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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Regular and semi regular tilings

In the following, tiling means covering the plane without leaving any gaps.

A regular tiling has regular polygons which are all the same. A semi-regular tiling has a mixture of different regular polygons.

Regular tilings

You can tile the plane with equilateral triangles, because each angle is 60¡ and six will fit around a point:

You can also tile the plane with squares, because each angle is 90¡ and four will fit around a point:

Regular pentagons have an angle of 108¡. If you try to tile with regular pentagons, three do not quite fit around a point because 3 x 108 = 324 and four then overlap because 4 x 108 = 432.


With regular hexagons, the plane can again be tiled, because each angle is 120¡ and three will fit around a point


All other regular polygons have angles which are greater than 120¡ and so three polygons will always overlap if you try to fit them around a point.

Semi-regular tilings

Only a limited number of regular polygons can be fitted together to tile the plane so that each vertex has the same tiling pattern around it.

These are called semi-regular tilings and include octagons and dodecagons (12 sided polygons) as well as the triangles, squares and hexagons of the regular tilings.

There are eight tilings in all, or nine if you include the fact that the tiling with triangles and hexagons has two forms which are mirror images of one another.

Squares and triangles tile two ways:


Squares and octagons can also tile:

Hexagons and triangles can tile two ways:

The second way has two versions which are mirror images:


Hexagons can also tile with squares:

Dodecagons (12-sided polygons) can tile with squares and hexagons:

Dodecagons can also tile with triangles:

 

 

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Sebastien Truchet and his tilings

Great advances in the foundations of modern mathematics were made in the seventeenth and eighteenth centuries. While mathematicians like Pascal, Fermat and Leibnitz were studying the new subject of probability, a Dominican priest named Sebastien Truchet published a work which explored the subject in graphical form by means of tilings.

Truchet is also the person who invented the point system to measure character sizes in typesetting which we now use to size characters in word-processors and publishing programs.

He took four simple tiles and looked at the ways to arrange them.

His little book “Memoir sur les Combinasions” was published in 1704. It is a wealth of patterns built up from a simple motif, which you can see here

http://gallica.bnf.fr/ark:/12148/bpt6k3486m.image.f526.langFR

You can get the plates from this here
http://jacques-andre.fr/faqtypo/truchet/truchet-planches.pdf

His method was publicised in a book by Dominique DOUAT in 1722:
Méthode pour faire une infinité de desseins différens avec des carreaux mi-partis de deux couleurs par une ligne diagonale : ou observations du Père Dominique Doüat Religieux Carmes de la Province de Toulouse sur un mémoire inséré dans l’Histoire de l’Académie Royale des Sciences de Paris l’année 1704, présenté par le Révérend Père Sébastien Truchet religieux du même ordre,

This book can be downloaded here
http://jacques-andre.fr/ed/douat.pdf

The tiles look like this:The letters are to identify them. As you can see, they are tiles that fit together in a regular square tiling.

Truchet looked at the combinations and then created designs based on these combinations.

Investigation

If you combine Truchet’s four tiles in pairs, how many combinations are there?

Truchet’s basic designs

The following designs are simple ones from Truchet’s plate 3.

How many different Truchet tiles are used in each one and which ones are they?

The next set from Plate 4 are more complicated designs.

Key to Truchet tilings

As might be expected, the tiles in plate 3 are much easier to analyse than those in plate 4.

Plate 3

Plate 3 tiles are very straight forward repeats, but even so generate a number of different tilings.

Plate 3 no 1

This tiling has two rows in the repetition, of the same set of tiles, but with the second shifted two tiles to the left. The row repeat is of four tiles, one each of A B C and D.

BCADBCADBCAD
ADBCADBCADBC
………
Plate 3 no 2

This has all rows the same, with tiles BC as the repeat in each row.

BCBCBCBCBCBC
………

Plate 3 no 3

This tiling has two rows in the repetition, with different pairs repeating in each row. The second row is the same as the rows in no 2.

