| %! |
| % If you're concerned that the cpu in your PostScript printer will atrophy |
| % from disuse, here is another Escher-like contribution to to keep it busy |
| % for a while. It uses PostScript color commands, but will still work on |
| % a monochrome printer (but isn't very pretty in black & white). |
| % |
| % The butterflies are arranged in a hexagonal grid (wallpaper group p6), |
| % and the moveto, lineto, curveto commands used to render the tesselation |
| % are redefined so as to impose a nonlinear transform that shrinks the |
| % infinite plane to an ellipse. This is a sleazy way to mimic Escher's |
| % "circle limit" sorts of things. |
| % |
| % The butterfly permimeter was made by imposing all the symmetry constraints |
| % on a path, and then that path was filled in using Adobe Illustrator |
| % |
| % The routines Xform and next_color are easy to change if you want to hack |
| % with them. The code was written to sacrifice efficiency for readability. |
| % |
| % Bob Wallis |
| % |
| % UUCP {sun,pyramid,cae780,apple}!weitek!wallis |
| |
| %statusdict begin waittimeout 6000 lt % if you have a slow printer, you |
| % {0 60 6000 setdefaulttimeouts} % might need to uncomment this |
| %if end |
| |
| /nlayers 1 def % 1 takes about 10 minutes on a LW+; 2 takes 4x longer |
| /warp 1 def % 1 -> ellipsoidal distortion; 0 -> flat Euclidean |
| /inch {72 mul} def |
| |
| /x4 152 def /y4 205.6 def % 6 fold rotation center of bfly |
| /x12 387.20 def /y12 403.84 def % 3 fold center of bfly |
| |
| /dx x4 x12 sub def % [dx,dy] = distance between the |
| /dy y4 y12 sub def % two fixed points above |
| |
| /Dm dx dup mul dy dup mul % magnitude of basis vectors of |
| add sqrt 3 sqrt mul % parallelogram lattice |
| def % = |dx,dy| * sqrt(3) |
| |
| /Da dy dx atan 30 add def |
| /D1x Dm Da cos mul def % [D1x, D1y] = basis vector vector #1 |
| /D1y Dm Da sin mul def % = [Dm,0] exp(j30) |
| |
| /Da dy dx atan 30 sub def |
| /D2x Dm Da cos mul def % [D2x, D2y] = basis vector vector #2 |
| /D2y Dm Da sin mul def % = [Dm,0] exp(-j30) |
| |
| /m { moveto} def |
| /L {lineto} def |
| /S {stroke} def |
| /c {curveto} def |
| /f {closepath fill} def |
| /F {closepath fill} def |
| /g { setgray} def |
| |
| /FillStroke { % fill interior & stroke black border |
| closepath gsave fill grestore 0 setgray stroke |
| } def |
| |
| % |
| % Description of 1 butterfly |
| % |
| /body { |
| 314.96 280.19 m |
| 383.4 261.71 445.11 243.23 513.52 224.68 c |
| 463.68 256.59 490.26 328.83 446.99 360.76 c |
| 423.71 347.32 397.08 339.7 367.07 337.9 c |
| 388.93 358.28 414.14 372.84 442.73 381.58 c |
| 426.68 398.18 394.07 389.7 387.2 403.84 c |
| 371.52 404.96 362.56 372.48 340.16 366.88 c |
| 346.88 396.01 346.88 425.12 340.16 454.24 c |
| 326.72 427.35 320 400.48 320 373.6 c |
| 270.71 352.1 221.44 411.23 168.88 384.02 c |
| 189.04 388.03 202.48 380.4 212.57 366.95 c |
