(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). 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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[ 409957, 11561]*) (*NotebookOutlinePosition[ 411176, 11597]*) (* CellTagsIndexPosition[ 411132, 11593]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ The Buffon needle problem revisited in a pedagogical perspective\ \>", "Title", FontFamily->"Arial"], Cell["Ivar Johannesen", "Author"], Cell["\<\ Oslo University College Faculty of Engineering Cort Adelers gt.30 N-0254 Oslo, Norway ivar.johannesen@iu.hio.no\ \>", "TextAboutAuthor"], Cell["\<\ Imagine marking the floor with many equally spaced parallel lines and a thin \ stick whose length exactly equals the distance L = 1 between the lines. If \ the stick is thrown on the floor, the stick may or may not cross one of the \ lines. The probability for a hit will involve \[Pi]. This is surprising since \ there are no circles involved, on the contrary all is typically linear. If we \ repeat the experiment many times, and keep track of the hits, we can get an \ estimate of the irrational number \[Pi]. We also consider sticks of length L > 1. This exercise can easily be done in \ a first year calculus course, where the students are challenged to consider \ concepts such as probability, definite integral, symmetry and inverse \ trigonometric function. The solution to this problem will therefore give many \ applications in a variety of fields in calculus. We go on throwing regular polygons of different sizes, increasing the number \ of edges and at last reach the ultimate goal: throwing circular objects. This \ paper illustrates the process of throwing sticks, polygons and circles \ analytically and graphically, and carrry out calculations for different n - \ gons. The result always include the number \[Pi], except when the circle is \ introduced! We will also see the circle result as a limiting value when n \ increases to infinity.\ \>", "Abstract", FontSlant->"Italic"], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell["\<\ The problem of throwing sticks on a set of parallel equidistant lines was \ first raised by the French naturalist and mathematician Georges Louis \ Leclerc Comte de Buffon in 1733, and later solved in 1777 by Buffon himself. \ Despite the linearity of the situation, the result gives us a method to \ compute the irrational number \[Pi]. For more than 250 years scientists have \ been intrigued by this puzzle, as can be seen by a quick search on the \ Internet. Many authors also extend the exercise to throwing regular polygons. \ In this paper I will consider regular polygons with n both even and odd. \ When the number of vertices is even, opposite vertices are situated on the \ diameter of the circumscribed circle. There are no diametrically opposed \ vertices in odd regular polygons, and therefore these n - gons offer more \ challenge for the students to handle. The length L of the stick is replaced \ by the diameter L of the circumscribed circle when regular polygons are \ considered.