ADADADADADAD
BCBCBCBCBCBC
……..

Plate 3 no 4

This has all rows the same, with all tiles used in the repeat in each row.

BCADBCADBCAD
……..

Plate 3 no 5

The two rows of the repeat have different double pairs in each row.

BBCCBBCCBBCC
AADDAADDAADD
……..

Plate 3 no 6

There is only one row repeated here, with each tile in turn forming the repeat in each row.

ABCDABCDABCD
……..

Plate 4

Plate 4 no 1

The line in this pattern is shifted to the left in each successive row. This is highlighted by the bold A.

DCBCDCACDCBC
CBCDCACDCACD
BCDCACDCACDC
CDCACDCACDCB
DCACDCACDCBC
CACDCACDCBCD
……..

Plate 4 no 2

The first line consists of a repeat of ABCD; then the next has each pair swapped (AB becomes BA and CD becomes DC). The pairs (BA and DC) are then swapped and the rule repeats for every two sets of lines. This means that the whole pattern repeats after four lines.

ABCDABCDABCD
BADCBADCBADC
DCBADCBADCBA
CDABCDABCDAB
……..

Plate 4 no 3

The first line consists of a repeat of ABCD; then the next swaps pairs so that BC is followed by DA. The same rule of the positions is then applied on subsequent lines. This means that the whole pattern repeats after four lines.

ABCDABCDABCD
BCDABCDABCDA
CDABCDABCDAB
DABCDABCDABC
……..

Plate 4 no 4

The first line consists of a repeat of ADCB as in the previous one, but then the next line is shifted one position to the right and the subsequent one back one position, and so on.. This means that the whole pattern repeats after two lines.

ADCBADCBADCB
BADCBADCBADC
ADCBADCBADCB
BADCBADCBADC
ADCBADCBADCB
……..

Plate 4 no 5

Although it is easy to see a set of stripes going diagonally across the design, it is the hardest of the set to analyse. Each line has a pair of CC and a pair of DD (if you look at the line wrapping round). In the line below a CC has DC directly underneath and DD has CD. If these are removed, as at the right, then the diagonal stripe appears more obvious, with CB changing to DA and then back to CB. It is then easy to see that there is a stripe where DD goes to CC and back to DD.

DCBCC DD CBCCD     . CB . . . . CB . . .

DADD CC DADDCC     DA . . . . DA . . . .

BCC DD CBCCDDC     B . . . . CB . . . . C

DD CC DADDCCDA     . . . . DA . . . . DA

C DD CBCCDDCBC     . . . CB . . . . CB .

CC DADDCCDADD     . . DA . . . . DA . .

……..

Plate 4 no 6

This is similar to plate 4 No 2 except that the first line consists of a repeat of ABDC; then the next has each pair swapped (AB becomes BA and DC becomes CD). The pairs (BA and CD) are then swapped and the rule repeats for every two sets of lines. This means that the whole pattern repeats after four lines.

ABDCABDCABDC
BACDBACDBACD
CDBACDBACDBA
DCABDCABDCAB
……..

New Truchet tiles

Truchet tiles have been developed in many ways. The following three developments are ideas on ways you can create new tilings. One of these is the route to Morphing tilings.

Development 1

Truchet’s original tiles rely on colouring half a square tile. As a first stage to morphing tiles, what happens when the colouring is removed and only the boundary remains?

The tiles are only of two types now:

This would seem to reduce the possibilities for designing, but converting Truchet designs yields new ones which often look like mazes:

Development 2

Another common way Truchet’s original tiles have been modified is to take them in pairs within the square unit tile and then make variations using the pair.

As with development method 1, there are only two tiles. They can be thought of a pair of the original Truchet tiles together with a pair of empty square tiles.

Development 3

Combining methods 1 and 2 gives a set of two tiles which can be developed a stage further, since the lines do not need to be straight.