| 216.72 350.85 209.23 341.46 190.1 338.79 c |
| 177.34 343.57 167.94 354.17 161.9 370.59 c |
| 176.06 305.52 132.02 274.05 152 205.6 c |
| 201.29 257.12 250.56 234.72 299.84 279.52 c |
| 288.64 266.08 284.16 252.64 286.4 239.2 c |
| 298.27 223.97 310.15 222.18 322.02 233.82 c |
| 328.62 249.28 328.51 264.74 314.96 280.19 c |
| FillStroke |
| } def |
| |
| /eyes { |
| 294.8125 238.3246 m |
| 296.9115 238.3246 298.6132 242.7964 298.6132 248.3125 c |
| 298.6132 253.8286 296.9115 258.3004 294.8125 258.3004 c |
| 292.7135 258.3004 291.0118 253.8286 291.0118 248.3125 c |
| 291.0118 242.7964 292.7135 238.3246 294.8125 238.3246 c |
| closepath gsave 1 g fill grestore 0 g S |
| |
| 319.5 241.1782 m |
| 321.7455 241.1782 323.5659 245.4917 323.5659 250.8125 c |
| 323.5659 256.1333 321.7455 260.4468 319.5 260.4468 c |
| 317.2545 260.4468 315.4341 256.1333 315.4341 250.8125 c |
| 315.4341 245.4917 317.2545 241.1782 319.5 241.1782 c |
| closepath gsave 1 g fill grestore 0 g S |
| 0 g |
| 296.875 242.0939 m |
| 297.4608 242.0939 297.9356 243.479 297.9356 245.1875 c |
| 297.9356 246.896 297.4608 248.2811 296.875 248.2811 c |
| 296.2892 248.2811 295.8143 246.896 295.8143 245.1875 c |
| 295.8143 243.479 296.2892 242.0939 296.875 242.0939 c |
| f |
| 0 g |
| 318.5 243.7707 m |
| 319.281 243.7707 319.9142 245.0766 319.9142 246.6875 c |
| 319.9142 248.2984 319.281 249.6043 318.5 249.6043 c |
| 317.719 249.6043 317.0858 248.2984 317.0858 246.6875 c |
| 317.0858 245.0766 317.719 243.7707 318.5 243.7707 c |
| f |
| } def |
| |
| /stripes { |
| 292 289 m |
| 252 294 241 295 213 279 c |
| 185 263 175 252 159 222 c |
| S |
| 285 313 m |
| 239 326 226 325 206 315 c |
| 186 305 164 278 161 267 c |
| S |
| 298 353 m |
| 262 342 251 339 237 355 c |
| 223 371 213 380 201 383 c |
| S |
| 330 288 m |
| 384 293 385 292 418 280 c |
| 451 268 452 264 473 247 c |
| S |
| 342 306 m |
| 381 311 386 317 410 311 c |
| 434 305 460 287 474 262 c |
| S |
| 345 321 m |
| 352 357 359 367 379 377 c |
| 399 387 409 385 426 382 c |
| S |
| 327.75 367.75 m |
| 336.5 392.25 333.682 403.348 335.25 415.5 c |
| S |
| 320 364.75 m |
| 322 361.75 323.5 360.5 326.25 360 c |
| 329 359.5 332 360.5 334 362.75 c |
| S |
| 316.25 356.5 m |
| 318.75 353.25 320 353 323.25 352.25 c |
| 326.5 351.5 329 352 331.5 353.25 c |
| S |
| 312.5 349 m |
| 316.75 345.5 318.25 344.5 321.25 343.75 c |
| 324.25 343 327 344 329.75 346 c |
| S |
| 310.75 340.75 m |
| 314.25 336.5 316.25 335.25 320 335.25 c |
| 323.75 335.25 327 336.5 329.25 338 c |
| S |
| 308.5 332 m |
| 311.75 328.5 312.5 327.25 317 327 c |
| 321.5 326.75 325.75 328.25 327.75 329.75 c |
| S |
| 305 322 m |
| 309.5 317.75 310.75 317 315 316.5 c |
| 319.25 316 322.25 318 324.75 320 c |
| S |
| 302.25 311 m |
| 307 307.5 307.75 306.25 312.75 306 c |
| 317.75 305.75 320 307.25 323.75 309.5 c |
| S |
| 301.25 298.25 m |
| 304.5 292.75 305.25 292 308.25 292 c |
