Typing shapes and combining types
In this paper, we provide a concise introduction to Sebastien Truchet's 17th-century combinatorial approach and Dominique Douat's notation, with the aim of modernising it for computer-based applications and contemporary interpretation.
The purpose of this paper is to create a stable framework for the combinatorial composition method trough typography, from a theoretical and practical point of view.
In this study different fields are involved, such as combinatorics, geometry and decoration, and typography. In order to define a general method, we briefly introduce the work of Sebastien Truchet and focus on the notation introduced by Dominique Douat. Both apply combinatorics to the tiling problem for aesthetic purposes, to identify the more pleasing tessellations rather than seeking new ways to divide the plane.
Finally we explain how Douat notation can be used nowadays with digital fonts and how it can be implemented.
Sébastien Truchet was a 17th-century French Dominican priest whose contributions spanned in different fields, including hydraulics, tilings and typography. Our paper begins with a trip near a canal, leads trough the untiled floor of a church and ends in a virtual typographic chase.
During a trip on the Canal d’Orleans, Truchet sees some ceramic tiles and decides to examine which compositions can be made with such tiles.
In order to be able to form pleasing designs and patterns by arranging these tiles, I first examined the number of ways in which they could be joined together in pairs […]1
In order to create a simple framework, we introduce a new lexicon, give an overview of the passages of the combinatorial method, and shortly discuss each of its passages.
This method is based on the combination of modules into nodes, which are then recombined to create the final compositions. Let's clarify the nomenclature of these elements3 in the Table 1.
Umberto Eco defines Creativity as a new combination of existing elements4 and the Truchet’s study relies on this: through the generation of nodes, he highlights the expressive potential of the modules.
For instance, if we were asked to create sixty designs in a 12x12 grid starting from a single tile, it would seem like a daunting challenge.
However, if we were asked to do the same but were also given an ordered catalog of 16 nodes to work with, the task would seem easier.
Despite the initial conditions remaining unchanged (we are still composing using the same tile), the second task appears much simpler.
Eco also writes:
That creativity is based on combinatorics seems obvious to me, and I have tried to show this in the course of my conversation. But the reasons for the final choice, for the repudiation of unsatisfactory solutions, to stop at the only one that seems perfect, is not yet clear to us.5
A preliminary simple combination of tiles helps us explore new possibilities, but we still need to choose and arrange them to create a final composition.
While Truchet is widely celebrated for his idea to apply combinatorics to decoration, there is another priest whose work is less known: Dominique Douat.
Dominique Douat examines how many and which compositions can be made with the Truchet tile.
He sets a new combinatorial method that uses a type notation for both combining tiles and creating compositions, and he also calculates all the possible compositions that can be made by taking the tiles n by n.6
P = number of permutations with repetitions
n = numbers of modules
k = class of the combination (number of total tiles)
Douat’s notation makes Truchet’s calculations seem obsolete, and his estimation of possible compositions also helps the decorator in terms of predictability of the final output.
Furthermore, he describes four generative rules for the final compositions, based on strings of four characters.
The notation introduced by Douat allows the decorator to compose tilings as a typographer composes his chase.
The chase is an essential component of the letterpress printing process, it is a frame that holds the type characters in place, and serves as a tool both for composing and aligning.
We can relate the motif of a square tile to the letter of a monospaced unicase typeface:
In a unicase typeface, uppercase and lowercase letters have the same height, and there are no ascenders or descenders.
In a monospaced typeface each character takes up the same horizontal space. This uniform character spacing is commonly used in coding, typewriters, and other applications where precise alignment is essential.
Significant advancements have occurred since Douat authored his book, and computers can process strings efficiently nowadays. Douat notation can be easily computed, facilitating both calculations and shape variations.
Any composition that Douat has written in the 18th century can now be typed on a computer, and with the appropriate font his work comes back to life as a vector7.
Moreover, it’s really easy to obtain variations in any composition by simply changing the shape of the font, keeping in mind that the decorator still needs to study the geometrical properties of his variation in order to obtain a coherent result.
We have exposed Douat notation and how to apply his method to computers. This process has different advantages, such as:
efficiency: the process runs really efficiently on the computer
alignment: the tiles result perfectly aligned
file weight: the text files are lighter then a vector file
vectorisation: texts can be easily vectorised with alignment preservation
flexibility: it is really easy to make variation of the tilings
This new approach also opens a new series of questions:
How do we analyse a module’s geometric properties?
How do we manage the complexity of a large numbers of combinations?
By which criteria can we catalogue our nodes?
How do we choose the nodes to compose?
Which generative rules can we use to create pleasant tilings?
For those interested in such questions, some studies can be found in the master thesis Un’infinità di disegni8 (An infinity of Compositions). In this work the combinatorial method has been studied using a binary system (with only two modules: 0, 1), and a new system for cataloguing nodes is established. Furthermore, a higher-level notation is introduced, based on the generating strings trough the conversion of a decimal number into a base-n number, where n represents the number of modules.
This notation was applied in the Blauer Schnipsel project, with the objective of designing textile compositions using industrial waste. For this project, a new cataloguing system was introduced, based on both aesthetics and ecological constraints9.
The string notation has practical application in type design, which is exemplified by its use in the GTL. Thanks to this tool numerous fonts were created during workshops conducted by the Scuola Open Source of Bari.
Finally, as a tribute Dominique Douat we present 256 nodes that he has written but never visually represented. These nodes can be useful to find the centre of orthogonal compositions10 (Fig. 13).
Cyril Stanley Smith and Pauline Boucher; The tiling Patterns of Sebastien Truchet and the Topology of structural Hierarchy. Leonardo, 20(4):373-385, 1987.
Sebastien Truchet, Memoire sur les combinations. Memoires de l’Academie Royale des sciences, Paris, 1704.
Umberto Eco, Combinatoria della creatività, 2004
Dominique Douat, Méthode pour faire une infinité de desseins différents avec des carreaux mi-partis de deux couleurs par une ligne diagonale : ou observations du Père Dominique Doüat, Religieux Carme 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, Académicien honoraire; Printed by Jacques Quillau, Paris, 1722.
Rocco Lorenzo Modugno, Un’infinità di disegni, ISIA Faenza, a.a. 2019-2020.
Adele Buffa, Rocco Modugno, Andrea Righetto; Blauer Schnipsel, co-generating a social fabric, Franzlab, Bolzano 2023.