Statement
Academic Study
Team

Hasan Caner Üretmen

Can Koçak
Softwares

Rhyno + Grasshopper

Arduino

Revit
A Generative System For Truchet Tiles
In today's architectural aesthetics, a stochastic appearance is tried to be obtained in various architectural elements such as floor coverings or facade panels. This is usually created by combining many elements in different colors and patterns. Mass production of different parts is costly, and determining how these parts will come together creates a separate work and documentation burden. Truchet patterns, on the other hand, can produce the same effect at a much lower cost by randomly bringing a uniform piece together. In this project, a productive system was created to create different types of truchet patterns and to represent these patterns visually.
Truchet Tiles
The truchet pattern, first described by Sébastien Truchet in 1704, is that a pattern on rotationally unsymmetrical square, triangular or hexagonal pieces comes together to form a larger pattern. This pattern, which is frequently encountered in data visualization and graphic design, also allows a pattern to be formed by randomly rotating one or several types of pieces. Figure 1 shows an example of a pattern created by rotating the same piece 90 degrees to the left in the pattern on the left. In the pattern on the right, the same piece can only be positioned in 2 different ways. However, a highly repetitive and chaotic pattern can be obtained.
Method
Figure 3: A total of 105 variations, including 30 unique ones that can be created on the square with 2 points on each side, and a total of 105 variations, which can be created on the hexagon with one point on each side.
In the first phase of the project, it was investigated what kind of patterns can form Truchet pattern and how these patterns can be generalized mathematically. In order for a tile to form a continuous pattern with other tiles where it randomly juxtaposes, the end point of a line on the tile must coincide with the start point of a line on the other tile. As long as this rule is maintained, patterns can be produced in infinite variations. All the alternate patterns are created by combining the points obtained by dividing all the edges of the selected smooth geometry (triangle, square or hexagon) separately into two pieces with different combinations. Below are the patterns that can be created on square and hexagon according to the mathematical generalization made by David A. Reimann in Figure 3.
Reimann defined the number of variations that can occur according to the number of points on the edge with the f function shown in Figure 4, and the number of variations that can occur according to the number of intersections of the lines with the g function. The value of m, which is given as an input to these functions, is the half of the number of the geometry and the number of points on an edge.
Figure 4: The f function, which gives the variations that may occur according to the number of points on the edge and the edge, and the g function, which gives the intersection number of the curves.
Figure 5: Table of variations that can occur according to the number of points on the edge and edge.
Representation of the Model in Digital Media
Digital Representation of Mathematical Model with Rhinoceros Grasshopper
After the mathematical generalization phase, a code was created to create, visualize and use these patterns electronically. The code created in the Grasshopper environment, which is the parametric modeling plugin of Rhinoceros 5, is based on the mathematical logic described above. The code shown in Figure 7 performs the process of matching the points obtained by dividing each edge of the square into identical parts with random Bezier curves and giving a certain thickness to these curves. By adjusting the tangent of Bezier curves to be perpendicular to the edges of the square, the lines with widths are provided to overlap in sidebyside arrivals.
Figure 7: Grasshopper code
d3
d2
Figure 8: Examples of pattern variations produced with Grasshopper code. The number of dots on the edge is expressed by the variable "d".
Figure 9: The pattern of a pattern randomly juxtaposed at 4 different angles on a square created with d3
Representation of Digital Model in Physical Environment
Creating a Physical Model with Arduino
After the mathematical generalization phase, a code was created to create, visualize and use these patterns electronically. We wanted to combine a pattern selected from products derived from digital representation and a pattern created by it with an interactive façade. In this context, it was aimed to create a physical model by reflecting the data received from the distance sensor with an arduino circuit to the step motor step.
Physical Model Design Principle
Result
In today's architectural aesthetics, a stochastic appearance is tried to be obtained in various architectural elements such as floor coverings or facade panels. This is usually created by combining many elements in different colors and patterns. Mass production of different parts is costly, and determining how these parts will come together creates a separate work and documentation burden. Truchet patterns, on the other hand, can produce the same effect at a much lower cost by randomly bringing a uniform piece together. In this project, we set out to create a productive system for creating different types of truchet patterns and to represent these patterns visually, with the help of Rhinoceros and its additive Grasshopper, we were able to produce infinite probability patterns through the algorithm we determined. The depth of the subject has enabled us to understand the number of options we can obtain with our understanding of the mathematical solution of the problem and the variety of products it offers. Because the products that we can obtain / obtain are defined geometrically. This removes us a limitation on materials and structural elements. For example, the patterns that can be obtained can be used in various architectural structural element surface patterns (ceramic coating) or can be included in the construction structure (pavilion carrier system or curtain wall).
References
[1] Reimann, David A. “Decorating regular tiles with arcs.” Bridges Coimbra:
Mathematics, Music, Art, Architecture, Culture (2011): 581584.
[2] Krawczyk, Robert J. “Truchet tilings revisited.” Proceedings of ISAMA 2011
(2011): 69.
[3] “Grand Central, Le Centquatre, Paris.” SWWS Presents the Work of Designer
Sebastien Wierinck / Sebastien Wierinck WorkShop  OnSite Studio / Grand
Central, Le Centquatre, Paris. N.p., n.d. Web. 04 June 2017.
[4] ‘‘Satin  Dijital Fabrications  Architectural Material Techniques’’ / Lisa Iwamoto
[5] https://en.wikipedia.org/wiki/Truchet_tiles
[6] http://www.cameronius.com/games/truchet/
[7] http://www.grasshopper3d.com/video/truchettiles