\ \>", "Text"], Cell[TextData[{ "This paper illustrates the process of throwing sticks, polygons and \ circles analytically and graphically, and carrry out calculations for \ different n - gons. The mathematics necessary is elementary and suitable for \ students in a first calculus course. The students will solve the necessary \ integrals and calculate the probabilities by hand before invoking ", StyleBox["Mathematica", FontSlant->"Italic"], ". \nThe introductory part of the lab considers sticks of length L = 1, the \ same unit length as the distance between lines. The idea is described in [1], \ including relevant ", StyleBox["Mathematica", FontSlant->"Italic"], " code for illustrations. 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In the parameter space (\[Theta], y) the graph of the function ", Cell[BoxData[ \(TraditionalForm\`y\ = \ 1\/2\ sin\ \[Theta]\)]], " will be the border line between areas representing hits and misses. In \ the next figure the misses are drawn in gray and the hits in black. 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On the other hand, arbitrary long sticks avoid \ hitting a line if the inclination is small enough. The probability for a \ stick of length L is given by the function ", StyleBox["probSticks:", FontFamily->"Arial"] }], "Text"], Cell[BoxData[ \(\(probSticks[L_] = \ \(4\/\[Pi]\) If[L\ < 1, \[Integral]\_0\%\(\[Pi]\/2\)\(L\/2\) Sin[\[Theta]] \[DifferentialD]\[Theta], \((\(1\/2\) \ \((\[Pi]\/2 - ArcSin[1\/L])\)\ + \ \[Integral]\_0\%\(ArcSin[1\/L]\)\(L\/2\) Sin[\[Theta]] \[DifferentialD]\[Theta])\)];\)\)], "Input",\ CellLabel->"In[60]:="], Cell[TextData[{ "The expression is interesting for several reasons. First we have a \ \"real\" situation in which an inverse trigonometric function arises \ naturally. Second, the definite integral that makes up the last term is \ noteworthy in that finding an antiderivative is easy, while evaluating it at \ the integral's endpoints requires a little more work. The students are \ encouraged to simplify cos(arc sin ", Cell[BoxData[ \(TraditionalForm\`1\/L\)]], ") and verify the simpler expression:" }], "Text"], Cell[BoxData[ \(probSticks[L_] := If[L \[LessEqual] 1, \(2\ L\)\/\[Pi], \(\(\(2\)\(\ \)\)\/\[Pi]\) \((\((L - \@\(L\^2 - \ 1\))\) + \ ArcCos[1\/L]\ )\)]\)], "Input", CellLabel->"In[31]:=", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "We see that the probabilities always involve the factor ", Cell[BoxData[ \(TraditionalForm\`1\/\[Pi]\)]], ". For ", Cell[BoxData[ \(TraditionalForm\`L \[LessEqual] \ 1\)]], "the graph is linear" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(probSticks\ [1]\)], "Input", CellLabel->"In[32]:=", Background->None], Cell[BoxData[ \(2\/\[Pi]\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[%]\)], "Input", CellLabel->"In[4]:=", Background->None], Cell[BoxData[ \(0.6366197723675814`\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["\<\ We summarize our results for sticks of any length by plotting the probability \ of hitting a line as a function of L. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[probSticks[L], {L, 0, 15}, PlotStyle \[Rule] Thickness[0.015], \[IndentingNewLine]Epilog \[Rule] {Blue, Dashing[{ .02}], Line[{{1, 0}, {1, 1}}]}, \[IndentingNewLine]AxesLabel -> TraditionalForm /@ {x, P[x]}];\)\)], "Input", CellLabel->"In[40]:=", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}, Background->None], 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This is the same as the \ diameter of the circumscribed circle. Due to symmetry it is enough to \ consider 0 \[LessEqual] \[Theta] \[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Pi]\/4\)\(.