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Hart \ \>", "Text", Evaluatable->False], Cell[TextData[{ "\n One important role of high-level mathematical software such as ", StyleBox["Mathematica", FontSlant->"Italic"], " is that it easily allows for the testing of experimental algorithms. Here \ we explore a method of finding a canonical form of a polyhedron. The \ canonical form is a polyhedron topologically equivalent to an input \ polyhedron, but with all edges tangent to a unit sphere and with the center \ of gravity of the tangent points being the origin. A variety of examples \ will illustrate this procedure. One application is that the algorithm \ constructs a geometrically self-dual polyhedron given one which is only \ combinatorially self-dual." }], "Text", Evaluatable->False], Cell[TextData[{ "A theorem of Schramm [Schramm 1992] states that for any given convex \ polyhedron or 3-connected planar graph, there is a topologically equivalent \ polyhedron with the following properties:\n\n 1) each edge is tangent to the \ unit sphere,\n\n 2) the center of gravity of the points of tangency is the \ origin.\n\n\n\nThe solution is unique, up to rotations and reflections of the \ sphere and so provides a kind of canonical representation. Although we will \ not pursue it here, the theorem is actually more general in that it allows \ the sphere to be replaced with an arbitrary smooth convex body, e.g., an \ egg-shaped quasi-ellipsoid. The theorem is closely related to the \ Koebe-Andreev-Thurston Circle Packing theorem for planar graphs. See Ziegler \ [Ziegler 1995, p. 118] for discussion and other references. \n\n\n\nFor \ example, if our input polyhedron is a geometrically distorted (but \ topologically unchanged) form of any Platonic or Archimedean solid, and we \ calculate its canonical form, our algorithm should output its undistorted \ form centered at the origin, with a midradius of 1, and with an arbitrary \ rotation. E.g., given any parallelepiped as input, the canonical form output \ is a cube of edge ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ". For an arbitrary polyhedron, the canonical form is a way of illustrating \ its combinatorial or topological structure, which often lets one immediately \ see and understand its structure and symmetry. \n\n\n\nAs an illustration, \ we will use the algorithm to find three geometrically self-dual polyhedra, \ given starting points which are only combinatorially self-dual. It follows \ from the above that the dual polyhedron also shares properties (1) and (2). \ Thus, for these three self-dual examples, we can make a compound of the \ polyhedron with its dual, i.e., with itself, in which both polyhedra have the \ same points of tangencies and at these points their edges cross each other's \ at right angles. \n\n\n\nThe proofs cited above are of existence and not \ constructive, so we are interested in an algorithm for determining the \ canonical form of an input polyhedron. The algorithm proposed here operates \ by relaxation to iteratively move the vertices of the given polyhedron along \ a trajectory which converges at the canonical form. Although we begin and end \ with a polyhedron, during this relaxation, the object defined by the vertices \ is likely not to be a polyhedron geometrically. Sets of points which belong \ to a face (combinatorially) are likely not to be coplanar. So we add a third \ condition for a solution, to the two above:\n\n\n\n3) the faces are planar.\n\ \n\n" }], "Text", Evaluatable->False], Cell["\<\ Algorithm \ \>", "Section", Evaluatable->False], Cell[TextData[{ "\nOur algorithm inputs a polyhedron and iteratively adjusts its vertices \ to slightly improve its conformance with the three conditions above. Within a \ couple of hundred iterations, it typically finds all the conditions are \ satisfied within a small tolerance, and stops. Three simple operations are \ all that is needed, corresponding to the three conditions:\n\n\n\n A) For \ each edge, the closest point to the origin is determined; call it p. If p \ lies at unit distance from the origin, condition (1) is satisfied for that \ edge. If not, a small fraction of p is added to the two vertices which define \ the edge, (in proportion to the sign and amount of the error) so that at the \ next iteration the edge will be closer to tangency with the unit sphere. \n\n \ B) The center of gravity of all the points p is determined. If it is zero, \ condition (2) is satisfied. If not, it is subtracted from all the vertices. \ \n\n C) For each face, if the vertices lie in some plane in space, \ condition (3) is satisfied. If not, a plane which approximates it is \ computed. Each vertex of the face is then moved along a normal towards the \ plane. \n\n\n\nIterating, it sometimes takes many steps for a correction at \ one part of the polyhedron to percolate its way around and equalize the \ conditions everywhere. On the examples below, between 50 and 100 iterations \ were sufficient to have all conditions satisfied within a tolerance of ", Cell[BoxData[ \(TraditionalForm\`10\^\(-8\)\)]], ". Although the individual steps are quite simple, this takes a minute or \ so in these examples and would take longer on more complex polyhedra. \ Refinement and optimization are left as future work.\n\nAny canonicalization \ program must begin with a data structure for representing polyhedra \ positioned in space. Our data structure is naturally of a two-part nature, \ consisting of the topological connectivity information (which remains \ constant) and a list of vertex coordinates (which is iteratively updated). We \ use v for the list of vertex coordinates, each a triple {x,y,z} of Cartesian \ components. The topological information is given as a list f of faces, where \ each face is a list of vertex indices (in either clockwise or \ counter-clockwise order). For example, a cube of edge 2 can be represented \ with the following v and f: \n" }], "Text", Evaluatable->False], Cell["\<\ vCube = {{-1, -1, -1}, {-1, -1, 1}, {-1, 1, -1}, {-1, 1, 1}, {1, \ -1, -1}, {1, -1, 1}, {1, 1, -1}, {1, 1, 1}}; \ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell["\<\ fCube = {{1, 2, 4, 3}, {5, 6, 8, 7}, {1, 2, 6, 5}, {3, 4, 8, 7}, {1, 3, 7, 5}, {2, 4, 8, 6}}; \ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[TextData[{ "\nA polyhedron in this ", StyleBox["v", FontSlant->"Italic"], " and ", StyleBox["f", FontSlant->"Italic"], " form can be displayed with the following view function, e.g. ", StyleBox["view[vCube, fCube]", "Input", FontFamily->"Courier"], ".\n" }], "Text", Evaluatable->False], Cell["\<\ view[v_, f_] := Show[Graphics3D[Polygon /@ Map[v[[#1]] & , f, {2}]], PlotRange-> All,Boxed \ -> False]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[TextData[{ "\nAs we will be manipulating edges, we write a function ", StyleBox["face2edges", "Input"], " which creates a list of edges from a given face, and a function ", StyleBox["f2e", "Input", FontFamily->"Courier"], " which creates a list of all edges from the list of faces, f. The use of \ ", StyleBox["Sort", FontWeight->"Bold"], " and ", StyleBox["Union", FontWeight->"Bold"], " eliminates duplicates, e.g., resulting in only twelve edges for the cube.\ \n" }], "Text", Evaluatable->False], Cell["\<\ face2edges[face_] := MapThread[Sort[{#1, #2}] & , {face, \ RotateLeft[face]}]; \ \>", "Input"], Cell["f2e[f_] := Union @@ face2edges /@ f", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["eCube = f2e[fCube]", "Input", PageWidth->Infinity, InitializationCell->True], Cell[BoxData[ \({{1, 2}, {1, 3}, {1, 5}, {2, 4}, {2, 6}, {3, 4}, {3, 7}, {4, 8}, {5, 6}, {5, 7}, {6, 8}, {7, 8}}\)], "Output"] }, Open ]], Cell[TextData[ "\nGiven any line, the closest point on that line to the origin will be at \ distance 1 from the origin if the line is tangent to the unit sphere, i.e., \ an edge satisfying condition (1) above. The following method finds the \ closest point on a line to the origin, when we know two points, P1 and P2, of \ the line. Use a parameter alpha to represent an arbitrary point on the line \ as P1 + alpha L, where L=P2-P1 is a vector parallel to the line. Observe \ that if p is the closest point, the vector p = P1 + alpha L is orthogonal to \ L. Expressing this as a dot product being zero and solving for alpha gives \ alpha = - L.P1 / L.L, which is used in the following function.\n"], "Text", Evaluatable->False], Cell[BoxData[ \(closest[{P1_, P2_}] := With[{L = P2 - P1}, P1 - \(L . P1\ L\)\/L . L]\)], "Input", PageWidth->Infinity, InitializationCell->True], Cell[TextData[ "\nThe following statement finds a list of these points corresponding to the \ list of edges, which in this example gives us the cube's twelve edge \ midpoints. The inner Map replaces indices into v with xyz coordinates.