| 311.25 292 313.75 293.75 315.75 295.75 c |
| S |
| } def |
| /nostrils { |
| 0 g |
| 304.062 227.775 m |
| 304.599 227.775 305.034 228.883 305.034 230.25 c |
| 305.034 231.616 304.599 232.724 304.062 232.724 c |
| 303.525 232.724 303.09 231.616 303.09 230.25 c |
| 303.09 228.883 303.525 227.775 304.062 227.775 c |
| f |
| 304.062 230.25 m |
| F |
| 309.562 228.275 m |
| 310.099 228.275 310.534 229.383 310.534 230.75 c |
| 310.534 232.116 310.099 233.224 309.562 233.224 c |
| 309.025 233.224 308.59 232.116 308.59 230.75 c |
| 308.59 229.383 309.025 228.275 309.562 228.275 c |
| f |
| } def |
| /thorax |
| { |
| 327.5 300 m |
| 316.5 283 315.5 275.5 308 277.5 c |
| 294 311.5 299 313.5 304 334 c |
| 309 354.5 315.5 362 322.5 372 c |
| 329.5 382 327.5 376.5 331 376 c |
| 334.5 375.5 339.1367 379.1109 339 369 c |
| 338.5 332 333.4999 324.5 330.5 311.5 c |
| 0 g S |
| } def |
| /spots { |
| next_color |
| 192 242.201 m |
| 202.1535 242.201 210.3848 251.0655 210.3848 262 c |
| 210.3848 272.9345 202.1535 281.799 192 281.799 c |
| 181.8465 281.799 173.6152 272.9345 173.6152 262 c |
| 173.6152 251.0655 181.8465 242.201 192 242.201 c |
| FillStroke |
| next_color |
| 447.5 250.2365 m |
| 459.6061 250.2365 469.4203 257.5181 469.4203 266.5 c |
| 469.4203 275.4819 459.6061 282.7635 447.5 282.7635 c |
| 435.3939 282.7635 425.5797 275.4819 425.5797 266.5 c |
| 425.5797 257.5181 435.3939 250.2365 447.5 250.2365 c |
| FillStroke |
| next_color |
| 401 369.1005 m |
| 409.5914 369.1005 416.5563 373.5327 416.5563 379 c |
| 416.5563 384.4673 409.5914 388.8995 401 388.8995 c |
| 392.4086 388.8995 385.4436 384.4673 385.4436 379 c |
| 385.4436 373.5327 392.4086 369.1005 401 369.1005 c |
| FillStroke |
| next_color |
| 249 348.2721 m |
| 261.4966 348.2721 271.6274 353.9707 271.6274 361 c |
| 271.6274 368.0293 261.4966 373.7279 249 373.7279 c |
| 236.5034 373.7279 226.3726 368.0293 226.3726 361 c |
| 226.3726 353.9707 236.5034 348.2721 249 348.2721 c |
| FillStroke |
| } def |
| |
| /ncolor 6 def |
| /cidx 0 def |
| |
| /next_color { |
| cidx ncolor div % hue |
| .75 % saturation (change these if you like) |
| .8 % lightness |
| sethsbcolor |
| /cidx cidx 1 add ncolor mod def |
| } def |
| |
| /cidx 0 def |
| |
| /max_r2 % radius^2 for center of outermost ring of butterflies |
| Dm nlayers mul 1.05 mul dup mul |
| def |
| |
| /max_radius max_r2 sqrt def |
| /max_radius_inv 1 max_radius div def |
| /Dm_inv 1 Dm div def |
| |
| % |
| % Ellipsoidal distortion, maps "nlayers" concentric rings of cells into |
| % an ellipse centered on page |
| |
| % D length of 1 basis vector separating hexagonal cells |
| % z0 center of 6-fold rotation = origin of shrink xform |
| % z' = (z - z0)/D new coord system |
| % |z'| = sqrt(x^2 + [(8.5/11)*y]^2) aspect ratio of paper |
| % z" = z' * a/M(|z'|) shrink by "a/M(|z|)" as fcn of radius |
| |
