and time had to be united. Here was a topic tailor-made for \ a 4D tool, but Albert was not a math buff, and built a machine that worked \ from locally available parts. We can say now that Einstein discovered \ Minkowski spacetime and the Lorentz transformation, the tools required to \ solve problems in special relativity.\ \>", "Text", FontFamily->"Times New Roman"], Cell["\<\ Today, quaternions are of interest to historians of mathematics. Vector \ analysis performs the daily mathematical routine that could also be done with \ quaternions. I personally think that there may be 4D roads in physics that \ can be efficiently traveled only by quaternions, and that is the path which \ is laid out in these web pages.\ \>", "Text", FontFamily->"Times New Roman"] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{462, 141}, WindowMargins->{{0, Automatic}, {Automatic, 20}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, None}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{57.5625, 50.375}, {36, 57.5625}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. 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StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], 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FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[TextData[StyleBox[ "The following styles are for linking automatically to the on-line help \ system."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], 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If a cell's FormatType matches the name of one of the styles defined \ below, then that style is applied between the cell's style and its own \ options."]], "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell[TextData[StyleBox[ "The style defined below is mixed in to any cell that is in an inline cell \ within another."]], "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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