\)\)\)]], " " }], "Text"], Cell[TextData[{ "For ", Cell[BoxData[ FormBox[ RowBox[{"L", " ", "\[LessEqual]", " ", RowBox[{ "1", " ", "we", " ", "always", " ", "have", " ", Cell[TextData[{ " y \[LessEqual] ", Cell[BoxData[ \(TraditionalForm\`\(\(1\/2\)\(.\)\)\)]] }]]}]}], TraditionalForm]]], "What about squares whose diameter is greater than 1? Since the \[Theta] \ parameter is restricted to [0, Pi /4], we must consider the limit L = ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ". If L increases beyond that value, there will always be hits with at \ least one side of the square. The curve dividing hits and misses, will exceed \ the value of 1/2 for all values of \[Theta] , and the plot in parameter \ space will be empty. 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+ \ \[Integral]\_0\%\(\[Pi]\/4 - \ arccos(1\/L)\)\(L\/2\) \(cos(\[Pi]\/4 - \[Theta])\) \[Theta]\ = \(1\/2\) \ \(arccos(1\/L)\) + \ \[Integral]\_\(arcCos(1\/L)\)\%\(\[Pi]\/4\)\ \ \(L\/2\) cos\ \ \[Theta] \[DifferentialD]\[Theta]\)]], "\n for ", Cell[BoxData[ \(TraditionalForm\`1\ < \ L\ < \ \(\(\@2\)\(.\)\(\ \)\)\)]], " This gives us the probability function" }], "Text"], Cell[BoxData[ \(probSquare[L_] := \ 8\/\[Pi]\ Piecewise[{{\(L\/2\) \(\[Integral]\_0\%\(\[Pi]\/4\)Cos[\ \[Theta]] \[DifferentialD]\[Theta]\), 0\ \[LessEqual] L\ \[LessEqual] 1}, {\((\(1\/2\) ArcCos[1\/L]\ + L\/2\ \(\[Integral]\_\(ArcCos[1\/L]\)\%\(\[Pi]\/4\)Cos[\ \[Theta]] \[DifferentialD]\[Theta]\))\), \ 1 < L\ \[LessEqual] \ \@2}, {\[Pi]\/8, True}}]\)], "Input", CellLabel->"In[33]:="], Cell[CellGroupData[{ Cell[BoxData[ \({probSquare[1], probSquare[\@2]}\)], "Input", CellLabel->"In[34]:=", Background->None], Cell[BoxData[ \({\(2\ \@2\)\/\[Pi], 1}\)], "Output", CellLabel->"Out[34]="] }, Open ]], Cell[CellGroupData[{ 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\[Rule] 4\/3}\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ "When ", Cell[BoxData[ FormBox[ RowBox[{\(2\/\@3\), " ", "<", "L", "<", FormBox[\(4\/3\), "TraditionalForm"]}], TraditionalForm]]], ", the triangle will cut the line when \[Theta] > ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Theta]\_1\)\(\ \)\)\)]], "is the solution of the equation ", Cell[BoxData[ \(TraditionalForm\`\(L\/2\) \((cos(\ \[Pi]\/3 - \[Theta]) + cos\ \[Theta]\ )\) = \ 1, \ given\ \ 0\ < \ \[Theta]\ < \ \[Pi]\/6\)]], ". For ", Cell[BoxData[ \(TraditionalForm\`L\ \ \[GreaterEqual] \ 4\/3\)]], "the triangle has to cross at least one line since then ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\ \[LessEqual] \ 0\)]], ". Notice that ", Cell[BoxData[ \(TraditionalForm\`4\/3 = \ \((2\/\@3)\)\^2\)]], "." }], "Text"], Cell[BoxData[ \(\(\(probTriangle[L_]\)\(:=\)\(\ \)\(If[ L \[LessEqual] 2\/\@3, \(3\/\[Pi]\) \(\[Integral]\_0\%\(\[Pi]\/3\)L\ Cos[\ x] \[DifferentialD]x\)]\)\(\ \)\)\)], "Input", CellLabel->"In[28]:="], Cell[CellGroupData[{ Cell[BoxData[ \({probTriangle[1], probTriangle[2\/\@3]}\)], "Input", CellLabel->"In[30]:=", Background->None], Cell[BoxData[ \({\(3\ \@3\)\/\(2\ \[Pi]\), 3\/\[Pi]}\)], "Output", CellLabel->"Out[30]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Pentagons", "Section"], Cell["The calculations follow the same outline as for triangles:", "Text"], Cell[BoxData[ \(y[x_, L_] := \(L\/2\) \((Cos[\[Pi]\/5 - x] + Cos[x])\)\)], "Input", CellLabel->"In[1]:="], Cell[CellGroupData[{ Cell[BoxData[ \({Solve[y[\[Pi]\/10, L] \[Equal] 1, L], Solve[y[0, L] \[Equal] 1, L]} // \ Flatten\)], "Input", CellLabel->"In[3]:=", Background->None], Cell[BoxData[ \({L \[Rule] 2\ \@\(2\/\(5 + \@5\)\), L \[Rule] 8\/\(5 + \@5\)}\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[TextData[{ "If ", Cell[BoxData[ \(TraditionalForm\`L\ \[GreaterEqual] \ 8\/\(5 + \@5\)\)]], ", the pentagon has to cut one line. Again we see that this limit is the \ square of the lower limit, as was the case with ", Cell[BoxData[ \(TraditionalForm\`n\ = \ 3. \)]], " For ", Cell[BoxData[ \(TraditionalForm\`\@\(8\/\(5 + \@5\)\) < \ L\ < \ \(\(8\/\(5 + \@5\)\)\(.