The last one of these has been used by many people to create very interesting tilings. It was invented by Cyril Stanley Smith who wrote an article about Truchet in the journal Leonardo, “The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy” Leonardo 20, pp 373-385 1987. Smith was a metallurgist and historian of technology at MIT who used Truchet Tilings to study crystals. One of the examples in the paper used this pair of quarter circle tilings. When other people have used it, the name Truchet has stuck to it.

When coloured these tilings look very much like patterns in nature.

This tiling only needs two colours. Is this always true for a tiling generated this way using the two “quarter circle” tiles?

Other Truchet tilings

These techniques open up a host of ways to create new tilings.

Truchet tilings do not have to be based on squares. They can be any of the regular or semiregular tilings.

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New tiles from old

The Cairo tile

This is a well known pentagonal tiling, frequently known as the Cairo tiling because it was said to be found paving the streets of Cairo. The tiling is with pentagons, although they are not regular pentagons (see regular tilings and why regular pentagons can’t tile).

Now, if we add a regular square tiling on top, we can see that the Cairo tiling fits in the framework in a regular way and could have been tiled using square tiles, suitably marked with a motif.
The tiles are all the same, but their orientation changes with a 90° rotation as you move along each row or each column.
Note how the ends of the lines of the motif on the square tile go to corners of the square and how this means that the lines continue on the next tile. This is the key to making new tiles.

Designing motifs

To make designs like the Cairo tile, we need to make sure that when we design the motifs, the lines at the borders of the framework tile (the square in the case of the Cairo tile) end at the same position when the framework tiles fit together.

Examples of corner endings

The first set of tiling motifs have ends of lines at the corners:

Some of these, because of the symmetry, have only one orientation. This means that there is only one complete tiling. With others like T1, and T3 having two different possible orientations and T4, T5, T6, T7 and T8 having four.

Note that T8 does not have a line at the top right corner, so that the four possible orientations cannot fit together in all cases without there being a line which does not join up. This does not mean that it can’t give interesting tilings with fully joined lines, though.

Examples of endings at a quarter of the sides

The next first set of tiling motifs have lines which terminate a quarter the way along each side from the corners.

All have endings at all quarter positions, except T13.

Examples of endings at the midpoints of sides

The next set of tiling motifs have ends of lines at the mid-points of the sides:

The first one, T17 has a centre of symmetry, but T18, T19, T20 and T21 have two possibilities for orientation, with T22 and T23 having four.

Other variations using a square

The following four also end a quarter of the way along a side, but only at one end of a diagonal.
It is also possible to divide the side up in other fractions, like the following examples showing division of a third.

Examples using other polygons

The same principles apply when using other polygons, although with triangles because there are an odd number of sides, it is much harder to design a motif.

Question
How many orientations are possible when tiling these four motifs?

Tilings using these tiling motifs

Some of the examples show just the use of one motif, while others show the use of mixtures. The mixtures obviously have to use the same division for the line endings. For example the following is a tiling created using T18, T19 and T20.

How the motifs are placed is a whole subject in itself.

From new tiles to morphing patterns

The methods used with Truchet tiles are applicable to changing a tiling, but there are more possibilities using these tiling motifs. See Morphing patterns for developing something completely new.

This can also lead to new types of Islamic patterns.

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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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Using the Morphing Patterns

The following are a few suggestions for new ways to explore patterns.

Morphing patterns are more complex than simply repeating patterns and so offer more possibilities for creativity.

Analysing the patterns

Everyone sees something different in morphing patterns. As the eye wanders over them it is possible to see the changes in different ways. Sometimes you can focus on a local pattern and sometimes you see a movement over a larger area which is hard to pin down. They are thus ideal for teaching analytical skills and being able to put discoveries into words. Seeing what is happening is only the first step in showing someone else what you can see. Do you need diagrams as well as words? Can you express what is happening in more than one way?

