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- %!
- % 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
- /eschersave save def % prevent left over side effects
- %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
- clear cleardictstack
- eschersave restore
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