Partition images into hexagons ?

Questions
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1

I am trying to partition images into hexagons. Is that possible in Mathematica?

2

The basic Image format is rectangular. ImagePartition only partitions Images into rectangular portions. You could however presumably make use of ImagePartition with various sizes and offsets to create a set of square crops at hexagonal grid points. Then these could be masked to remove the areas outside of the hexagonal region in each–leaving those areas transparent. This then could be used as an in-effect hexagonal partition. Of course the devil is in the details…

3

It is a bit not clear what you mean. Say you mean a simple overlap…

h[x_, y_] := Polygon[Table[.1 {Cos[2 Pi k/6] + x, Sin[2 Pi k/6] + y}, {k, 6}]]
hex = Graphics[{EdgeForm[Directive[Black, Thick]], FaceForm[], LightBlue, 
   Table[h[3 i + 3 ((-1)^j + 1)/4, Sqrt[3]/2 j], {i, 3}, {j, 10}]},  ImageSize -> 512]

sdf34rqefdavadfv.png
sdf34rqefdavadfv.png512Ă—513 14 KB

ImageMultiply[{ExampleData[{"TestImage", "Lena"}], Rasterize@hex}]

43reqfadvcvadfv.jpg
43reqfadvcvadfv.jpg512Ă—512 115 KB

Or maybe you’d like something (adopted from here ) fancier…

Manipulate[SeedRandom[1];
 Graphics[{EdgeForm[Black], Texture[ExampleData[{"TestImage", "Lena"}]], 
   GeometricTransformation[#, 
      Composition[TranslationTransform@RandomReal[.1 s {-1, 1}, 2], 
       RotationTransform[s RandomReal[{-Pi, Pi}], Mean@First[#]]]] & /@
     Cases[Normal@hex, 
     Polygon[p_, ___] :> Polygon[p, VertexTextureCoordinates -> p], Infinity]}], {s, 0, 1}]

sadf5w4y6juritkjhfgdnbfdv.gif

4

The solutions proposed by Vitally are very interesting
A question : We have only two level of hexagones.
Do you have a simple solution for 3, 4 or 5 levels?

5

I am not sure what “3, 4 or 5 levels” would look like - is there a link to a picture somewhere?

6

This model is the theory of Christaller or Lösch to explain hierarchy of cities
An image illustrating these theories:

Captured’écran2014-11-14à17.18.19.png
Captured’écran2014-11-14à17.18.19.png390×410 31.5 KB

7

If one can build such a figure from polygons in Mathematica, then it is easy to use the same procedure - just this simple line will make texture mapping (partitioning):

Cases[ ..... , Polygon[p_, ___] :> Polygon[p, VertexTextureCoordinates -> p], Infinity]

no matter what complex arrangement of polygons we deal with.

8

Thanks for you reply