\n"], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell["closest /@ Map[vCube[[#1]] & , eCube, {2}]", "Input", PageWidth->Infinity, InitializationCell->True], Cell[BoxData[ \({{\(-1\), \(-1\), 0}, {\(-1\), 0, \(-1\)}, {0, \(-1\), \(-1\)}, { \(-1\), 0, 1}, {0, \(-1\), 1}, {\(-1\), 1, 0}, {0, 1, \(-1\)}, {0, 1, 1}, {1, \(-1\), 0}, {1, 0, \(-1\)}, {1, 0, 1}, {1, 1, 0}}\)], "Output"] }, Open ]], Cell[TextData[ "\nFor step (B) of the algorithm, we subtract from each vertex the mean of \ these points of tangency.\n"], "Text", Evaluatable->False], Cell["\<\ recenter[v_, e_] := With[{centroid = Plus @@ closest /@ Map[v[[#1]] & , e, {2}]/Length[e]}, (#1 - centroid & ) /@ v]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[TextData[{ "\nFor step (A) we ", StyleBox["Scan", FontWeight->"Bold"], " through the edges, compute the distance of the tangent point from the \ origin, and make a correction to the two endpoints in proportion to how the \ distance differs from unity. The parameter 0.5 is experimental; it is \ designed to move the points slowly, preventing large unstable dynamics.\n" }], "Text", Evaluatable->False], Cell["\<\ tangentify[v_, e_] := Module[{newV = v, t, c}, Scan[(t = closest[v[[#1]]]; c = 0.5*(1 - Sqrt[t . t])*t; newV[[#1[[1]]]] += c; newV[[#1[[2]]]] += c; ) & , e]; newV]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[TextData[ "\nFor step (C) we characterize an approximating face plane by its unit \ normal vector, n, and its distance from the origin. A fast approximation to \ the normal is just to sum the normals at each face vertex, obtained by taking \ the cross product of the two edge vectors meeting there. The distance is \ obtained by projecting the centroid of the vertices on a unit vector in that \ direction. (A least-squares fit of a plane to the points might perform \ better, but would be slower.) Again, a stabilizing factor is employed, here \ 0.2. Future work might optimize these factors.\n"], "Text", Evaluatable->False], Cell["\<\ unit[x_] := With[{mag2 = x . x}, If[mag2 != 0, x/Sqrt[mag2], \ x]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ cross[{ax_, ay_, az_}, {bx_, by_, bz_}] := {ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx}\ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ approxNormal[face_] := unit[Plus @@ MapThread[unit[cross[#1 - #2, #2 - #3]] & , {face, RotateLeft[face, 1], RotateLeft[face, 2]}]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell["\<\ planarize[v_, f_] := Module[{newV = v, faceXYZ, n, centroid}, Scan[(faceXYZ = v[[#1]]; n = approxNormal[faceXYZ]; centroid = Plus @@ faceXYZ/Length[faceXYZ]; if[n . centroid < 0, n = -n]; Scan[newV[[#1]] += 0.2*n . (centroid - v[[#1]])*n & , #1]; ) & , f]; newV]\ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ "\nAssembling the above pieces into an iterative loop gives the following \ canonicalization function. As a sanity test, we verify that the procedure \ operates on a cube of edge 2 to give a cube of edge ", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ", i.e., with edges tangent to the unit sphere.\n" }], "Text", Evaluatable->False], Cell["\<\ canonicalize[v_, f_] := Module[{newV = N[v], e = f2e[f], oldV, maxChange}, Do[oldV = newV; newV = tangentify[newV, e]; newV = recenter[newV, e]; newV = planarize[newV, f]; maxChange = Max[Abs[oldV - newV]]; If[maxChange < 10.^(-8), Break[]], {i, 80}]; Print[\"Solved within \", maxChange]; newV]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[CellGroupData[{ Cell["canonicalize[vCube, fCube]", "Input", PageWidth->Infinity, InitializationCell->True], Cell[BoxData[ InterpretationBox[ \("Solved within "\[InvisibleSpace]3.98792110445356229`*^-13\), SequenceForm[ "Solved within ", 3.9879211044535623*^-13], Editable->False]], "Print"], Cell[BoxData[ \({{\(-0.707106781186547461`\), \(-0.707106781186547461`\), \(-0.707106781186547461`\)}, {\(-0.707106781186547461`\), \(-0.707106781186547461`\), 0.707106781186547461`}, { \(-0.707106781186547461`\), 0.707106781186547461`, \(-0.707106781186547461`\)}, {\(-0.707106781186547461`\), 0.707106781186547461`, 0.707106781186547461`}, { 0.707106781186547461`, \(-0.707106781186547461`\), \(-0.707106781186547461`\)}, {0.707106781186547461`, \(-0.707106781186547461`\), 0.707106781186547461`}, { 0.707106781186547461`, 0.707106781186547461`, \(-0.707106781186547461`\)}, {0.707106781186547461`, 0.707106781186547461`, 0.707106781186547461`}}\)], "Output"] }, Open ]], Cell["\<\ Examples \ \>", "Section", Evaluatable->False], Cell[TextData[ "\nWe begin with a polyhedron which John Conway has named a \ \"hermaphrodite.\" It is composed of an n-gonal prism joined to an n-gonal \ pyramid, and so has an n-gon for a base, n rectangular sides attached to the \ base, and n triangular faces which meet at an apex. Note that it is \ combinatorially self-dual. Numbering the vertices from 1 to n around the \ base, then from n+1 to 2n around the juncture, and numbering the apex 2n+1, \ the following construction functions are straightforward.\n\n\n"], "Text", Evaluatable->False], Cell["\<\ vHermaphrodite[n_] := N[Join[Table[{Cos[(2*Pi*i)/n], Sin[(2*Pi*i)/n], -1}, {i, n}], Table[{Cos[(2*Pi*i)/n], Sin[(2*Pi*i)/n], 0}, {i, n}], {{0, 0, 1}}]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell["\<\ fHermaphrodite[n_] := Join[{Range[n], {1, n + 1, 2*n, n}, {n + 1, 2*n, 2*n + 1}}, Table[{i, i + 1, n + i + 1, n + i}, {i, n - 1}], Table[{n + i, n + i + 1, 2*n + 1}, {i, n - 1}]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True], Cell[CellGroupData[{ Cell["view[vHermaphrodite[7], fHermaphrodite[7]]; ", "Input", PageWidth->Infinity, InitializationCell->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.11194 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def 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o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00 o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00 o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00o@400?d1003m0@00 o@400?d1003m0@00\ \>"], ImageRangeCache->{{{0, 252.125}, {287, 0}} -> {0.0652213, -0.000250661, 0.00328322, 0.00328322}}], Cell["\<\ Figure 2. \"Hermaphrodite\" now in canonical form is not only \ combinatorially self- dual, but also geometrically self-dual.\ \>", "Text"] }, Open ]], Cell["\<\ The canonical form is of special interest because it is geometrically \ self-dual. Conveniently (because 7 is odd) we can position its dual by \ simple inversion through the origin, i.e., negating all the vertex \ coordinates. With the original vertices numbered from 1 to 15 and the dual's \ vertices numbered from 16 to 30, a compound of the canonical form with its \ dual is produced as follows. Note that every edge crosses an edge of the \ dual at right angles. 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L 0 0 0 r F P 0 g s .29841 .50046 m .45059 .41147 L .29841 .50046 L 0 0 0 r F .74819 .55116 m .70483 .60793 L .51619 .32945 L .443 .425 .744 r F .7476 .61923 m .30922 .57105 L .33864 .43207 L .663 .742 .909 r F .51619 .32945 m .74819 .55115 L p .74819 .55116 L .443 .425 .744 r F P 0 g s .74881 .47951 m .74819 .55115 L p .77747 .55926 L 0 0 .461 r F P 0 g s .77748 .55926 m .77747 .55923 L p .77747 .55926 L 0 0 .461 r F P 0 g s .77747 .55923 m .74881 .47951 L p .77747 .55926 L 0 0 .461 r F P 0 g s .65795 .43574 m .74881 .47951 L .74819 .55115 L p .663 .742 .909 r F P 0 g s .51619 .32945 m .65794 .43573 L p .70401 .49464 L 0 .152 .677 r F P 0 g s .74819 .55115 m .51619 .32945 L p .77747 .55926 L 0 .152 .677 r F P 0 g s .7476 .61923 m .59446 .4283 L .65795 .43574 L .663 .742 .909 r F .45062 .41147 m .65793 .43573 L .51619 .32945 L .548 .018 0 r F .65794 .43573 m .65794 .43573 L p .65793 .43573 L F P 0 g s .65794 .43573 m .51619 .32945 L p .65793 .43573 L .548 .018 0 r F P 0 g s .77747 .55926 m .83641 .63545 L .74819 .55115 L p 0 .152 .677 r F P 0 g s .74819 .55115 m .74819 .55116 L p .65795 .43574 L .663 .742 .909 r F P 0 g s .74819 .55116 m .7476 .61923 L p .72755 .57818 L .663 .742 .909 r F P 0 g s .72755 .57818 m .65795 .43574 L .74819 .55116 L .663 .742 .909 r F .83641 .63545 m .77453 .71078 L p .74819 .55116 L .443 .425 .744 r F P 0 g s .74819 .55115 m .74819 .55115 L p .74819 .55116 L .443 .425 .744 r F P 0 g s .74819 .55115 m .83641 .63545 L p .74819 .55116 L .443 .425 .744 r F P 0 g s .83641 .63545 m .77747 .55926 L p .79531 .65809 L .969 .753 .558 r F P 0 g s .83641 .63545 m .81129 .74664 L p .78409 .7201 L 0 0 .265 r F P 0 g s .77453 .71078 m .80856 .82648 L .81129 .74663 L p 0 0 .461 r F P 0 g s .71537 .68506 m .59811 .9246 L .77454 .71081 L .519 .368 .616 r F .77452 .7108 m .83641 .63545 L p .73593 .79803 L 0 0 .265 r F P 0 g s .79531 .65809 m .81129 .74661 L .83641 .63545 L .969 .753 .558 r F .81129 .74664 m .8113 .7466 L p .81129 .74661 L F P 0 g s .8113 .7466 m .83641 .63545 L p .81129 .74661 L .969 .753 .558 r F P 0 g s .80856 .82648 m .81129 .74663 L p .73788 .90124 L .944 .956 .736 r F P 0 g s .78984 .84163 m .73593 .79803 L .78409 .7201 L 0 0 .265 r F .24657 .73868 m .2526 .76919 L .35618 .77836 L .931 .731 .61 r F .47094 .98993 m .30936 .80668 L .35621 .77834 L .836 .515 .445 r F .35618 .77833 m .35621 .77834 L p .30936 .80668 L F P 0 g s .73787 .90125 m .73805 .87028 L p .78984 .84163 L 0 0 .265 r F P 0 g s .73787 .90125 m .68839 .95803 L p .69877 .89202 L 0 0 .265 r F P 0 g s .59811 .9246 m .70121 .87653 L p .77448 .71087 L .519 .368 .616 r F P 0 g s .68839 .95803 m .57891 .97409 L p .50582 .87736 L .601 .326 .472 r F P 0 g s .68839 .95803 m .50582 .87736 L .52572 .85931 L .601 .326 .472 r F .68839 .95803 m .52572 .85931 L .61848 .82162 L F .7012 .87656 m .68839 .95803 L p .61848 .82162 L F P 0 g s .73787 .90125 m .68839 .95803 L p .64844 .94223 L .647 .761 .932 r F P 0 g s .68839 .95803 m .57892 .97409 L p .64844 .94223 L .647 .761 .932 r F P 0 g s .51619 1.13578 m .73787 .90126 L p .70118 .87654 L .16 0 .01 r F P 0 g s .78984 .84163 m .7012 .87656 L .73593 .79803 L 0 0 .265 r F .70121 .87653 m .70118 .87654 L p .73787 .90126 L .16 0 .01 r F P 0 g s .57891 .9741 m .51619 1.13578 L p .70118 .87654 L .16 0 .01 r F P 0 g s .53035 .7448 m .39055 .81953 L .40871 .84132 L .737 .532 .616 r F .52421 .7388 m .39055 .81953 L .53035 .7448 L F .36742 .79901 m .35621 .77834 L .37399 .79968 L .836 .515 .445 r F .36742 .79901 m .37399 .79968 L .46763 .912 L F .46763 .912 m .47094 .98993 L p .36742 .79901 L F P 0 g s .2296 .65291 m .3175 .62512 L .30817 .56423 L .932 .901 .773 r F .2296 .65291 m .24474 .7334 L p .24843 .74808 L 0 0 .156 r F P 0 g s .33601 .69845 m .29766 .63139 L .23958 .70337 L .931 .731 .61 r F .24657 .73868 m .23958 .70337 L .19087 .76372 L F .33601 .69845 m .23958 .70337 L .24657 .73868 L F .22958 .65292 m .19087 .76372 L p .29766 .63139 L F P 0 g s .24657 .73868 m .19087 .76372 L .2526 .76919 L .931 .731 .61 r F .26007 .56565 m .29841 .50047 L p .26552 .5733 L 0 0 0 r F P 0 g s .26907 .56544 m .2296 .65291 L .30817 .56423 L .932 .901 .773 r F .26007 .56565 m .26552 .5733 L .22959 .65289 L 0 0 0 r F .22958 .65292 m .22959 .65289 L p .22959 .65289 L F P 0 g s .22959 .65289 m .26007 .56565 L p .22959 .65289 L 0 0 0 r F P 0 g s .24964 .74676 m .2362 .68003 L .2296 .65291 L 0 0 .156 r F .21346 .56716 m .22959 .65289 L p .26907 .56544 L .932 .901 .773 r F P 0 g s .22959 .65289 m .2296 .65291 L p .26907 .56544 L .932 .901 .773 r F P 0 g s .22959 .65289 m .2296 .65291 L p .2362 .68003 L 0 0 .156 r F P 0 g s .19087 .76372 m .22958 .65292 L p .2296 .65291 L 0 0 0 r F P 0 g s .51619 .32945 m .4506 .41146 L p .45062 .41147 L .548 .018 0 r F P 0 g s .33864 .43207 m .45061 .41147 L p .7476 .61923 L .663 .742 .909 r F P 0 g s .7476 .61923 m .45061 .41147 L .59446 .4283 L .663 .742 .909 r F .74819 .55115 m .7476 .61923 L p .77748 .55926 L 0 0 .461 r F P 0 g s .7476 .61923 m .79355 .64832 L .77748 .55927 L 0 0 .461 r F .74819 .55116 m .77452 .7108 L .70483 .60793 L .443 .425 .744 r F .81129 .74663 m .8142 .6614 L p .77453 .71078 L 0 0 .461 r F P 0 g s .77453 .71078 m .77452 .7108 L p .74819 .55115 L .443 .425 .744 r F P 0 g s .6171 .44789 m .65795 .43574 L .75376 .57411 L .849 .48 .352 r F .6171 .44789 m .75376 .57411 L .74101 .58212 L F .65795 .43574 m .74881 .47951 L .77747 .55923 L p F P 0 g s .77747 .55923 m .77748 .55926 L p .77746 .55924 L .849 .48 .352 r F P 0 g s .65795 .43574 m .77747 .55923 L .77746 .55924 L .849 .48 .352 r F .70854 .48802 m .77747 .55926 L .70401 .49464 L 0 .152 .677 r F .77746 .55924 m .77747 .55926 L p .70854 .48802 L F P 0 g s .7608 .62759 m .77453 .71077 L .8142 .6614 L 0 0 .461 r F .77747 .55926 m .77746 .55924 L p .79531 .65809 L .969 .753 .558 r F P 0 g s .79531 .65809 m .77746 .55924 L .75689 .57215 L .969 .753 .558 r F .8142 .6614 m .77748 .55926 L p .77748 .55927 L 0 0 .461 r F P 0 g s .79355 .64832 m .8142 .6614 L .77748 .55927 L 0 0 .461 r F .77452 .7108 m .66325 .66237 L .70483 .60793 L .443 .425 .744 r F .71147 .66452 m .79531 .65809 L .75689 .57215 L .969 .753 .558 r F .30795 .79971 m .73824 .83642 L .73787 .90125 L .647 .761 .932 r F .31211 .79343 m .73824 .83642 L .30795 .79971 L F .25527 .77534 m .2296 .65291 L .24843 .74808 L 0 0 .156 r F .37837 .95511 m .26755 .84199 L .27104 .8554 L .653 .146 0 r F .26532 .83336 m .19087 .76372 L p .37838 .95513 L F P 0 g s .25128 .75507 m .34659 .74141 L .27104 .85539 L .361 .374 .739 r F .25128 .75507 m .26456 .69997 L .34659 .74141 L F .29841 .50046 m .21346 .56716 L p .30817 .56423 L .932 .901 .773 r F P 0 g s .27547 .70549 m .31633 .50738 L .23778 .68644 L .472 .317 .589 r F .23776 .68647 m .23778 .68644 L p .25128 .75507 L .361 .374 .739 r F P 0 g s .23778 .68644 m .28764 .6043 L p .26456 .69997 L .361 .374 .739 r F P 0 g s .23778 .68644 m .26456 .69997 L .25128 .75507 L .361 .374 .739 r F .2296 .65291 m .26579 .6226 L .23776 .68647 L 0 0 0 r F .26552 .5733 m .28764 .6043 L .2296 .65291 L F .22959 .65289 m .2296 .65291 L .22958 .65292 L F .26552 .5733 m .2296 .65291 L .22959 .65289 L F .23776 .68643 m .21346 .56716 L p .2362 .68003 L 0 0 .156 r F P 0 g s .23776 .68647 m .2362 .68003 L .24964 .74676 L 0 0 .156 r F .23776 .68643 m .2362 .68003 L .23776 .68647 L F .26552 .5733 m .30243 .50201 L .28764 .6043 L 0 0 0 r F .34659 .74141 m .34659 .74141 L .27104 .85539 L .361 .374 .739 r F .27104 .8554 m .27104 .85539 L p .26529 .83334 L 0 0 .156 r F P 0 g s .27104 .85539 m .25845 .78802 L p .2607 .80293 L 0 0 .156 r F P 0 g s .2607 .80293 m .26529 .83334 L .27104 .85539 L 0 0 .156 r F .25845 .78802 m .73787 .90125 L .64844 .94223 L .647 .761 .932 r F .51496 .88945 m .64844 .94223 L .57891 .97409 L F .25845 .78802 m .51496 .88945 L .57891 .97409 L F .57892 .97409 m .57891 .97409 L p .64844 .94223 L F P 0 g s .57891 .97409 m .47094 .98993 L p .44049 .961 L .647 .761 .932 r F P 0 g s .51619 1.13578 m .57891 .9741 L p .46761 .91198 L .643 .2 .183 r F P 0 g s .46764 .91198 m .46761 .91198 L p .57891 .9741 L .643 .2 .183 r F P 0 g s .47094 .98993 m .3784 .95513 L p .44049 .961 L .647 .761 .932 r F P 0 g s .3784 .95513 m .37839 .95512 L p .44049 .961 L .647 .761 .932 r F P 0 g s .44049 .961 m .44049 .961 L .37839 .95512 L .647 .761 .932 r F .37838 .95513 m .51619 1.13578 L p .47904 .96464 L .643 .2 .183 r F P 0 g s .46761 .91198 m .37839 .95512 L .47904 .96464 L .643 .2 .183 r F .37839 .95512 m .37838 .95513 L .47904 .96464 L F .25845 .78802 m .57891 .97409 L .44049 .961 L .647 .761 .932 r F .69861 .60874 m .73854 .78183 L .65161 .74453 L .821 .683 .711 r F .73788 .90124 m .73787 .90125 L p .73824 .83642 L .647 .761 .932 r F P 0 g s .51619 1.13578 m .73787 .90126 L p .73788 .90124 L .944 .956 .736 r F P 0 g s .73788 .90124 m .78268 .76047 L .73832 .82294 L .944 .956 .736 r F .58893 .802 m .5889 .802 L p .61885 .74578 L .662 .538 .702 r F P 0 g s .25845 .78802 m .7384 .80828 L .73824 .83642 L .647 .761 .932 r F .73788 .90124 m .73832 .82294 L .51619 1.13578 L .944 .956 .736 r F .61885 .74578 m .47749 .69316 L .66324 .66238 L .662 .538 .702 r F .25845 .78802 m .61751 .7299 L .7384 .80828 L .647 .761 .932 r F .61751 .7299 m .73855 .78184 L .7384 .80828 L F .73854 .78184 m .51619 1.13578 L p .73832 .82294 L .944 .956 .736 r F P 0 g s .78268 .76047 m .73855 .78184 L .73832 .82294 L .944 .956 .736 r F .73855 .78184 m .73854 .78184 L .73832 .82294 L F .73854 .78183 m .73855 .78184 L p .61751 .7299 L .647 .761 .932 r F P 0 g s .81129 .74664 m .78984 .84163 L p .78409 .7201 L 0 0 .265 r F P 0 g s .81129 .74663 m .81129 .74661 L p .73787 .90126 L .944 .956 .736 r F P 0 g s .68467 .71903 m .73855 .78184 L p .81125 .74663 L .969 .753 .558 r F P 0 g s .81129 .74661 m .78268 .76047 L .73787 .90126 L .944 .956 .736 r F .68467 .71903 m .81129 .74664 L .79531 .65809 L .969 .753 .558 r F .81129 .74661 m .8142 .6614 L p .78268 .76047 L .944 .956 .736 r F P 0 g s .78984 .84163 m .81129 .74664 L p .81125 .74663 L .969 .753 .558 r F P 0 g s .73855 .78184 m .78984 .84163 L p .81125 .74663 L .969 .753 .558 r F P 0 g s .73855 .78184 m .78984 .84163 L p .7384 .80828 L .647 .761 .932 r F P 0 g s .73824 .83642 m .7384 .80828 L .78984 .84163 L .647 .761 .932 r F .47746 .69314 m .34065 .71681 L .31751 .62513 L .804 .667 .712 r F .68866 .6291 m .47746 .69315 L .42908 .67257 L .663 .742 .909 r F .47756 .62401 m .63741 .53866 L .5703 .46181 L .742 .523 .597 r F .42908 .67257 m .3175 .62512 L .31357 .5995 L .663 .742 .909 r F .27547 .70549 m .36942 .54051 L .31633 .50738 L .472 .317 .589 r F .39017 .45565 m .47777 .42289 L .53534 .49144 L .565 .257 .406 r F .39017 .45565 m .53534 .49144 L .51185 .51134 L F .47756 .62401 m .5703 .46181 L .4316 .57933 L .742 .523 .597 r F .57715 .54122 m .57824 .5539 L p .39017 .45565 L .565 .257 .406 r F P 0 g s .29378 .71473 m .37584 .54452 L .27547 .70549 L .472 .317 .589 r F .3872 .45676 m .39017 .45565 L .57824 .5539 L .565 .257 .406 r F .37584 .54452 m .36942 .54051 L .27547 .70549 L .472 .317 .589 r F .31636 .50737 m .31633 .50738 L p .36942 .54051 L F P 0 g s .28764 .6043 m .31634 .50738 L p .31641 .5074 L .565 .257 .406 r F P 0 g s .57824 .5539 m .43156 .57934 L p .31641 .5074 L .565 .257 .406 r F P 0 g s .43156 .57934 m .4316 .57933 L p .34661 .7414 L .654 .53 .701 r F P 0 g s .49958 .64541 m .63741 .53866 L .47756 .62401 L .742 .523 .597 r F .68262 .63701 m .47746 .69315 L .68866 .6291 L .663 .742 .909 r F .68262 .63701 m .66324 .66238 L .47746 .69315 L F .66325 .66237 m .66324 .66238 L p .68262 .63701 L F P 0 g s .47749 .69316 m .47748 .69315 L .66325 .66238 L .662 .538 .702 r F .47748 .69315 m .47746 .69315 L p .66324 .66238 L .663 .742 .909 r F P 0 g s .47749 .69316 m .47747 .69315 L .47748 .69315 L .662 .538 .702 r F .47748 .69315 m .47746 .69315 L p .52421 .7388 L .737 .532 .616 r F P 0 g s .25845 .78802 m .34661 .7414 L p .60454 .72434 L .647 .761 .932 r F P 0 g s .25845 .78802 m .60454 .72434 L .61751 .7299 L .647 .761 .932 r F .62071 .66425 m .69861 .60874 L .65161 .74453 L .821 .683 .711 r F .71147 .66452 m .75689 .57215 L .69861 .60874 L .969 .753 .558 r F .65794 .43573 m .71503 .47854 L p .70401 .49464 L 0 .152 .677 r F P 0 g s .77748 .55926 m .8142 .6614 L p .65795 .43574 L .849 .48 .352 r F P 0 g s .71503 .47854 m .77746 .55924 L p .70854 .48802 L 0 .152 .677 r F P 0 g s .49958 .64541 m .56202 .70609 L .65842 .56273 L .742 .523 .597 r F .5703 .46181 m .57034 .46182 L p .4506 .41146 L .548 .018 0 r F P 0 g s .8142 .6614 m .69857 .60872 L p .60223 .45232 L .849 .48 .352 r F P 0 g s .6171 .44789 m .8142 .6614 L .60223 .45232 L .849 .48 .352 r F .57034 .46182 m .60657 .45103 L .4506 .41146 L .548 .018 0 r F .45061 .41147 m .56419 .39057 L p .59446 .4283 L .663 .742 .909 r F P 0 g s .65793 .43573 m .65795 .43574 L p .59446 .4283 L .663 .742 .909 r F P 0 g s .45062 .41147 m .60657 .45103 L .65793 .43573 L .548 .018 0 r F .57032 .4618 m .56419 .39057 L p .60223 .45232 L .849 .48 .352 r F P 0 g s .60223 .45232 m .69857 .60872 L .57034 .46182 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Combining the canonical hermaphrodite with itself \ properly rotated gives a compound. The duality of the two is evidenced by the fact that the edges cross at right angles. The crossing points all lie on the unit \ sphere.\ \>", "Text"] }, Open ]], Cell["\<\ A second example Conway has called the \"antihermaphrodite,\" composed of \ half an n-gonal antiprism joined to half an n-gonal trapezohedron (the dual \ of an antiprism). It has an n-gon for a base, n triangular sides attached to \ the base, and n kite-shaped faces which meet at an apex. It is also \ self-dual, so in its canonical form, it makes an attractive compound with \ itself. Numbering the vertices as above, the following code is analogous. We \ do not view this before canocalization however, as this funcion only provides \ a combinatorial polyhedron, in which the 4-sided faces are not planar. After \ canonicalization we have a true geometric polyhedron with planar faces. \ \>", "Text", Evaluatable->False], Cell["\<\ vAntiHermaphrodite[n_] := N[Join[Table[{Cos[2*Pi*i/n], Sin[2*Pi*i/n], -1}, {i, n}], Table[{Cos[2*Pi*(i + 0.5)/n], Sin[2*Pi*(i + 0.5)/n], 0}, {i, n}], {{0, 0, 1}}]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ fAntiHermaphrodite[n_] := Join[{Range[n], {1, 2*n, n}, {1, n + 1, 2*n + 1, 2*n}}, Table[{i, i + 1, n + i}, {i, n - 1}], Table[{i, n + i, 2*n + 1, n + i - 1}, {i, 2, n}]]\ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell["\<\ fA7 = fAntiHermaphrodite[7]; vA7c = \ canonicalize[vAntiHermaphrodite[7], fA7]; view[vA7c, 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1.20848 L F .69458 1.26304 m .69447 1.26309 L p .573 1.28215 L .653 .754 .924 r F P 0 g s .58027 1.21798 m .47996 1.29678 L .573 1.28215 L .575 .267 .409 r F .45252 1.30107 m .45252 1.30109 L p .47841 1.25501 L F P 0 g s .47841 1.25501 m .45252 1.30109 L .47996 1.29678 L .575 .267 .409 r F .41016 1.22389 m .41877 1.28745 L .45252 1.30107 L .833 .468 .365 r F .45252 1.30109 m .45252 1.30107 L p .41877 1.28745 L F P 0 g s .38959 1.27565 m .45252 1.30109 L .31558 1.28736 L .653 .754 .924 r F .45251 1.30109 m .31558 1.28736 L p .45252 1.30109 L F P 0 g s .31558 1.28736 m .33421 1.19281 L .39292 1.21683 L .828 .632 .638 r F .41016 1.22389 m .31879 1.12746 L p .33421 1.19281 L .833 .468 .365 r F P 0 g s .41016 1.22389 m .31558 1.28736 L p .43311 1.18018 L .828 .632 .638 r F P 0 g s .33421 1.19281 m .34999 1.25965 L .41016 1.22389 L .833 .468 .365 r F .23225 1.1073 m .25006 1.05653 L p .25559 1.13709 L .828 .632 .638 r F P 0 g s .25726 1.16135 m .31879 1.12746 L .25007 1.05651 L p .947 .722 .565 r F P 0 g s .25007 1.05651 m .25006 1.05649 L p .25006 1.05653 L .947 .722 .565 r F P 0 g s .25007 1.05651 m .25006 1.05653 L .25726 1.16135 L .947 .722 .565 r F .25006 1.05653 m .23225 1.1073 L p .23362 1.08561 L .985 .898 .579 r F P 0 g s .25726 1.16135 m .23225 1.1073 L p .25559 1.13709 L .828 .632 .638 r F P 0 g s .25726 1.16135 m .25559 1.13709 L .28278 1.17178 L .828 .632 .638 r F .25006 1.05649 m .17934 .98348 L p .22825 1.09512 L .947 .722 .565 r F P 0 g s .25006 1.05649 m .22825 1.09512 L .25006 1.05653 L .947 .722 .565 r F .23225 1.1073 m .215 1.11856 L p .23362 1.08561 L .985 .898 .579 r F P 0 g s .215 1.11856 m .23225 1.1073 L .25726 1.16135 L p .956 .633 .367 r F P 0 g s .17934 .98348 m .215 1.11856 L p .25726 1.16135 L .947 .722 .565 r F P 0 g s .215 1.11856 m .23023 1.17624 L .25726 1.16135 L p .947 .722 .565 r F P 0 g s .31558 1.28736 m .24563 1.21752 L p .25293 1.18222 L .956 .633 .367 r F P 0 g s .24563 1.21752 m .24562 1.2175 L p .24562 1.21748 L .956 .633 .367 r F P 0 g s .24563 1.21752 m .24562 1.21748 L .25293 1.18222 L .956 .633 .367 r F .23023 1.17624 m .24562 1.21748 L p .25341 1.17993 L .833 .468 .365 r F P 0 g s .25726 1.16135 m .31558 1.28736 L p .25293 1.18222 L .956 .633 .367 r F P 0 g s .25726 1.16135 m .23023 1.17624 L p .25341 1.17993 L .833 .468 .365 r F P 0 g s .25293 1.18222 m .215 1.11856 L .25726 1.16135 L .956 .633 .367 r F .31558 1.28736 m .25007 1.0565 L .35264 1.0993 L .828 .632 .638 r F .25726 1.16135 m .25341 1.17993 L .33421 1.19281 L .833 .468 .365 r F .24562 1.21748 m .24563 1.21751 L p .25341 1.17993 L F P 0 g s .73704 1.04277 m .38699 1.11365 L .17934 .98348 L .658 .749 .917 r F .24999 1.05642 m .17934 .98348 L p .38699 1.11365 L F P 0 g s .21326 1.10596 m .24022 .98174 L .23362 1.08561 L .985 .898 .579 r F .21324 1.10599 m .2072 1.06297 L .21326 1.10596 L F .2072 1.06297 m .24022 .98174 L .21326 1.10596 L F .24562 1.21748 m .23023 1.17624 L .215 1.11856 L p 0 0 .396 r F P 0 g s .17361 1.1456 m .215 1.11856 L p .25293 1.18222 L .956 .633 .367 r F P 0 g s .24562 1.2175 m .17361 1.1456 L p .25293 1.18222 L .956 .633 .367 r F P 0 g s .215 1.11856 m .17934 .98348 L p .21245 1.10035 L 0 0 .396 r F P 0 g s .21324 1.10599 m .21326 1.10596 L .215 1.11856 L .985 .898 .579 r F .21326 1.10596 m .23362 1.08561 L .21503 1.11852 L F .21326 1.10596 m .21503 1.11852 L .215 1.11856 L F .21245 1.10035 m .24562 1.21748 L .215 1.11856 L 0 0 .396 r F .215 1.11856 m .17361 1.1456 L p .21324 1.10599 L .985 .898 .579 r F P 0 g s .25006 1.05653 m .22825 1.09512 L .25726 1.16135 L .947 .722 .565 r F .2072 1.06297 m .2251 1.06799 L .23555 1.14388 L 0 0 .396 r F .79759 1.04921 m .23004 .90483 L .23008 .90475 L .658 .749 .917 r F .23004 .90483 m .22996 .90481 L .23003 .90478 L F .23004 .90483 m .23003 .90478 L .23008 .90475 L F .79759 1.04921 m .23001 .90473 L .33539 .84677 L F .20185 1.02504 m .20632 .98265 L .23002 .90473 L .985 .898 .579 r F .42156 .79932 m .50341 .33436 L p .32485 .85257 L .277 0 .264 r F P 0 g s .50341 .33436 m .23002 .90473 L p .32485 .85257 L .277 0 .264 r F P 0 g s .23001 .90473 m .28228 .8235 L p .33539 .84677 L .658 .749 .917 r F P 0 g s .19918 1.00613 m .17934 .98348 L .23001 .90473 L p 0 0 .18 r F P 0 g s .23001 .90473 m .23002 .90472 L p .23001 .90475 L 0 0 .18 r F P 0 g s .23001 .90473 m .23001 .90475 L .19918 1.00613 L 0 0 .18 r F .23001 .90475 m .23002 .90473 L p .20632 .98265 L .985 .898 .579 r F P 0 g s .17934 .98348 m .20185 1.02504 L .2072 1.06297 L 0 0 .396 r F .17934 .98348 m .2072 1.06297 L .21245 1.10035 L F .2072 1.06297 m .23002 .90473 L .24022 .98174 L .985 .898 .579 r F .17361 1.1456 m .20185 1.02504 L .2072 1.06297 L F .22996 .90481 m .23001 .90473 L p .23008 .90475 L .658 .749 .917 r F P 0 g s .20185 1.02504 m .23002 .90473 L .2072 1.06297 L .985 .898 .579 r F .19918 1.00613 m .21017 .94615 L p .20185 1.02504 L F P 0 g s .23002 .90473 m .23001 .90475 L p .32485 .85257 L .277 0 .264 r F P 0 g s .2301 .90485 m .23005 .90493 L .23004 .90483 L .658 .749 .917 r F .79759 1.04921 m .24132 .99007 L .2301 .90485 L F .32485 .85257 m .23001 .90475 L .23009 .92989 L .277 0 .264 r F .19918 1.00613 m .22828 .99942 L .23019 .95844 L .124 .055 .5 r F .31558 1.28736 m .2457 1.21759 L p .38959 1.27565 L .653 .754 .924 r F P 0 g s .38959 1.27565 m .2457 1.21759 L .2457 1.21754 L .653 .754 .924 r F .38959 1.27565 m .24563 1.21752 L .24563 1.21751 L F .21245 1.10035 m .23555 1.14388 L .24562 1.21748 L 0 0 .396 r F .24563 1.21751 m .24562 1.21748 L p .23555 1.14388 L F P 0 g s .25341 1.17993 m .24563 1.21751 L .26963 1.2272 L .833 .468 .365 r F .38959 1.27565 m .24562 1.2175 L .34999 1.25965 L .653 .754 .924 r F .33421 1.19281 m .25341 1.17993 L .26963 1.2272 L .833 .468 .365 r F .34999 1.25965 m .26963 1.2272 L .50246 1.90558 L F .33421 1.19281 m .26963 1.2272 L .34999 1.25965 L F .28278 1.17178 m .25559 1.13709 L .25007 1.0565 L .828 .632 .638 r F .34999 1.25965 m .41877 1.28745 L .41016 1.22389 L .833 .468 .365 r F .39292 1.21683 m .35264 1.0993 L 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Combining the antihermaphrodite in canonical form with \ itself properly rotated gives a compound. Again the edges cross at right angles and the crossing points all lie on the unit sphere, illustrating self-duality\ \>", "Text"] }, Open ]], Cell["\<\ As a final example, we create a \"tetrahedrally stellated icosahedron\" (TSI) \ by choosing four faces of an icosahedron in the face planes of a tetrahedron \ and covering them with low triangular pyramids. If the height of the pyramid \ is chosen properly, the pyramid faces become an extensions of the adjacent \ icosahedron faces. The resulting polyhedron has twelve kite-shaped faces and \ four triangular faces, and is an interesting model of chiral tetrahedral \ symmetry, i.e., it has four 3-fold axes and three 2-fold axes, but no mirror \ planes. Examining its net shows that the TSI is combinatorially self-dual. \ We construct it from the vertices of an icosahedron and a surrounding \ tetrahedron, manually writing the vertex indices from a pencil sketch: \ \>", "Text", Evaluatable->False], Cell["\<\ vTSI = With[{t = 1.618, d = 1.1708}, {{-1, t, 0}, {1, t, 0}, {-1, -t, 0}, {1, -t, 0}, {0, 1, -t}, {0, -1, -t}, \ {0, 1, t}, {0, -1, t}, {-t, 0, -1}, {-t, 0, 1}, {t, 0, -1}, {t, 0, 1}, {d, d, d}, {d, -d, -d}, {-d, d, -d}, {-d, -d, d}}]; \ \>", "Input", PageWidth->Infinity, InitializationCell->True, FontFamily->"Courier", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["\<\ fTSI = {{1, 2, 5, 15}, {1, 7, 13, 2}, {3, 4, 14, 6}, {3, 4, 8, 16}, \ {5, 6, 9, 15}, {5, 6, 14, 11}, {7, 8, 16, 10}, {7, 8, 12, 13}, {9, 10, 1, 15}, {9, 10, 16, 3}, {11, 12, 13, 2}, {11, 12, 4, 14}, {1, 7, 10}, {2, 5, 11}, {3, 6, 9}, {4, 8, 12}}; \ \>", "Input", PageWidth->Infinity, 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L .44179 .33714 L .797 .795 .85 r F .37627 .36858 m .52632 .34495 L p .44179 .33714 L .544 .859 .997 r F P 0 g s .48889 .23645 m .52632 .34495 L p .52632 .34495 L .797 .795 .85 r F P 0 g s .44349 .33729 m .52632 .34495 L .37591 .48741 L .797 .795 .85 r F .48889 .23645 m .52632 .34495 L .44349 .33729 L F .48889 .23645 m .44179 .33714 L .4296 .33601 L F .48889 .23645 m .4296 .33601 L .39041 .33239 L F .44179 .33714 m .4296 .33601 L .35234 .32887 L .544 .859 .997 r F .35234 .32887 m .48889 .23645 L p .29906 .42137 L .797 .795 .85 r F P 0 g s .68019 .32072 m .63711 .28299 L p .55412 .3294 L .544 .859 .997 r F P 0 g s .63711 .28299 m .63711 .28299 L p .57949 .31521 L .544 .859 .997 r F P 0 g s .55412 .3294 m .63711 .28299 L .57949 .31521 L .544 .859 .997 r F .48889 .23645 m .52632 .34495 L p .57535 .31753 L .486 .61 .9 r F P 0 g s .52632 .34495 m .68019 .32072 L p .55412 .3294 L .544 .859 .997 r F P 0 g s .52632 .34495 m .55412 .3294 L .44179 .33714 L .544 .859 .997 r F .68019 .32072 m .68788 .46579 L p .48973 .40397 L .679 .629 .792 r F P 0 g s .52632 .34495 m .68019 .32072 L p .48973 .40397 L .679 .629 .792 r F P 0 g s .57949 .31521 m .71943 .45263 L .73169 .44752 L .486 .61 .9 r F .48889 .23645 m .57535 .31753 L .57949 .31521 L F .57535 .31753 m .71943 .45263 L .57949 .31521 L F .57535 .31753 m .52632 .34495 L .71943 .45263 L F .69912 .34681 m .73169 .44752 L .75475 .40405 L F .63711 .28299 m .63711 .28299 L .69912 .34681 L F .63711 .28299 m .73169 .44752 L .69912 .34681 L F .77002 .48085 m .77815 .42814 L p .73169 .44752 L F P 0 g s .77815 .42814 m .68019 .32072 L p .69912 .34681 L .599 .096 0 r F P 0 g s .77815 .42814 m .77815 .42814 L p .73169 .44752 L .486 .61 .9 r F P 0 g s .77815 .42814 m .77815 .42814 L p .73169 .44752 L .431 .431 .758 r F P 0 g s .77815 .42814 m .793 .33194 L p .73169 .44752 L .486 .61 .9 r F P 0 g s .73169 .44752 m .77815 .42814 L .73169 .44752 L .486 .61 .9 r F .73169 .44752 m .71943 .45263 L .77002 .48085 L F .88206 .54206 m .77815 .42814 L p .68788 .46579 L .431 .431 .758 r F P 0 g s .68788 .46579 m .77815 .42814 L .73169 .44752 L .431 .431 .758 r F .69513 .60254 m .78309 .57408 L p .78309 .57408 L .325 .137 .465 r F P 0 g s .73144 .64395 m .69513 .60254 L .78309 .57408 L .325 .137 .465 r F .805 .73041 m .66477 .73412 L p .72201 .6567 L .585 .467 .685 r F P 0 g s .66477 .73412 m .69513 .60254 L p .73144 .64395 L .325 .137 .465 r F P 0 g s .60647 .53748 m .62613 .52017 L .69513 .60254 L .679 .629 .792 r F .73144 .64395 m .59708 .54575 L .62613 .52017 L .585 .467 .685 r F .805 .73041 m .72201 .6567 L .73144 .64395 L F .72201 .6567 m .59708 .54575 L .73144 .64395 L F .63243 .87431 m .66477 .73412 L p .66477 .73412 L .325 .137 .465 r F P 0 g s .63243 .87431 m .66477 .73412 L .68236 .75812 L .325 .137 .465 r F .805 .73041 m .66477 .73412 L p .66477 .73412 L .521 .181 .331 r F P 0 g s .69621 .77701 m .66477 .73412 L .69621 .77701 L .521 .181 .331 r F .805 .73041 m .66477 .73412 L .69621 .77701 L F .66477 .73412 m .66477 .73412 L p .73144 .64395 L .325 .137 .465 r F P 0 g s .805 .73041 m .73144 .64395 L .75782 .60826 L .585 .467 .685 r F .78309 .57408 m .805 .73041 L p .75782 .60826 L F P 0 g s .805 .73041 m .82861 .69475 L p .79039 .59343 L 0 0 .434 r F P 0 g s .88206 .54206 m .82861 .69475 L p .80813 .64046 L .325 .137 .465 r F P 0 g s .82861 .69475 m .82861 .69475 L p .80813 .64046 L 0 0 .434 r F P 0 g s .78309 .57408 m .88206 .54206 L p .80813 .64046 L .325 .137 .465 r F P 0 g s .82541 .56512 m .82351 .48789 L .77815 .42814 L 0 0 .434 r F .80813 .64046 m .79039 .59343 L .82861 .69475 L F .78309 .57408 m .78066 .50223 L .68788 .46579 L .431 .431 .758 r F .78309 .57408 m .805 .73041 L p .78066 .50223 L 0 0 .434 r F P 0 g s .78066 .50223 m .78309 .57408 L .78066 .50223 L 0 0 .434 r F .78309 .57408 m .78309 .57408 L p .68788 .46579 L .431 .431 .758 r F P 0 g s .63243 .87431 m .70081 .78329 L .71575 .80367 L .325 .137 .465 r F .73043 .8237 m .805 .73041 L p .69621 .77701 L .521 .181 .331 r F P 0 g s .63243 .87431 m .68733 .7649 L .70081 .78329 L .325 .137 .465 r F .69621 .77701 m .70081 .78329 L .73043 .8237 L .521 .181 .331 r F .73043 .8237 m .63243 .87431 L p .67078 .84179 L .325 .137 .465 r F P 0 g s .74226 .80816 m .68935 .82605 L p .80021 .73205 L .325 .137 .465 r F P 0 g s .63243 .87431 m .73144 .64395 L .78309 .57408 L .325 .137 .465 r F .53419 .73758 m .48439 .8244 L p .57543 .77883 L .521 .181 .331 r F P 0 g s .6174 .75783 m .6174 .75783 L .57543 .77883 L .521 .181 .331 r F .6174 .75783 m .53419 .73758 L .57543 .77883 L F .34019 .77578 m .48439 .8244 L p .48439 .8244 L .695 .485 .599 r F P 0 g s .48439 .8244 m .45374 .749 L .34019 .77578 L .695 .485 .599 r F .41478 .65318 m .53419 .73758 L p .45374 .749 L .827 .543 .508 r F P 0 g s .34019 .77578 m .41478 .65318 L p .29579 .66487 L .862 .714 .697 r F P 0 g s .2923 .5666 m .41478 .65318 L p .41478 .65318 L .827 .543 .508 r F P 0 g s .25961 .66843 m .34019 .77578 L p .29579 .66487 L .862 .714 .697 r F P 0 g s .22323 .78175 m .25961 .66843 L p .29815 .73146 L .827 .543 .508 r F P 0 g s .25961 .66843 m .34019 .77578 L .36765 .84512 L p .801 .357 0 r F P 0 g s .48439 .8244 m .63243 .87431 L p .57543 .77883 L .695 .485 .599 r F P 0 g s .63243 .87431 m .66477 .73412 L p .60008 .76649 L .695 .485 .599 r F P 0 g s .66477 .73412 m .53419 .73758 L p .6174 .75783 L .521 .181 .331 r F P 0 g s .66477 .73412 m .53419 .73758 L .54707 .58979 L p .585 .467 .685 r F P 0 g s .57543 .77883 m .54707 .58979 L .48138 .62126 L .695 .485 .599 r F .63243 .87431 m .60008 .76649 L .57543 .77883 L F .60008 .76649 m .54707 .58979 L .57543 .77883 L F .68733 .7649 m .78309 .57408 L .70081 .78329 L .325 .137 .465 r F .66477 .73412 m .6174 .75783 L .69621 .77701 L .521 .181 .331 r F .48439 .8244 m .48439 .8244 L p .6174 .75783 L F P 0 g s .48439 .8244 m .45141 .88188 L p .4965 .84818 L .521 .181 .331 r F P 0 g s .45141 .88188 m .36765 .84512 L p .45232 .83009 L .827 .543 .508 r F P 0 g s .45141 .88188 m .36765 .84512 L p .44421 .87136 L .084 .556 .904 r F P 0 g s .48439 .8244 m .34019 .77578 L .36765 .84512 L p .684 .247 .208 r F P 0 g s .53419 .73758 m .48439 .8244 L p .45374 .749 L .827 .543 .508 r F P 0 g s .57543 .77883 m .48138 .62126 L .48439 .8244 L .695 .485 .599 r F .48439 .8244 m .45141 .88188 L p .45374 .749 L .827 .543 .508 r F P 0 g s .63243 .87431 m .48439 .8244 L p .48439 .8244 L .684 .247 .208 r F P 0 g s .50252 .85999 m .4965 .84818 L .45141 .88188 L .521 .181 .331 r F .52185 .89796 m .52185 .89796 L p .50252 .85999 L .684 .247 .208 r F P 0 g s .52185 .89796 m .63243 .87431 L p .48439 .8244 L .684 .247 .208 r F P 0 g s .48439 .8244 m .48439 .8244 L .50252 .85999 L .684 .247 .208 r F .48439 .8244 m .50252 .85999 L .52185 .89796 L F .73043 .8237 m .63243 .87431 L p .73043 .8237 L .501 .907 .895 r F P 0 g s .69621 .77701 m .50252 .85999 L .73043 .8237 L .521 .181 .331 r F .50252 .85999 m .45141 .88188 L .50769 .87015 L F .50252 .85999 m .50769 .87015 L .73043 .8237 L F .63243 .87431 m .52185 .89796 L p .73043 .8237 L .501 .907 .895 r F P 0 g s .52605 .86632 m .50769 .87015 L .52464 .87698 L .521 .181 .331 r F .52605 .86632 m .52464 .87698 L .64806 .92676 L F .64806 .92676 m .73043 .8237 L p .52605 .86632 L F P 0 g s .45141 .88188 m .52185 .89796 L p .50769 .87015 L .521 .181 .331 r F P 0 g s .52185 .89796 m .64806 .92676 L p .50769 .87015 L .521 .181 .331 r F P 0 g s .39897 .92424 m .52185 .89796 L p .52185 .89796 L .684 .247 .208 r F P 0 g s .52185 .89796 m .48439 .8244 L .44421 .87136 L .684 .247 .208 r F .52185 .89796 m .45141 .88188 L p .44421 .87136 L .084 .556 .904 r F P 0 g s .52185 .89796 m .44421 .87136 L .39897 .92424 L .684 .247 .208 r F .50252 .85999 m .48439 .8244 L .52185 .89796 L F .69621 .77701 m .4965 .84818 L .50252 .85999 L .521 .181 .331 r F .36765 .84512 m .45374 .749 L .45232 .83009 L .827 .543 .508 r F .36765 .84512 m .36765 .84512 L p .45232 .83009 L F P 0 g s .36765 .84512 m .39897 .92424 L p .48439 .8244 L .684 .247 .208 r F P 0 g s .25961 .66843 m .36765 .84512 L .32045 .76794 L .801 .357 0 r F .45374 .749 m .42631 .68153 L .34019 .77578 L .695 .485 .599 r F .41478 .65318 m .34019 .77578 L p .48138 .62126 L F P 0 g s .32045 .76794 m .45374 .749 L .36765 .84512 L .827 .543 .508 r F .71575 .80367 m .80813 .64046 L .80021 .73205 L .325 .137 .465 r F .70081 .78329 m .80813 .64046 L .71575 .80367 L F .88206 .54206 m .82861 .69475 L p .82861 .69475 L .969 .856 .675 r F P 0 g s .82351 .48789 m .82351 .48789 L p .82615 .59494 L .969 .856 .675 r F P 0 g s .82612 .59362 m .82615 .59494 L .82861 .69475 L .969 .856 .675 r F .82351 .48789 m .82615 .59494 L .82612 .59362 L F .82612 .59362 m .77815 .42814 L .78066 .50223 L 0 0 .434 r F .82541 .56512 m .77815 .42814 L .82612 .59362 L F .82694 .62695 m .78066 .50223 L .79039 .59343 L F .82351 .48789 m .88206 .54206 L p .82861 .69475 L .969 .856 .675 r F P 0 g s .85591 .65352 m .82612 .59362 L .82694 .62695 L 0 0 .434 r F .82612 .59362 m .78066 .50223 L .82694 .62695 L F .82861 .69475 m .85591 .65352 L p .79039 .59343 L F P 0 g s .82861 .69475 m .82861 .69475 L p .82351 .48789 L .969 .856 .675 r F P 0 g s .82861 .69475 m .805 .73041 L .73043 .8237 L p .606 .846 .478 r F P 0 g s .82861 .69475 m .79517 .79026 L p .80813 .64046 L .325 .137 .465 r F P 0 g s .82351 .48789 m .82861 .69475 L .80088 .72932 L .969 .856 .675 r F .85591 .65352 m .82541 .56512 L .82612 .59362 L 0 0 .434 r F .38253 .81552 m .44421 .87136 L .36765 .84512 L .084 .556 .904 r F .69621 .77701 m .6174 .75783 L .4965 .84818 L .521 .181 .331 r F .6174 .75783 m .48439 .8244 L .4965 .84818 L F .22323 .78175 m .39535 .66622 L .39663 .75712 L .827 .543 .508 r F .39535 .66622 m .41478 .65318 L .39663 .75712 L F .39663 .75712 m .41478 .65318 L .45374 .749 L F .29815 .73146 m .25961 .66843 L .41478 .65318 L F .25961 .66843 m .2923 .5666 L p .41478 .65318 L F P 0 g s .1728 .55277 m .26906 .56267 L .27212 .52837 L .862 .714 .697 r F .2923 .5666 m .27848 .4571 L p .34515 .47784 L .797 .795 .85 r F P 0 g s .27848 .4571 m .1728 .55277 L p .40368 .49606 L .862 .714 .697 r F P 0 g s .27848 .4571 m .27848 .4571 L p .26781 .57663 L .987 .845 .469 r F P 0 g s .25961 .66843 m .26906 .56267 L .26781 .57663 L .987 .845 .469 r F .26906 .56267 m .27848 .4571 L .26781 .57663 L F .2923 .5666 m .41478 .65318 L p .42138 .57835 L .75 .613 .706 r F P 0 g s .26906 .56267 m .41478 .65318 L .42138 .57835 L .862 .714 .697 r F .1728 .55277 m .26647 .59163 L .26906 .56267 L F .35234 .32887 m .26939 .38501 L p .30039 .37465 L .068 0 0 r F P 0 g s .27848 .4571 m .26939 .38501 L p .26732 .44505 L .987 .845 .469 r F P 0 g s .27848 .4571 m .27848 .4571 L p .34515 .47784 L .797 .795 .85 r F P 0 g s .27848 .4571 m .26939 .38501 L p .29906 .42137 L .797 .795 .85 r F P 0 g s .34515 .47784 m .27848 .4571 L .29906 .42137 L .797 .795 .85 r F .1728 .55277 m .27848 .4571 L p .27848 .4571 L .895 .982 .804 r F P 0 g s .24626 .42234 m .1728 .55277 L p .27848 .4571 L .895 .982 .804 r F P 0 g s .26295 .44034 m .27848 .4571 L .27848 .4571 L .895 .982 .804 r F .24626 .42234 m .27848 .4571 L .26295 .44034 L F .22323 .78175 m .23785 .65614 L .23464 .65433 L .987 .845 .469 r F .21387 .6426 m .1728 .55277 L p .23464 .65433 L .801 .357 0 r F P 0 g s .21387 .6426 m .22323 .78175 L p .23464 .65433 L .987 .845 .469 r F P 0 g s .1728 .55277 m .25961 .66843 L p .23785 .65614 L .801 .357 0 r F P 0 g s .23955 .6571 m .25961 .66843 L .26732 .44505 L .987 .845 .469 r F .22323 .78175 m .25961 .66843 L p .23955 .6571 L F P 0 g s .23785 .65614 m .25961 .66843 L .23464 .65433 L .801 .357 0 r F .25961 .66843 m .25961 .66843 L p .23785 .65614 L F P 0 g s .23785 .65614 m .26732 .44505 L .26295 .44034 L .987 .845 .469 r F .22323 .78175 m .23955 .6571 L .23785 .65614 L F .23955 .6571 m .26732 .44505 L .23785 .65614 L F .25961 .66843 m .25961 .66843 L p .23955 .6571 L F P 0 g s .23464 .65433 m .23785 .65614 L .23464 .65433 L .801 .357 0 r F .1728 .55277 m .23785 .65614 L .23464 .65433 L F .36765 .84512 m .22323 .78175 L p .32045 .76794 L .827 .543 .508 r F P 0 g s .21387 .6426 m .31942 .80319 L .2823 .75539 L .801 .357 0 r F .31942 .80319 m .30225 .78826 L .30398 .78331 L F .30225 .78826 m .30225 .78826 L .30398 .78331 L F .30225 .78826 m .22323 .78175 L .21387 .6426 L p .247 .322 .743 r F P 0 g s .30225 .78826 m .24128 .70256 L p .30398 .78331 L .801 .357 0 r F P 0 g s .36765 .84512 m .22323 .78175 L .30225 .78826 L p .084 .556 .904 r F P 0 g s .33186 .81401 m .36765 .84512 L .30225 .78826 L .084 .556 .904 r F .23464 .65433 m .33186 .81401 L .21387 .6426 L .801 .357 0 r F .33186 .81401 m .31942 .80319 L .21387 .6426 L F .23464 .65433 m .25961 .66843 L .39897 .92424 L F .25961 .66843 m .27848 .4571 L .26732 .44505 L .987 .845 .469 r F .25961 .66843 m .26781 .57663 L .25961 .66843 L F .26781 .57663 m .27848 .4571 L .25961 .66843 L F .26647 .59163 m .41478 .65318 L .26906 .56267 L .862 .714 .697 r F .26647 .59163 m .29579 .66487 L .41478 .65318 L F .63711 .28299 m .48889 .23645 L p .57949 .31521 L .486 .61 .9 r F P 0 g s .44179 .33714 m .55412 .3294 L .59272 .27777 L .544 .859 .997 r F .44179 .33714 m .57088 .2752 L .35234 .32887 L F .50152 .26705 m .48889 .23645 L .63711 .28299 L p .492 0 0 r F P 0 g s .57088 .2752 m .60793 .25743 L .5522 .27301 L .544 .859 .997 r F .57088 .2752 m .5522 .27301 L .35234 .32887 L F .63711 .28299 m .63711 .28299 L p .58299 .27663 L .492 0 0 r F P 0 g s .50152 .26705 m .63711 .28299 L .58299 .27663 L .492 0 0 r F .44179 .33714 m .59272 .27777 L .57088 .2752 L .544 .859 .997 r F .70081 .78329 m .78309 .57408 L .80813 .64046 L .325 .137 .465 r F .85591 .65352 m .82861 .69475 L p .82861 .69475 L .606 .846 .478 r F P 0 g s .44393 .83636 m .52632 .86433 L .52185 .89796 L .084 .556 .904 r F .52632 .86433 m .52427 .87974 L .52185 .89796 L F .52185 .89796 m .52185 .89796 L p .52632 .86433 L F P 0 g s .48439 .8244 m .48138 .62126 L .42631 .68153 L .695 .485 .599 r F .60008 .76649 m .66477 .73412 L .54707 .58979 L F .39752 .82061 m .52774 .80999 L .53155 .66506 L .586 .605 .838 r F .52774 .80999 m .52774 .80999 L .52922 .75376 L F .52774 .80999 m .52922 .75376 L .53155 .66506 L F .78309 .57408 m .78309 .57408 L p .71446 .49603 L .585 .467 .685 r F P 0 g s .75782 .60826 m .71446 .49603 L .78309 .57408 L .585 .467 .685 r F .78309 .57408 m .73144 .64395 L .78309 .57408 L .325 .137 .465 r F .7695 .54746 m .82351 .48789 L .82351 .48789 L .969 .856 .675 r F .7695 .54746 m .82351 .48789 L .80088 .72932 L F .7695 .54746 m .82351 .48789 L .7695 .54746 L F .72201 .6567 m .66477 .73412 L .59708 .54575 L .585 .467 .685 r F .75782 .60826 m .62613 .52017 L .71446 .49603 L F .73144 .64395 m .62613 .52017 L .75782 .60826 L F .45137 .46585 m .60647 .53748 L .54707 .58979 L .679 .629 .792 r F .46934 .43687 m .60647 .53748 L .45137 .46585 L F .68788 .46579 m .68788 .46579 L p .52632 .34495 L .486 .61 .9 r F P 0 g s .52632 .34495 m .68788 .46579 L .71943 .45263 L .486 .61 .9 r F .68788 .46579 m .68788 .46579 L p .62613 .52017 L .679 .629 .792 r F P 0 g s .62613 .52017 m .60647 .53748 L .62613 .52017 L .585 .467 .685 r F .54707 .58979 m .54707 .58979 L p .60647 .53748 L .679 .629 .792 r F P 0 g s .68236 .75812 m .66477 .73412 L .73144 .64395 L .325 .137 .465 r F .54707 .58979 m .54707 .58979 L p .66477 .73412 L .695 .485 .599 r F P 0 g s .57088 .2752 m .59272 .27777 L .60793 .25743 L .544 .859 .997 r F .4296 .33601 m .35234 .32887 L .35234 .32887 L F .4296 .33601 m .44179 .33714 L .4296 .33601 L .797 .795 .85 r F .44179 .33714 m .35234 .32887 L .4296 .33601 L .544 .859 .997 r F .4296 .33601 m .37591 .48741 L .34515 .47784 L .797 .795 .85 r F .44349 .33729 m .37591 .48741 L .4296 .33601 L F .44179 .33714 m .44179 .33714 L .4296 .33601 L .544 .859 .997 r F .39041 .33239 m .34515 .47784 L .29906 .42137 L .797 .795 .85 r F .4296 .33601 m .34515 .47784 L .39041 .33239 L F .50152 .26705 m .48889 .23645 L .35234 .32887 L p .068 0 0 r F P 0 g s .53517 .27776 m .5522 .27301 L .52412 .27424 L .544 .859 .997 r F .53517 .27776 m .52412 .27424 L .32472 .28303 L F .32472 .28303 m .35234 .32887 L p .53517 .27776 L F P 0 g s .50152 .26705 m .32472 .28303 L p .5522 .27301 L .544 .859 .997 r F P 0 g s .50152 .26705 m .50152 .26705 L p .58034 .29215 L .662 .357 .445 r F P 0 g s .45106 .31298 m .50152 .26705 L .35234 .32887 L .068 0 0 r F .35234 .32887 m .35234 .32887 L p .50152 .26705 L F P 0 g s .50152 .26705 m .32472 .28303 L p .47796 .2885 L .662 .357 .445 r F P 0 g s .47796 .2885 m .50152 .26705 L .47796 .2885 L .662 .357 .445 r F .58034 .29215 m .50152 .26705 L .47796 .2885 L F .57949 .31521 m .73169 .44752 L .63711 .28299 L .486 .61 .9 r F .63711 .28299 m .60793 .25743 L p .55412 .3294 L .544 .859 .997 r F P 0 g s .71141 .61154 m .59989 .43502 L .57851 .47578 L .855 .596 .546 r F .23785 .65614 m .26295 .44034 L .21387 .6426 L .987 .845 .469 r F .25961 .66843 m .29579 .66487 L .26647 .59163 L .862 .714 .697 r F .29584 .63976 m .29584 .63976 L p .23612 .64183 L .24 .112 .497 r F P 0 g s .27212 .52837 m .42138 .57835 L .40368 .49606 L .862 .714 .697 r F .26906 .56267 m .42138 .57835 L .27212 .52837 L F .42796 .50361 m .42796 .50361 L p .46934 .43687 L .797 .795 .85 r F P 0 g s .42138 .57835 m .42796 .50361 L .40368 .49606 L .862 .714 .697 r F .42138 .57835 m .42796 .50361 L .42138 .57835 L F .42796 .50361 m .42796 .50361 L p .54707 .58979 L .75 .613 .706 r F P 0 g s .48124 .4418 m .29584 .63976 L .42804 .65395 L .622 .481 .669 r F .30873 .58903 m .23612 .64183 L .29584 .63976 L .24 .112 .497 r F .46575 .43639 m .46575 .43639 L .35716 .39846 L .662 .357 .445 r F .46575 .43639 m .58034 .29215 L .35716 .39846 L F .46575 .43639 m .35716 .39846 L .35716 .39846 L F .49622 .44703 m .57851 .47578 L .6613 .31793 L F .48124 .4418 m .42804 .65395 L .57851 .47578 L .622 .481 .669 r F .53155 .66506 m .53155 .66506 L p .42804 .65395 L .586 .605 .838 r F P 0 g s .82861 .69475 m .73967 .80903 L .74962 .79325 L .606 .846 .478 r F .72846 .69263 m .77532 .7612 L .74962 .79325 L .969 .856 .675 r F .82861 .69475 m .74962 .79325 L .79014 .74271 L .606 .846 .478 r F .71141 .61154 m .77925 .7563 L .72846 .69263 L .969 .856 .675 r F .79517 .79026 m .73043 .8237 L p .77659 .76307 L .325 .137 .465 r F P 0 g s .79517 .79026 m .7802 .75833 L p .80021 .73205 L .325 .137 .465 r F P 0 g s .52337 .88651 m .73967 .80903 L .73043 .8237 L .501 .907 .895 r F .52398 .88197 m .73967 .80903 L .52337 .88651 L F .73043 .8237 m .73043 .8237 L p .73967 .80903 L F P 0 g s .73043 .8237 m .73043 .8237 L p .82861 .69475 L .606 .846 .478 r F P 0 g s .79517 .79026 m .73043 .8237 L p .73043 .8237 L .501 .907 .895 r F P 0 g s .73043 .8237 m .73967 .80903 L .82861 .69475 L .606 .846 .478 r F .73967 .80903 m .73967 .80903 L .73043 .8237 L .501 .907 .895 r F .73967 .80903 m .79517 .79026 L .73043 .8237 L F .82861 .69475 m .79517 .79026 L p .80088 .72932 L .969 .856 .675 r F P 0 g s .80088 .72932 m .82861 .69475 L .80088 .72932 L .969 .856 .675 r F .45374 .749 m .42631 .68153 L .42631 .68153 L .695 .485 .599 r F .39752 .82061 m .53155 .66506 L .42804 .65395 L .586 .605 .838 r F .39513 .65042 m .42804 .65395 L .39513 .65042 L F .41478 .65318 m .41478 .65318 L p .29579 .66487 L .862 .714 .697 r F P 0 g s .39752 .82061 m .42804 .65395 L .39513 .65042 L .586 .605 .838 r F .57851 .47578 m .47216 .65869 L .542 .62294 L .622 .481 .669 r F .57851 .47578 m .57851 .47578 L p .542 .62294 L F P 0 g s .42804 .65395 m .47216 .65869 L .57851 .47578 L .622 .481 .669 r F .22801 .54643 m .26295 .44034 L .24083 .45927 L .987 .845 .469 r F .21387 .6426 m .26295 .44034 L .22801 .54643 L F .26939 .38501 m .24626 .42234 L p .26295 .44034 L F P 0 g s .26295 .44034 m .26732 .44505 L .26939 .38501 L .987 .845 .469 r F .30039 .37465 m .24626 .42234 L .27848 .4571 L .895 .982 .804 r F .26939 .38501 m .24626 .42234 L p .30039 .37465 L .068 0 0 r F P 0 g s .32472 .28303 m .24626 .42234 L p .30039 .37465 L .895 .982 .804 r F P 0 g s .24626 .42234 m .24626 .42234 L p .30039 .37465 L .895 .982 .804 r F P 0 g s .24626 .42234 m .1728 .55277 L p .22801 .54643 L .24 .112 .497 r F P 0 g s .35716 .39846 m .58034 .29215 L .47796 .2885 L .662 .357 .445 r F .35234 .32887 m .32472 .28303 L p .27848 .4571 L .895 .982 .804 r F P 0 g s .40781 .35235 m .30039 .37465 L .26393 .41853 L .068 0 0 r F .43145 .33083 m .30039 .37465 L .40781 .35235 L F .35716 .39846 m .47796 .2885 L .43145 .33083 L .662 .357 .445 r F .35716 .39846 m .51617 .30251 L p .40781 .35235 L .068 0 0 r F P 0 g s .40781 .35235 m .26393 .41853 L .35716 .39846 L .068 0 0 r F .35716 .39846 m .35716 .39846 L p .43145 .33083 L .662 .357 .445 r F P 0 g s .51617 .30251 m .35234 .32887 L .30039 .37465 L .068 0 0 r F .32472 .28303 m .35716 .39846 L p .43145 .33083 L .662 .357 .445 r F P 0 g s .43145 .33083 m .47796 .2885 L .47796 .2885 L .662 .357 .445 r F .43145 .33083 m .47796 .2885 L .32472 .28303 L F .51617 .30251 m .43145 .33083 L .40781 .35235 L .068 0 0 r F .23932 .46956 m .24083 .45927 L .20368 .49107 L .987 .845 .469 r F .21387 .6426 m .23932 .46956 L .20368 .49107 L F .24626 .42234 m .20368 .49107 L p .26295 .44034 L F P 0 g s .3223 .53561 m .24626 .42234 L .22801 .54643 L .24 .112 .497 r F .24626 .42234 m .24626 .42234 L p .30039 .37465 L .068 0 0 r F P 0 g s .30039 .37465 m .24626 .42234 L .26393 .41853 L .068 0 0 r F .33498 .48571 m .24626 .42234 L .3223 .53561 L .24 .112 .497 r F .32472 .28303 m .24626 .42234 L p .33498 .48571 L F P 0 g s .24626 .42234 m .20368 .49107 L p .26393 .41853 L .068 0 0 r F P 0 g s .35716 .39846 m .32472 .28303 L p .33498 .48571 L .24 .112 .497 r F P 0 g s .20368 .49107 m .35716 .39846 L p .26393 .41853 L .068 0 0 r F P 0 g s .43559 .42585 m .29584 .63976 L .48124 .4418 L .622 .481 .669 r F .41854 .4199 m .35716 .39846 L .3223 .53561 L F .51617 .30251 m .3223 .53561 L .29584 .63976 L F .30873 .58903 m .22801 .54643 L .23612 .64183 L .24 .112 .497 r F .3223 .53561 m .22801 .54643 L .30873 .58903 L F .35716 .39846 m .20368 .49107 L p .29584 .63976 L .622 .481 .669 r F P 0 g s .3934 .52744 m .33498 .48571 L .3223 .53561 L .24 .112 .497 r F .3934 .52744 m .40373 .52456 L p .40373 .52456 L .763 .623 .706 r F P 0 g s .40373 .52456 m .57851 .47578 L p .40373 .52456 L .763 .623 .706 r F P 0 g s .40373 .52456 m .57851 .47578 L .57851 .47578 L .763 .623 .706 r F .55508 .5702 m .57851 .47578 L .71141 .61154 L F .3934 .52744 m .57851 .47578 L .55508 .5702 L F .47216 .65869 m .53155 .66506 