Steps to Morphing tilings

Morphing tilings are sets of tilings which change from one tiling pattern to another. This can happen in many ways:

  • as a related set of tilings which change like an animation
  • as tilings where the change is seen as you move around the page

To create them you need to go through the following stages:

  1. take a regular tiling
  2. add a motif to each tile to create a new tile
  3. change the motif to create a morphing tiling

This is an example of just one type:

Take a square grid:

Add a motif:


Create a set of motif changes:


Apply the changes to the tiling and remove the guiding square tiling:


The motif can change in a number of other ways too:

  • the motif can be replaced
  • by a variation of the motif (for example a rotation)
  • by replacing it with another related motif
  • the motif can be distorted by transforming or positioning the points.

Combining these with the way in which the motifs are mathematically related when they are changed gives rise to millions of new tilings.

The following are some tiling examples, and there are more in subsequent posts

Example 1

This is a replacement morphing. There are four tiles, based on four orientations of a motif:

This set of tilings is based on a Truchet Tiling and the four tiles are available as a Truetype font.

Replacement is then made cyclically:

A B C D becomes B C D A becomes C D A B becomes D A B C.

So the set of tilings is:

Questions

Why are the first and third a similar looking pair, and second and fourth also?
How are the pairs similar?
What other possibilities could you use as a replacement?

Example 2

This is a different type of replacement morphing. There are three types of motif, each of which can exist in two orientations. In the replacement, each tile is substituted by an equivalent tile showing a different motif.


Example 3

This is an example of a tiling where the motif is changed as the tile is placed.

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4 Responses to Steps to Morphing tilings

  1. Jeannine Carson says:

    I love this blog! it has been very helpful as I am a painter who is beginning to use tessellation in my work. Is there a book you could recommend on this subject? Or a place to buy the templates?

    • John Sharp says:

      Hi Jeannine

      I don’t know that this is published in books. I think Escher did one.

      When I transferred the file to WordPress, I was not able to upload the font files which are mentioned in a later post.
      They are Truetype, and I am not sure if Macs can handle them. If so I can email them.
      I generated most actual patterns by programming.
      Let me know what software you use and how skilled you are in vector drawing.

      John S

      • Jeannine Carson says:

        I am currently looking into what software I would need to design these type of patterns. Photoshop? Also very broad palette is necessary for me as a painter. Any suggestions how to start with software for vector drawing( beginner)?

      • John Sharp says:

        I suggest you use Inkscape. You need to setup a grid and make sure you snap to the corners you need.
        I would then use Photoshop or the Gimp (free software as is Inkscape) to fill with the colours you want after exporting.
        If you have trouble, please come back to me

        John S

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