Try looking for the changes in some of the tilings we have already explored. Which patterns change in one direction and which change in two directions? Can you identify a motif and how it is changing?

Colouring the patterns

Because the way the lines in the patterns interact, colouring them is not a simple process of deciding on colours to use in one part and then continuing using the same colours. Each pattern presents its own problems. Colouring changes the whole appearance of the morphing and creative use of colour can add many variations.

Some patterns require only two colours, others more. Some can be coloured in two colours in one part and then require three or four elsewhere.

How far can you use colours to analyse the patterns?

Can you colour adjacent polygons to alter the morphing?

The Using the Morphing Tilings downloads post has more information on colouring morphing patterns

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Islamic art and tilings

Islamic art, while including representational images, is highly developed as a decorative art. Their design and use of tilings developed into a very sophisticated form of art over thousands of years. The artisans who developed the tilings worked with mathematicians both to develop new designs and also efficient ways to produce them.

The Persian mathematician Abu’l Wefa (940 to 998) wrote a book describing the discussions that took place.

Islamic patterns as new tiles from old

This is a common Islamic pattern. It is derived from a visual proof of Pythagoras’ Theorem by Abu’l Wefa.


The steps to creating the motif are as follows:
1. Take a square and join corners to the mid-points of the sides as here.
2. Rub out part of the lines to make right angled triangles.

This motif is chiral (it exists in right and left-handed forms) so the design is now created by successive reflections. Thus four together look like this (the blue lines show the lines of symmetry):

If a square tile is made up from reflections of the original motif (as above), then it can be used to build up the pattern.

Investigation

Using the following version of the motif:


1. What is the area of the square in the centre if the large square has side one unit?
2. Why do the lines forming the central square intersect on the circles?

Some more motifs in Islamic tilings

The above motif can be modified in many ways. For example, the red lines show how to construct one of the most common islamic tilings.


Using successive reflections of the red tile above, the pattern can be built up as below:

A slightly different motif:
makes this pattern when reflected:

Can you see how it is tiled to create the pattern? This pattern is often found in tiling pavements. One of the most famous examples of this pattern can be seen at the Taj Mahal in India.

Morphing Islamic Tilings

There are many ways to morph Islamic patterns. The following set of morphings is based on one of the motifs that we have seen before. The central square increases from a point until it becomes the square that is the size of the tile itself:
The following three tilings are all based on this morphed set.

What is the difference between them?
Which is most like the original tiling?
What types of symmetry are there?

The Taj Mahal pavement tile can be morphed from the original star shape to an irregular octagon:

This then gives a morphed tiling like this:

What other types of tilings can you make?

Morphing Stellated Islamic Patterns

Many Islamic patterns are “stellated”. These star shaped designs are often said to have arisen since the mathematicians working with the artisans were astronomers as well. The following method for creating a stellated pattern is ideal for morphing.

Step 1 – Make a triangular grid.

Step 2 – Draw circles at the intersections of the grid. The radius of the circle does not matter but they should all be the same.
Step 3 – Divide the  circumference of the circle into twelve.
Step 4 – Join points in a star shape like this:
Step 5 – Do this for all the circles and then rub out the construction lines.

Here is the stellated pattern above coloured in:

Can you see sets of overlapping hexagons?
What other closed polygons are there?

Morphing the designs

To change the design change the radius of the circle:

When you enlarge the radius, the central “hole” gets larger and gives a lace effect:

But making it a lot larger causes overlaps and the “hole” gets smaller again.


The following magnified part of this design shows it is unlike anything in normal Islamic art and almost looks like basket weaving:

Systematically varying the size of the circle on the same grid, gives a tiling that morphs as it goes down the image:

Investigations

The picture below uses a rhombic grid and only divides the circle into six points:


Investigate making morphing tilings with this system.

The diagonals of each rhombus are in the ratio of 1:2. Investigate the morphing affect of changing the shape of the rhombus.

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