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
You can get the plates from this here
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
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.
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 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.
Plate 3 no 2
This has all rows the same, with tiles BC as the repeat in each row.
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.
Plate 3 no 4
This has all rows the same, with all tiles used in the repeat in each row.
Plate 3 no 5
The two rows of the repeat have different double pairs in each row.
Plate 3 no 6
There is only one row repeated here, with each tile in turn forming the repeat in each row.
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.
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.
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.
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.
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.
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.
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:
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.
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.