L .542 .62294 L .622 .481 .669 r F .53155 .66506 m .3934 .52744 L p .55508 .5702 L .763 .623 .706 r F P 0 g s .71141 .61154 m .71141 .61154 L p .57851 .47578 L .855 .596 .546 r F P 0 g s .55508 .5702 m .71141 .61154 L .53155 .66506 L .763 .623 .706 r F .55508 .5702 m .53155 .66506 L .53155 .66506 L F .53155 .66506 m .53155 .66506 L p .47216 .65869 L .622 .481 .669 r F P 0 g s .53155 .66506 m .61754 .58426 L p .542 .62294 L .622 .481 .669 r F P 0 g s .61754 .58426 m .57851 .47578 L p .542 .62294 L .622 .481 .669 r F P 0 g s .36765 .84512 m .36765 .84512 L p .32045 .76794 L .801 .357 0 r F P 0 g s .36765 .84512 m .39897 .92424 L p .25961 .66843 L .801 .357 0 r F P 0 g s .39752 .82061 m .3934 .52744 L .30225 .78826 L .586 .605 .838 r F .3925 .79571 m .53155 .66506 L p .47216 .65869 L .622 .481 .669 r F P 0 g s .41314 .82591 m .52185 .89796 L .44421 .87136 L .084 .556 .904 r F .44393 .83636 m .52185 .89796 L .41314 .82591 L F .52185 .89796 m .39897 .92424 L p .73043 .8237 L .501 .907 .895 r F P 0 g s .52398 .88197 m .58683 .84507 L .74962 .79325 L .501 .907 .895 r F .52922 .75376 m .59102 .84373 L .74962 .79325 L .784 .771 .841 r F .74962 .79325 m .74962 .79325 L p .58683 .84507 L .501 .907 .895 r F P 0 g s .52398 .88197 m .74962 .79325 L .73967 .80903 L .501 .907 .895 r F .69621 .77701 m .69621 .77701 L .70081 .78329 L .521 .181 .331 r F .52922 .75376 m .74962 .79325 L .72846 .69263 L .784 .771 .841 r F .52632 .86433 m .58683 .84507 L .52398 .88197 L .501 .907 .895 r F .52632 .86433 m .64806 .92676 L p .59102 .84373 L .784 .771 .841 r F P 0 g s .52632 .86433 m .59102 .84373 L .52922 .75376 L .784 .771 .841 r F .52632 .86433 m .66573 .79874 L p .58683 .84507 L .501 .907 .895 r F P 0 g s .66573 .79874 m .74962 .79325 L p .58683 .84507 L .501 .907 .895 r F P 0 g s .74962 .79325 m .59102 .84373 L .74962 .79325 L .784 .771 .841 r F .59102 .84373 m .59102 .84373 L .74962 .79325 L F .64806 .92676 m .74962 .79325 L p .59102 .84373 L F P 0 g s .74962 .79325 m .74962 .79325 L p .59102 .84373 L .784 .771 .841 r F P 0 g s .41234 .82564 m .66573 .79874 L .39752 .82061 L .586 .605 .838 r F .44393 .83636 m .52632 .86433 L .44393 .83636 L .084 .556 .904 r F .41314 .82591 m .52632 .86433 L .41234 .82564 L .586 .605 .838 r F .52632 .86433 m .66573 .79874 L p .41234 .82564 L F P 0 g s .3925 .79571 m .52632 .86433 L .44393 .83636 L .084 .556 .904 r F .52632 .86433 m .52632 .86433 L .44393 .83636 L F .52632 .86433 m .52632 .86433 L p .41314 .82591 L .586 .605 .838 r F P 0 g s .3925 .79571 m .52632 .86433 L p .52632 .86433 L .084 .556 .904 r F P 0 g s .3925 .79571 m .44393 .83636 L .41314 .82591 L .084 .556 .904 r F .41314 .82591 m .44421 .87136 L .38253 .81552 L F .36765 .84512 m .36765 .84512 L p .44421 .87136 L F P 0 g s .39897 .92424 m .52632 .86433 L p .41314 .82591 L .586 .605 .838 r F P 0 g s .39752 .82061 m .41314 .82591 L .39752 .82061 L .586 .605 .838 r F .39897 .92424 m .41314 .82591 L .39752 .82061 L F .73043 .8237 m .64806 .92676 L p .73967 .80903 L .606 .846 .478 r F P 0 g s .74962 .79325 m .79517 .79026 L p .73967 .80903 L .501 .907 .895 r F P 0 g s .77925 .7563 m .79517 .79026 L .77532 .7612 L .969 .856 .675 r F .77925 .7563 m .77532 .7612 L .72846 .69263 L F .71141 .61154 m .71141 .61154 L .72846 .69263 L F .53155 .66506 m .72125 .65832 L .53155 .66506 L .784 .771 .841 r F .7695 .54746 m .79517 .79026 L .71141 .61154 L .969 .856 .675 r F .74094 .57896 m .71141 .61154 L .71141 .61154 L F .7695 .54746 m .71141 .61154 L .74094 .57896 L F .76164 .55613 m .82351 .48789 L .63257 .37272 L .855 .596 .546 r F .76164 .55613 m .7695 .54746 L .76164 .55613 L F .7695 .54746 m .82351 .48789 L .76164 .55613 L F .7695 .54746 m .80088 .72932 L .79014 .74271 L .969 .856 .675 r F .72125 .65832 m .85591 .65352 L .72125 .65831 L .784 .771 .841 r F .72125 .65832 m .72125 .65831 L .53155 .66506 L F .52922 .75376 m .85591 .65352 L .53155 .66506 L F .53155 .66506 m .72125 .65831 L .67211 .62323 L F .71141 .61154 m .61754 .58426 L p .72125 .65831 L F P 0 g s .74962 .79325 m .66573 .79874 L p .71141 .61154 L .969 .856 .675 r F P 0 g s .74962 .79325 m .85591 .65352 L p .82861 .69475 L .606 .846 .478 r F P 0 g s .82861 .69475 m .79014 .74271 L .82861 .69475 L .606 .846 .478 r F .79014 .74271 m .74962 .79325 L .82861 .69475 L F .79014 .74271 m .74962 .79325 L .74962 .79325 L F .79517 .79026 m .74962 .79325 L p .74962 .79325 L .969 .856 .675 r F P 0 g s .77532 .7612 m .79517 .79026 L .74962 .79325 L .969 .856 .675 r F .74962 .79325 m .85591 .65352 L p .72846 .69263 L .784 .771 .841 r F P 0 g s .52922 .75376 m .52922 .75376 L .53155 .66506 L .784 .771 .841 r F .42804 .65395 m .3925 .79571 L .47216 .65869 L .622 .481 .669 r F .42804 .65395 m .39513 .65042 L .42804 .65395 L F .39513 .65042 m .42804 .65395 L .3934 .52744 L .586 .605 .838 r F .53155 .66506 m .53155 .66506 L p .67211 .62323 L .784 .771 .841 r F P 0 g s .53155 .66506 m .3934 .52744 L p .42804 .65395 L .586 .605 .838 r F P 0 g s .3934 .52744 m .6613 .31793 L .58034 .29215 L .662 .357 .445 r F .76164 .55613 m .61554 .40518 L .71141 .61154 L .855 .596 .546 r F .57851 .47578 m .57851 .47578 L p .61554 .40518 L .662 .357 .445 r F P 0 g s .57851 .47578 m .61554 .40518 L .6613 .31793 L .662 .357 .445 r F .61554 .40518 m .51617 .30251 L .59989 .43502 L .855 .596 .546 r F .61554 .40518 m .59989 .43502 L .71141 .61154 L F .48124 .4418 m .48124 .4418 L .57851 .47578 L .622 .481 .669 r F .49622 .44703 m .57851 .47578 L .49622 .44703 L .662 .357 .445 r F .57851 .47578 m .57851 .47578 L .49622 .44703 L F .48124 .4418 m .57851 .47578 L .57851 .47578 L .622 .481 .669 r F .3934 .52744 m .57851 .47578 L .49622 .44703 L .662 .357 .445 r F .3934 .52744 m .57851 .47578 L p .57851 .47578 L F P 0 g s .57851 .47578 m .51617 .30251 L p .48124 .4418 L .622 .481 .669 r F P 0 g s .29584 .63976 m .3223 .53561 L .29584 .63976 L .622 .481 .669 r F .29584 .63976 m .3934 .52744 L p .30873 .58903 L .24 .112 .497 r F P 0 g s .51617 .30251 m .43559 .42585 L .48124 .4418 L .622 .481 .669 r F .76164 .55613 m .63257 .37272 L .61554 .40518 L .855 .596 .546 r F .82351 .48789 m .73178 .39178 L .6613 .31793 L F .73178 .39178 m .6613 .31793 L .6613 .31793 L F .73178 .39178 m .6613 .31793 L .6613 .31793 L F .77815 .42814 m .77815 .42814 L p .82351 .48789 L 0 0 .434 r F P 0 g s .88206 .54206 m .77815 .42814 L p .76529 .42689 L .599 .096 0 r F P 0 g s .82351 .48789 m .88206 .54206 L p .76529 .42689 L .599 .096 0 r F P 0 g s .69572 .354 m .69912 .34681 L .64365 .29244 L .599 .096 0 r F .73178 .39178 m .69912 .34681 L .69572 .354 L F .6613 .31793 m .75667 .42606 L p .64365 .29244 L F P 0 g s .82541 .56512 m .82351 .48789 L .82541 .56512 L 0 0 .434 r F .85591 .65352 m 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Combining the canonical TSI with a rotated version of \ itself gives a compound, again illustrating self-duality.\ \>", "Text"] }, Open ]], Cell["\<\ Conclusions and future work \ \>", "Section", Evaluatable->False], Cell[TextData[ "\nA polyhedron canonicalization algorithm has been presented and \ demonstrated with a range of examples. See [Hart, 1997] for additional \ examples. Empirically, the algorithm has always been observed to converge, \ although we have no proof that it should do so generally. One application \ demonstrated is to construct geometrically self-dual polyhedra which form \ attractive compounds with themselves. Three-dimensional virtual reality \ models of these and other examples are available on the electronic version of \ this journal.\n\n\n\nThere are many directions for extending this work, e.g., \ faster algorithms, provably convergent algorithms, or tangency to an \ arbitrary, nonspherical, smooth, convex solid. It seems likely that a \ \"primal-dual\" algorithm which simultaneously relaxes both the original \ polyhedron and its dual should be feasible and have certain advantages.\n"], "Text", Evaluatable->False], Cell["\<\ Acknowledgment \ \>", "Section", Evaluatable->False], Cell[TextData[ "\nJohn Conway told me of this theorem, the need for a construction \ algorithm, and the self-dual hermaphrodite and antihermaphrodite polyhedra.\n\ "], "Text", Evaluatable->False], Cell["\<\ References \ \>", "Section", Evaluatable->False], Cell[TextData[{ "\nCoxeter, H.S.M. , ", StyleBox["Regular Polytopes,", FontSlant->"Italic"], " Macmillan, 1963, (Dover reprint, 1973).\n\nZiegler, G. M., ", StyleBox["Lectures on Polytopes", FontSlant->"Italic"], ", Springer-Verlag, 1995.\n\nSchramm, O., \"How to Cage an Egg,\" ", StyleBox["Inventiones Math", FontSlant->"Italic"], ". 107, 1992, pp. 543-560.\n\nHart, G. 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