Morphing Patterns

When you create normal patterns, you place multiple versions of a motif in a regular way. The variety of patterns produced in this way is extraordinary as you can see everyday by looking at textiles, carpets, wallpapers and tiled floors.

Morphing patterns add a whole new dimension by making the motif change or “morph” as it is repeated. There are many ways to morph a particular motif and each of these can be combined in many ways. These pages show some of the possibilities.

The technique is to keep the points at the edge of the tile fixed but to move points within the tile in a regular way.

Creating a simple morphing
This is how to create a simple morphing using the motif T2 in the New Tiles from Old post. It has corner endings. The central square can be morphed from centre of the tiling square (in black) to the corners of the square like this:

This morphing is a regular, linear expansion of the square, that is to say that the corners of the square part of the motif are spaced at regular intervals along the diagonals of the tiling square.

To draw them, use squared paper or graph paper with 10 squares for each tile.

A set of these morphed tiles is in the Downloads post to cut out for assembly into a morphing tiling.

This is a tiling where the morphed motif is arranged in rows using the order

T2a T2b T2c T2d T2e T2f

Colouring the tiling

This tiling requires at least three colours. This colouring highlights the change in size of the squares:

This colouring also uses three colours, but the emphasis is totally different.

Click here to find out more about colouring and using morphing tilings to explore the four colour theorem. (See post below for more.)

Different arrangements
The rows do not have to be repeated. You can morph down as well as across:

The order of the morphed motif is shifted one along in each row:

T2a T2b T2c T2d T2e T2f
T2b T2c T2d T2e T2f T2a
T2c T2d T2e T2f T2a T2b
T2d T2e T2f T2a T2b T2c
T2e T2f T2a T2b T2c T2d
T2f T2a T2b T2c T2d T2e

This is one three colour version of the tiling:

Random placings

Although a morphing tiling relies on the motif changing across the page, you can place the morphed motifs randomly to create a tiling. With six variations, throw a die to decide which one to place.

The following tiling has been created using this random set of the morphed motifs:

T2c T2e T2e T2d T2c T2e
T2a T2d T2e T2d T2a T2f
T2b T2d T2e T2b T2d T2c
T2e T2d T2c T2f T2c T2f
T2e T2f T2d T2d T2e T2a
T2a T2b T2c T2d T2d T2b

It can still be coloured in using only three colours:

Other examples of morphed motifs
These motifs are morphed using regular changes. There are more examples below using changes which are not regular, but changed by non-linear functions.

The swastika motif
This is motif T21 from the New Tiles from Old page. It has corner endings and four-fold rotational symmetry, but also has chirality (occurs in left and right handed forms):

The morphing for the left-handed version flattens the swastika shape into a cross. This could be continued all the way to the right handed form.

Using a mixture of alternate left and right handed versions, different morphing tilings can be produced.

(pictures missing in following)

This one is composed of columns of the morphing T21a to T21f with a complete reversal back to the right handed form.

This one starts and ends with T21f, but what happens in between?

These two tilings are interesting because if they are coloured in an illusion is produced. The first one appears to be curved:

The second one has curvature at either ends, but is not as pronounced.

The Y motif

Motif T8 on the New Tiles from Old page has an axis of symmetry on one diagonal of the square, and can be placed four ways. Which means it can be placed in many different combinations, like these four.

Can you see how the one on the right can be used to create the Cairo tiling?

One way to morph it is to move across a diagonal:

Investigate your own morphing tilings

Look at other motifs on the on the New Tiles from Old post, and work out ways in which they can be morphed.

Not all the motifs can be morphed.

Make a table to show which can and cannot.

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