\)\)\)]], ", there will always be hits if \[Theta] > ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\)]], " is the solution to the equation ", Cell[BoxData[ \(TraditionalForm\`\(L\/2\) \((cos(\[Pi]\/5 - \[Theta]) + cos\ \[Theta])\) = \ 1, given\ \ 0\ < \ \[Theta]\ < \ \[Pi]\/10\)]], "." }], "Text"], Cell[BoxData[ \(probPentagon[L_] := \ If[L \[LessEqual] \ \@\(8\/\(5 + \@5\)\), \ \(5\/\[Pi]\) \ \(\[Integral]\_0\%\(\[Pi]\/5\)L\ Cos[x] \[DifferentialD]x\)]\)], "Input", CellLabel->"In[121]:="], Cell[CellGroupData[{ Cell[BoxData[ \({ToRadicals[probPentagon[1]], probPentagon[\@\(8\/\(5 + \@5\)\)]}\)], "Input", CellLabel->"In[132]:=", Background->None], Cell[BoxData[ \({\(5\ \@\(1\/2\ \((5 - \@5)\)\)\)\/\(2\ \[Pi]\), \(5\ \((\(-1\) + \ \@5)\)\)\/\(2\ \[Pi]\)}\)], "Output", CellLabel->"Out[132]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2n+1 - gons", "Section"], Cell[BoxData[ \(y[x_, L_] := \(L\/2\) \((Cos[\[Pi]\/n - x] + Cos[x])\)\)], "Input", CellLabel->"In[39]:="], Cell[TextData[{ "\nValue of ", Cell[BoxData[ \(TraditionalForm\`L\)]], " where all polygons in the most symmetric position will hit a line:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[y[\[Pi]\/\(2\ n\), L] \[Equal] 1, L]\)], "Input", CellLabel->"In[134]:=", Background->None], Cell[BoxData[ \({{L \[Rule] Sec[\[Pi]\/\(2\ n\)]}}\)], "Output", CellLabel->"Out[134]="] }, Open ]], Cell[TextData[{ "Value of", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(L\)\)\)]], " where every polygon hits a line, independent of rotation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[y[0, L] \[Equal] 1, L]\)], "Input", CellLabel->"In[135]:=", Background->None], Cell[BoxData[ \({{L \[Rule] 2\/\(1 + Cos[\[Pi]\/n]\)}}\)], "Output", CellLabel->"Out[135]="] }, Open ]], Cell[TextData[{ "For higher order n - gons where n is odd we always hit a line when ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{Cell[TextData[Cell[BoxData[ \(TraditionalForm\`L\ \[GreaterEqual] \ 2\/\(1 + Cos(\[Pi]\/n)\)\ = \ \((1\/\(cos(\[Pi]\/\(2 \ n\))\))\)\^2\)]]]], "."}]}], TraditionalForm]]], " For ", Cell[BoxData[ \(TraditionalForm\`\(\(L\)\(\ \)\(\[LessEqual]\)\(\ \ \)\(1\/\(cos(\[Pi]\/\(2 n\))\)\)\(\ \)\)\)]], "the probability is proportional to L, and for ", Cell[BoxData[ \(TraditionalForm\`1\/\(cos(\[Pi]\/\(2 n\))\) \[LessEqual] \ L\ \[LessEqual] \ \ \((1\/\(cos(\[Pi]\/\(2 n\))\))\)\^2\)]], " there will always be a hit if ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\ \[GreaterEqual] \ \[Theta]\_1\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\_1\)]], " is a solution of the equation ", Cell[BoxData[ \(TraditionalForm\`\(L\/2\) \((cos(\ \[Pi]\/n - \[Theta]) + cos\ \[Theta])\) = \ 1, given\ \ 0\ < \ \[Theta]\ < \ \[Pi]\/\(2 n\)\)]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Tossing coins", "Section"], Cell[TextData[{ "Suppose a penny with diameter ", Cell[BoxData[ \(TraditionalForm\`L\)]], " is thrown on a ruled surface. 