(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 1320152, 21907]*) (*NotebookOutlinePosition[ 1320815, 21930]*) (* CellTagsIndexPosition[ 1320771, 21926]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Rigid Square Twists", "Title"], Cell[TextData[{ "4/26/03-4/27/03 by Thomas Hull\nOK. Yesterday I managed to animate a \ single vertex folding rigidly. And at the request of sarah-marie, I also did \ an example that doesn't work: a degree 4 vertex with plane angles all 90 \ degrees and all mountain creases. It's cool to see how it self-intersects! \ Those animations are all in the \"Square 1 vertex rigid.nb\" file.\n\nNow I'm \ going to try starting the arduous process of getting the homogeneous \ coordinate matricies and such for the complete, multivertex square twist \ crease pattern.\n\nFirst we need to define a function that will turn a \ homogeneous coordinates polygon into a regular 3D polygon which ", StyleBox["Mathematica", FontSlant->"Italic"], " can draw:\n" }], "Text"], Cell[BoxData[{ \(\(Clear[dum, n];\)\), "\[IndentingNewLine]", \(Trunc[dum_, n_] := Table[Drop[dum[\([k]\)], \(-1\)], {k, 1, n}]\)}], "Input"], Cell["\<\ Then we gotta define the polygons for the square twist crease pattern. (F1 \ is the one that will remain fixed during the folding process.) 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04oooooo0015ooooo`03o`000?oooooooooo00_ooooo00?o0000ooooooooool0:Oooool5o`0005?o oooo0016ooooo`03o`000?oooooooooo00[ooooo00?o0000ooooooooool09Oooool4o`0005Sooooo 0017ooooo`;o00002_ooool00ol0003oooooooooo`0Qooooo`Co0000G?ooool004Wooooo00?o0000 ooooooooool01oooool00ol0003oooooooooo`0Mooooo`Co0000H?ooool004[ooooo00?o0000oooo ooooool01_ooool00ol0003oooooooooo`0Hooooo`Go0000I?ooool004_ooooo0_l00006ooooo`03 o`000?oooooooooo01Cooooo1?l0001Yooooo`00COooool00ol0003oooooooooo`03ooooo`03o`00 0?oooooooooo013ooooo1?l0001]ooooo`00C_ooool00ol0003oooooooooo`02ooooo`03o`000?oo oooooooo00cooooo1?l0001aooooo`00Coooool2o`0000;ooooo00?o0000ooooooooool02?ooool4 o`0007Gooooo001Aooooo`03o`000?oooooo000000Gooooo1Ol0001iooooo`00D_ooool2o`000003 ooooool0003o000000;o0000O_ooool005?ooooo0_l00022ooooo`00\ \>"], ImageRangeCache->{{{0, 214.125}, {270.688, 0}} -> {0.101962, -0.000440009, \ 0.00357663, 0.00357663}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Now let's define the fundamental matrix transformations we'll need. These \ are all right out of the \"Modelling folding in three dimensions using affine \ transformations\" paper that I wrote with sarah-marie. They are: B = translate by the vertex {vx,vy,vz} H = rotate the y-z plane by angle x A = rotate the x-y plane by angle x \[Chi] = the composition of these which models folding along an arbitrary \ crease. Here, x = the folding angle (= \[Pi] - the dihedral angle of the \ fold), a = the \"position angle\", which is how much you have to rotate the \ x-axis by to move it in the position of the crease (assuming the vertex is at \ the origin), and {vx,vy} are the coordinates of the vertex the crease is \ connected to. (Yes, some creases are connected to 2 vertices (remember, we \ don't count vertices on the boundary as real vertices), and in those cases we \ can pick either, as long as we're being consistent...). Then \[Chi] does the \ following: translate the vertex to the origin, rotate the crease line to the \ x-axis, rotate the y-z plane by the folding angle, rotate the x-y plane so \ the crease line is back in its original position, and finally translate the \ vertex back to where it came from.