## 4 Computational Design Workshop
### Case Study 02: TIA of Intersecting Cones  
The TIA of intersecting cones investigation also included a historical precedent, the Truchet flat vault. The Truchet flat vault was developed in 1704 to improve the Abeille vault (Figure 13a). Truchet voussoirs actualized the alternating contact faces' rotation of the Abeille voussoirs as undulations of concave and convex double-curved surfaces. Thus, the Truchet voussoir configuration was challenging to fabricate until the introduction of contemporary six-axis CAD/CAM tools [32].  
Figure 11: TIA material system based on tetrahedra as a geodesic dome, physical working model [31].  
Figure 12: TIA material system based on tetrahedra as a geodesic dome, conceptual proposals [31].  
Figure 10: TIA material system based on tetrahedra as a geodesic dome, parametric variations [31].  
Figure 13: (a) Truchet flat vault; (b) TIA of intersecting cones, truncated top and bottom views.  
For the same reason, the design workshop task asked students to translate concave and convex surfaces into single-curvature surfaces and derive them from the intersection of cones (Figure 13b). Students' interpretation of the underlying geometric logic of the TIA of intersecting cones resulted in two procedural logics for a material system.  
The first procedural logic started with a grid of intersecting cones of radius r and height H (Figure 14a). The grid checkered pattern defined cone types A (light green) and B (dark green). The center C of each cone was at the distance r in the x and y directions from its neighbors. Each A cone intersected four adjacent B cones and was tangent to the next set of A cones. The same applied to B cones. Next, each A cone was trimmed by two B cones in the x direction (Figure 14b) and trimmed by two B cones in the y direction (Figure 14c). Finally, a set of trimming planes, as the offset of the original grid, truncated each cone at its mid-height H (Figure 14e) and some distance below it (Figure 14f). The resulting material system retained the TIA principle for planar or single-curvature input surfaces but not for double-curved ones (Figure 15a). As the final step, students developed design proposals speculating architectural applications of the material system (Figure 15b).  
The second procedural logic started with a square grid. Each side of a grid cell was a tangent that defined a circle, the middle of each cone (Figure 16a). In this way, the grid cell  
Figure 14: TIA material system based on intersected, truncated cones, the first procedural logic: (**a**) array of cone bases of radius r and heights H, (**b**) cone type A (light) trimmed by cones type B (dark) in x-direction, (**c**) cone type A (light) trims cones type B (dark) in y-direction, (**d**) intersecting cones TIA, (**e**) trimming plane at cones’ mid-height, (**f**) intersecting truncated cones TIA.  
Figure 15: TIA material system based on intersected cones, the first procedural logic: (**a**) single-curved surface [33]; (**b**) architectural speculations [33].  
sides remained tangents that defined a closed curve even for irregular grid cells (Figure 16b). The rest of the algorithmic sequence followed the first procedural logic: the checkered grid pattern defined cone types A and B; each A cone was trimmed by two B cones in the x direction and trimmed by two B cones in the y direction; two planes, grid offsets, truncated the cones at their mid-heights and some distance below it. Since the geometric rules of this procedural logic allowed for the deformation of the cones, the resulting material system retained the TIA principle for all surface curvatures (Figure 17).