wallis@weitek.WEITEK.COM (08/16/89)
% 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