\ \>", "Text"], Cell[BoxData[{ \(\(B[vx_, vy_, vz_] = {{1, 0, 0, vx}, {0, 1, 0, vy}, \ {0, 0, 1, vz}, \ {0, 0, 0, 1}};\)\), "\[IndentingNewLine]", \(\(H[ x_] = {{1, 0, 0, 0}, {0, Cos[x], \(-Sin[x]\), 0}, {0, Sin[x], Cos[x], 0}, \ {0, 0, 0, 1}};\)\), "\[IndentingNewLine]", \(\(A[ x_] = {{Cos[x], \(-Sin[x]\), 0, 0}, {Sin[x], Cos[x], 0, 0}, {0, 0, 1, 0}, \ {0, 0, 0, 1}};\)\), "\[IndentingNewLine]", \(\(\[Chi][x_, a_, vx_, \ vy_] := B[vx, vy, 0] . A[a] . H[x] . A[\(-a\)] . B[\(-vx\), \(-vy\), 0];\)\)}], "Input"], Cell["\<\ Now, Robert Lang's dihedral angle equations should still work here, \ meaning that one dihedral angle should determine all of the others. The hard \ part is figuring out which equation to use where, given the MV assignment I \ want to follow in this version of the square twist... But waitaminute. We only have to keep track of two angles: \ \[Gamma]1=\[Gamma]3 and \[Gamma]4 = 2\[Pi]-\[Gamma]2. Now, \[Gamma]1 is \ parameterized by t, so I'll just use t for that. I'll let \[Gamma]4 be the \ other one.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{\(Clear[t, a1, a2]\), "\n", \(a1 = \[Pi]/4;\), "\n", \(a2 = \[Pi]/2;\), "\n", RowBox[{ RowBox[{ RowBox[{ StyleBox[ RowBox[{"\[Gamma]", StyleBox["4", "Input"]}]], StyleBox["[", "Input"], StyleBox["t_", "Input"], StyleBox["]", "Input"]}], StyleBox["=", "Input"], RowBox[{ StyleBox["Simplify", "Input"], StyleBox["[", "Input"], RowBox[{ RowBox[{ StyleBox["2", "Input"], "\[Pi]"}], "-", \(ArcCos[ Cos[t] - \(\((Sin[t])\)^2*Sin[a1]*Sin[a2]\)\/\(1 - \ \((Cos[a1]*Cos[a2] + Sin[a1]*Sin[a2]*Cos[t])\)\)]\)}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{ StyleBox[ RowBox[{"\[Gamma]", StyleBox["2", "Input"]}]], StyleBox["[", "Input"], StyleBox["t_", "Input"], StyleBox["]", "Input"]}], StyleBox["=", "Input"], RowBox[{ StyleBox["Simplify", "Input"], StyleBox["[", "Input"], RowBox[{ RowBox[{ StyleBox["2", "Input"], "\[Pi]"}], "-", \(\[Gamma]4[t]\)}], "]"}]}], ";"}]}]}]], "Input"], Cell[BoxData[ \(Null\^5\)], "Output"] }, Open ]], Cell["\<\ Now then, the matrix computations that handled faces F2, F9, and F8 in the \ single vertex experiment should be exactly the same here (except we have to \ add the translation vertex). The other faces, F3, F4, F5, F6, and F7 will be determined by an artibrary \ vertex-avoiding path that we take from F1 to these faces. Of course, we want \ to keep this as simple as possible. So here are the paths I'll use: for F3: F1 -> F2 -> F3 for F4: F1-> F2 -> F3 -> F4 for F5: F1 -> F2 -> F3 -> F4 -> F5 for F6: F1 -> F2 -> F9 -> F6 for F7: F1 -> F8 -> F7 \ \>", "Text"], Cell["\<\ I'll use this space to test whether or not any of this is working...\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(t = 2;\)\[IndentingNewLine] \(NewF[2] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, \ 1/4, 1/2] . \(F[2]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[9] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, 1/4, 1/2] . \[Chi][\[Pi] - t, \(-\[Pi]\)/4, 1/4, 1/2] . \(F[9]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[8] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, 1/4, 1/2] . \[Chi][\[Pi] - t, \(-\[Pi]\)/4, \ 1/4, 1/2] . \[Chi][\[Pi] - \[Gamma]2[t], \[Pi]/4, 1/4, 1/2] . \(F[8]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[3] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, \ 1/4, 1/2] . \[Chi][\[Pi] - \[Gamma]2[t], \ 3 \[Pi]/2, \ 1/2, \ 1/4] . \(F[3]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[4] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, \ 1/4, 1/2] . \[Chi][\[Pi] - \[Gamma]2[t], \ 3 \[Pi]/2, \ 1/2, \ 1/4] . \[Chi][\[Pi] - t, \ 0, \ 1/2, \ 1/4] . \(F[4]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[5] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, \ 1/4, 1/2] . \[Chi][\[Pi] - \[Gamma]2[t], \ 3 \[Pi]/2, \ 1/2, \ 1/4] . \[Chi][\[Pi] - t, \ 0, \ 1/2, \ 1/4] . \[Chi][\[Pi] - \((2 \[Pi] - t)\), 0, 3/4, 1/2] . \(F[5]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[6] = Table[\[Chi][\[Pi] - \[Gamma]4[t], 3 \[Pi]/2, \ 1/4, 1/2] . \[Chi][\[Pi] - t, \(-\[Pi]\)/4, 1/4, 1/2] . \[Chi][\[Pi] - \((2 \[Pi] - t)\), \ 7 \[Pi]/4, \ 1/2, \ 3/4] . \(F[6]\)[\([k]\)], {k, 1, 4}];\)\[IndentingNewLine] \(NewF[7] = Table[\[Chi][2 \[Pi] - \((\[Pi] - t)\), \ \[Pi], \ 1/4, \ 1/2] . \[Chi][\[Pi] - t, \ \[Pi], \ 1/2, 3/4] . \(F[7]\)[\([k]\)], \ {k, \ 1, 4}];\)\[IndentingNewLine] \(NewF[1] = F[1];\)\[IndentingNewLine] 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