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Background: I am working on a program that produces design patterns ( using wallpaper- and frieze-group theory ). This is for example a 'generating region' for a frieze or a wallpaper.

   Graphics[Polygon[{{.1, .1}, {.3, .8}, {1, .1}, {.5, .5}}]]

In reality this piece of Mathematica Graphics code is the result of many ( matrix- ) calculations.

Usually I want this in a larger size, for example 1 by 2:

   Graphics[{Polygon[{{.1, .1}, {.3, .8}, {1, .1}, {.5, .5}}], 
    Polygon[{{1.1, .1}, {1.3, .8}, {2, .1}, {1.5, .5}}]}]

Currently I work as follows: ===pseudocode=== follows:

  Map[ CalculateBaseMotif[#1,#2] &, 
       Flatten[Map[# &, Table[{ii, jj}, {ii, 1, lenX}, {jj, 1, lenY}]], 1]

So CalculateBaseMotif is calculated over and over. While all I want is to transtlate the result of

   G=Graphics[Polygon[{{.1, .1}, {.3, .8}, {1, .1}, {.5, .5}}]].

Summarizing: I make a ( complicated ) graphic G requiring many calculations of width W and height H. Then I want to produce a ( final ) graphic like so:

  GGGG
  GGGG

thus having width 4 x W and height 2 x H in the most efficient manner.

Question: How to define and work with temporary graphics data ?

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2 Answers 2

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You can use Translate for this, e.g.

With[{w = 5, h = 4, gr = Polygon[RandomReal[1, {20, 2}]]},
 Graphics[Translate[gr, Tuples[{Range[w], Range[h]}] - 1]]]

Mathematica graphics

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  • $\begingroup$ This will speed up things nicely. $\endgroup$ May 18, 2012 at 12:11
  • $\begingroup$ I like the Tuples iterator. Would have gone for a Table out of old ingrained procedural habit. $\endgroup$
    – Yves Klett
    May 18, 2012 at 12:16
  • $\begingroup$ There's a problem with geometric transformations in graphics: ToBoxes processes them using some recusrive function, so if we try to make more than ~$IterationLimit/2 copies (4096 by default) of an object using e.g. GeometricTransformation, then it will fail. I hit this problem a few days ago, in 3D: Graphics3D@ GeometricTransformation[Sphere[], TranslationTransform /@ N@Tuples[Range[14], {3}]]. Strangely Translate doesn't have the problem... $\endgroup$
    – Szabolcs
    May 18, 2012 at 12:18
  • $\begingroup$ Since Range is Listable you could write: Tuples @ Range @ {w, h} (+1) $\endgroup$
    – Mr.Wizard
    May 18, 2012 at 12:26
  • $\begingroup$ @Szabolcs Do you think this $iterationLimit/2 limit should be reported as a bug? $\endgroup$
    – Jens
    May 18, 2012 at 14:14
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Just for completeness, I checked whether there is a failure due to $IterationLimit if one re-casts the tiling using GeometricTransformation. Although there is indeed a problem in 3D graphics if you tile more than 4096 (by default) 3d objects, as mentioned by @Szabolcs in the comment to Heike's answer, that fortunately doesn't seem to happen for the 2D graphics in this question.

Here is a test where I decided to use the at function from this answer, defined as follows

at[position_, angle_: 0][obj_] := 
 GeometricTransformation[obj, 
  Composition[
   TranslationTransform[position], 
   RotationTransform[angle]
 ]
]

With[
 {
  w = 8, h = 8,
  gr = {Orange, Disk[{0, 0}, {.1, .2}]}
 },
 Graphics[
  Map[(gr // at[#, ArcTan[Sin[#[[1]]]/Cos[#[[2]]]]]) &,
   Tuples[{Range[w], Range[h]}] - 1]]]

tiling

If you increase w and h to 70 both, it takes much longer but still works.

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