| % At the max radius, we want the shrunk ellipse to be "W" units wide so it |
| % just fits our output format - solve for scale factor "a" |
| |
| % zmax = n+0.5 for n layers of cells |
| % zmax * [a/M(zmax)] = W 1/2 width of output on paper |
| % a = M(zmax)*W/zmax solve for "a" |
| |
| %/M{dup mul 1 add sqrt}bind def % M(u) = sqrt(1+|u|^2) = one possible shrink |
| /M { 1.5 add } bind def % M(u) = (1.5+|u|) = another possible one |
| /W 3.8 inch def % 1/2 width of ellipse |
| /zmax 0.5 nlayers add def % radius at last layer of hexagons |
| /a zmax M W mul zmax div def % a = M(zmax)*W/zmax |
| |
| /Xform { % [x0,y0] = ctr ellipse |
| Matrix transform |
| /y exch def |
| /x exch def |
| /z x dup mul y .773 mul dup mul add sqrt def % ellipse radius |
| /Scale a z M div def % z=a/M(|z|) |
| x Scale mul x0 add % magnify back up |
| y Scale mul y0 add % [x0+x*s, y0+y*s] |
| } def |
| |
| |
| /Helvetica findfont 8 scalefont setfont |
| 4.25 inch 0.5 inch moveto |
| (RHW) stringwidth pop -0.5 mul 0 rmoveto |
| (RHW) show % autograph |
| |
| warp 1 eq { % redefine commands to use Xform |
| /moveto { Xform //moveto} def |
| /lineto { Xform //lineto} def |
| /curveto { |
| Xform 6 -2 roll |
| Xform 6 -2 roll |
| Xform 6 -2 roll |
| //curveto |
| } def |
| }if |
| |
| |
| /bfly { % paint 1 butterfly |
| next_color body |
| 1 setgray eyes |
| stripes |
| 0 setgray nostrils |
| 0.5 setgray thorax next_color |
| spots |
| } def |
| |
| /x0 x4 def % center |
| /y0 y4 def |
| |
| /T1matrix % xlate to center of image |
| x0 neg y0 neg matrix translate |
| def |
| |
| /Smatrix % scale so that 1 basis vector = 1.0 |
| Dm_inv dup matrix scale |
| def |
| |
| /HexCell { % 6 butterflys rotated about center of |
| /cidx 0 def % 6 fold symmetry |
| /color 0 def |
| /T2matrix dx dy matrix translate def |
| 0 60 300 { |
| /angle exch def |
| /Rmatrix angle matrix rotate def |
| /Matrix % translate, rotate, scale - used by Xform |
| T1matrix Rmatrix matrix concatmatrix |
| T2matrix matrix concatmatrix |
| Smatrix matrix concatmatrix |
| def |
| gsave |
| warp 0 eq % then may use usual PostScript machinery |
| { % else using Xform |
| x0 y0 translate angle rotate |
| .5 dup scale |
| dx x0 sub dy y0 sub translate |
| } if |
| bfly |
| next_color |
| grestore |
| } for |
| } def |
| |
| |
| %320 x4 sub 240 y4 sub translate |
| 4.25 inch x4 sub 5.5 inch y4 sub translate |
| |
| |
| 0 setlinewidth |
| /N 2 def |
| N neg 1 N { |
| /i exch def % translate to |
| N neg 1 N { % i*D1 + j*D2 |
| /j exch def % and draw HexCell |
| gsave |
| /dx i D1x mul j D2x mul add def % translate HexCell by |
| /dy i D1y mul j D2y mul add def % [dx,dy] |
| /r2 dx dup mul dy dup mul add def % r^2 = |dx,dy|^2 |
| r2 max_r2 lt % inside radius? |
| { % yes |
| 1 r2 max_r2 div sub sqrt 2 div |
| setlinewidth % make skinnier lines |
| HexCell % 6 butterflies |
| } |
| if |
| grestore |
| } for |
| } for |
| |
| showpage |