Mathematica notebook for calculating a Bryant cousin minimal surface
- To: mathgroup at smc.vnet.net
- Subject: [mg49284] Mathematica notebook for calculating a Bryant cousin minimal surface
- From: "Roger L. Bagula" <rlbtftn at netscape.net>
- Date: Sun, 11 Jul 2004 02:16:17 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Based on Pascal Collin's paper: "The Geometry of Finite Topology Bryant Surfaces" L^4 group geometery and U(1)*SU(2) and the quaternion are entires isomorphous in group terms ( that is the elemets of each can be transformed by simple comformal transformations to the other. I'm not really satisfied with this method. (*********************************************************************** 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 at 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[ 7155, 351]*) (*NotebookOutlinePosition[ 8054, 380]*) (* CellTagsIndexPosition[ 8010, 376]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(Clear[f, g, C0, A0, D0, B0, dF, F, W, M, t]\)], "Input"], Cell[BoxData[ \( (*\ a\ possible\ \ solution\ to\ the\ Bryant\ cousin\ equations*) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(A0[z] = 1/z\)], "Input"], Cell[BoxData[ \(1\/z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B0[z] = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(C0[z] = z^2\)], "Input"], Cell[BoxData[ \(z\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D0[z] = z\)], "Input"], Cell[BoxData[ \(z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(F = {{A0[z], B0[z]}, {C0[z], D0[z]}}\)], "Input"], Cell[BoxData[ \({{1\/z, 0}, {z\^2, z}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[F]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(dF = D[F, z]\)], "Input"], Cell[BoxData[ \({{\(-\(1\/z\^2\)\), 0}, {2\ z, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[dF]\)], "Input"], Cell[BoxData[ \(\(-\(1\/z\^2\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(F1 = MatrixPower[F, \(-1\)]\)], "Input"], Cell[BoxData[ \({{z, 0}, {\(-z\^2\), 1\/z}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[F1]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(M = F1 . dF\)], "Input"], Cell[BoxData[ \({{\(-\(1\/z\)\), 0}, {3, 1\/z}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(W = {{g*f, \(-f\)*g^2}, {f, \(-f\)*g}}\)], "Input"], Cell[BoxData[ \({{f\ g, \(-f\)\ g\^2}, {f, \(-f\)\ g}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f = Simplify[M[\([2, 1]\)]]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = M[\([1, 1]\)] - f*g\)], "Input"], Cell[BoxData[ \(\(-3\)\ g - 1\/z\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[a == 0, g]\)], "Input"], Cell[BoxData[ \({{g \[Rule] \(-\(1\/\(3\ z\)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g = Simplify[M[\([1, 1]\)]/f]\)], "Input"], Cell[BoxData[ \(\(-\(1\/\(3\ z\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(dF\)], "Input"], Cell[BoxData[ \({{\(-\(1\/z\^2\)\), 0}, {2\ z, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Det[dF] == 0, z]\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g\)], "Input"], Cell[BoxData[ \(\(-\(1\/\(3\ z\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = Expand[f*\((1 - g^2)\)/2]\)], "Input"], Cell[BoxData[ \(3\/2 - 1\/\(6\ z\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y1 = Expand[I*\((1 + g^2)\)/2]\)], "Input"], Cell[BoxData[ \(I\/2 + I\/\(18\ z\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(z1 = Expand[f*g]\)], "Input"], Cell[BoxData[ \(\(-\(1\/z\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(xa = Integrate[x1, {z, 0.1, t}]\)], "Input"], Cell[BoxData[ \(\(-1.8166666666666666`\) + \(\(1.`\[InvisibleSpace]\) + 0.`\ t + 9\ t\^2\)\/\(6\ \((\(0.`\[InvisibleSpace]\) + t)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ya = Integrate[y1, {z, 0.1, t}]\)], "Input"], Cell[BoxData[ \(0.505555555555555535`\ I + \(\(-1.`\)\ I + 0.`\ I\ t + 9\ I\ t\^2\)\/\(18\ \((\(0.`\[InvisibleSpace]\) + t)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(za = Integrate[z1, {z, 0.1, t}]\)], "Input"], Cell[BoxData[ \(\(-2.30258509299404545`\) - Log[\(0.