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appears on the scene, the result \ does not involve \[Pi]!\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Summary for the case L = 1", "Section"], Cell[TextData[{ "In this paper we have extended the Buffon needle problem to include \ polygons thrown on a ruled floor, and calculated the probabilities for hits \ for various values of the diameter ", Cell[BoxData[ \(TraditionalForm\`L\)]], " of the circumscribed circle. Each time the answer involved the irrational \ nuimber \[Pi], and therefore indicated a simulation to estimate the value of \ this famous number. For ", Cell[BoxData[ \(TraditionalForm\`L\ = \ 1\)]], " we summarize the results ( the stick counts as 2- gon):" }], "Text"], Cell[TextData[{ "n= 2: \t p = ", Cell[BoxData[ \(TraditionalForm\`2\/\[Pi]\)]], " \t\tN[p] = 0.63662\n\nn= 3:\tp = ", Cell[BoxData[ \(TraditionalForm\`\(3 \@ 3\)\/\[Pi]\)]], "\t\tN[p] = 0.82699\n\nn = 4: p= ", Cell[BoxData[ \(TraditionalForm\`\(2 \@ 2\)\/\[Pi]\)]], "\t\tN[p] = 0.90032\n\nn = 5:\tp = ", Cell[BoxData[ \(TraditionalForm\`\(5\ \@\(\(1\/2\) \((5 - \@5)\)\)\)\/\[Pi]\)]], "\tN[p] = 0.93549\n\nn= 6:\tp = ", Cell[BoxData[ \(TraditionalForm\`3\/\[Pi]\)]], "\t\t\tN[p] = 0.95493\n\n n= 8:\tp = ", Cell[BoxData[ \(TraditionalForm\`\(4\ \@\(2 - \@2\)\)\/\[Pi]\)]], "\t\tN[p]= 0.97450\n\nn = 12: p = ", Cell[BoxData[ \(TraditionalForm\`\(3 \(\@ 2\) \((\@3 - 1)\)\)\/\[Pi]\)]], "\tN[p] = 0.98862\n\n n = \[Infinity]:\t p = 1\t\t\tN[p]= 1.0000" }], "Text"], Cell[TextData[{ "For each value of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " we find p(n) = ", Cell[BoxData[ \(TraditionalForm\`\(n\ \(sin(\[Pi]\/n)\)\)\/\[Pi]\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\(sin(\[Pi]\/n)\)\/\(\[Pi]\/n\)\)]], ", and therefore ", Cell[BoxData[ \(TraditionalForm\`lim\+\(n \[Rule] \[Infinity]\)\)]], " p(n) = 1. Increasing the number of vertices in the regular polygon to \ infinity, we therefore reach the result for tossing circles on the ruled \ floor.\n\nFor even n- gons we found the border line to be ", Cell[BoxData[ \(TraditionalForm\`y\ = \ \(\(L\/2\)\(\ \)\(cos\)\(\ \)\((\[Pi]\/n - \ \[Theta])\)\(\ \ \)\)\)]], "for ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(0 \[LessEqual] \ \[Theta] \[LessEqual] \ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(2 \[Pi]\)\/n\)]], ". All n-gons would cut a line if ", Cell[BoxData[ \(TraditionalForm\`L\ \[GreaterEqual] \ 1\/\(cos(\[Pi]\/n)\)\)]], ".When n\[Rule]\[Infinity] this gives ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(y\ = \ L\/2\)\)\)]], " independent of \[Theta] and always hits when ", Cell[BoxData[ \(TraditionalForm\`L\ \[GreaterEqual] \ 1\)]], ". This is in agreement with the circular case.\nFor odd n - gons we found \ ", Cell[BoxData[ \(TraditionalForm\`y\ = \ \ \(L\/2\) \((cos(\[Pi]\/n\ - \[Theta]) + \ cos\ \[Theta])\), \ 0\ < \ \[Theta]\ < \ \(\(\[Pi]\/n\)\(.\)\)\)]], " All n- gons would cut a line when ", Cell[BoxData[ \(L\ \[GreaterEqual] \ 2\/\(cos \((\[Pi]\/n)\) + 1\)\)], FontSize->14], ". As n\[Rule]\[Infinity], this again is in accordance with the circles.\n\ So the result of throwing pennies is fully compatible with the limiting \ results obtained by studying ", Cell[BoxData[ \(TraditionalForm\`n\)]], " - gons for large ", Cell[BoxData[ \(TraditionalForm\`n\)]], ". 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