`\[InvisibleSpace]\) + t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = Exp[r]*\((Cos[t0] + I*Sin[t0])\)\)], "Input"], Cell[BoxData[ \(E\^r\ \((Cos[t0] + I\ Sin[t0])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ga = ParametricPlot3D[{Re[xa], Re[ya], Re[za]}, {r, \(-1\), 1}, {t0, \(-Pi\), Pi}, \ PlotPoints -> {30, 30}, \ Axes -> False, Boxed -> False]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ga, ViewPoint -> {0.196, \ 3.336, \ 0.531}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ga, ViewPoint -> {3.322, \ \(-0.360\), \ 0.531}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Show[ga, ViewPoint -> {0.045, \ 0.001, \ 3.383}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1920}, {0, 1060}}, WindowSize->{1610, 630}, WindowMargins->{{142, Automatic}, {Automatic, 123}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00/0004/0B`000003509H?ocokD<f@V[7b<5:@?l0040004/0B`000003509H04/ 02d5X5k/02H20@4101P00BL?00400ATLNm000000000000010000000000000000 0000000000000002000000@210D00000\>" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 76, 1, 27, "Input"], Cell[1788, 52, 103, 2, 27, "Input"], Cell[CellGroupData[{ Cell[1916, 58, 44, 1, 27, "Input"], Cell[1963, 61, 38, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2038, 67, 42, 1, 27, "Input"], Cell[2083, 70, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2155, 76, 44, 1, 27, "Input"], Cell[2202, 79, 38, 1, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2277, 85, 42, 1, 27, "Input"], Cell[2322, 88, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2394, 94, 69, 1, 27, "Input"], Cell[2466, 97, 56, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2559, 103, 39, 1, 27, "Input"], Cell[2601, 106, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2673, 112, 45, 1, 27, "Input"], Cell[2721, 115, 68, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2826, 121, 40, 1, 27, "Input"], Cell[2869, 124, 50, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2956, 130, 60, 1, 27, "Input"], Cell[3019, 133, 61, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3117, 139, 40, 1, 27, "Input"], Cell[3160, 142, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3232, 148, 44, 1, 27, "Input"], Cell[3279, 151, 65, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3381, 157, 71, 1, 27, "Input"], Cell[3455, 160, 72, 1, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3564, 166, 60, 1, 27, "Input"], Cell[3627, 169, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3699, 175, 56, 1, 27, "Input"], Cell[3758, 178, 50, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3845, 184, 49, 1, 27, "Input"], Cell[3897, 187, 68, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4002, 193, 62, 1, 27, "Input"], Cell[4067, 196, 54, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4158, 202, 35, 1, 27, "Input"], Cell[4196, 205, 68, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4301, 211, 55, 1, 27, "Input"], Cell[4359, 214, 36, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4432, 220, 34, 1, 27, "Input"], Cell[4469, 223, 35, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4541, 229, 34, 1, 27, "Input"], Cell[4578, 232, 54, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4669, 238, 63, 1, 27, "Input"], Cell[4735, 241, 55, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4827, 247, 63, 1, 27, "Input"], Cell[4893, 250, 56, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4986, 256, 49, 1, 27, "Input"], Cell[5038, 259, 47, 1, 40, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5122, 265, 64, 1, 27, "Input"], Cell[5189, 268, 169, 3, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5395, 276, 64, 1, 27, "Input"], Cell[5462, 279, 163, 3, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5662, 287, 64, 1, 27, "Input"], Cell[5729, 290, 98, 2, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5864, 297, 69, 1, 27, "Input"], Cell[5936, 300, 66, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6039, 306, 201, 4, 27, "Input"], Cell[6243, 312, 132, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6412, 320, 81, 1, 27, "Input"], Cell[6496, 323, 132, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6665, 331, 86, 1, 27, "Input"], Cell[6754, 334, 132, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6923, 342, 81, 1, 27, "Input"], Cell[7007, 345, 132, 3, 26, "Output"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)