## 4 Computational Design Workshop
### Case Study 01: TIA of Tetrahedra  
The TIA of tetrahedra investigation was particularly interesting due to its historical precedent, the Abeille flat vault, which predated the notions of a TIA, computational architecture, and a material system for several centuries. [PERSON] patented the flat vault in 1699 to provide a stone ceiling for the lower storey and pavement for the upper storey with identical voussoirs [29] (Figure 5a). The unique voussoir geometry and the rotation of neighboring voussoirs by ninety degrees allow for a simple method of their mutual arrangement. Each voussoir is carried by two neighboring ones through its protruding cuts while supporting two others on its sloped cuts [21]. The Abeille flat vault is a TIA of regular tetrahedra planarly truncated at their center and some distance below it (Figure 5b). Throughout history, the Abeille Vault has rarely been used due to the high amount of horizontal thrust that required the demanding construction of boundary constraints, like buttresses or massive walls [30]. Students' investigations developed four distinct procedural logics to formulate a material system, leading to unique outcomes, limitations, and diverse architectural applications. The first procedural logic derived the tetrahedron from the diagonals of a cube's top and bottom faces and generated the assembly as an array of tetrahedra (Figure 6). Thus, the procedural logic solely dealt with the assembly as the propagation of parts to create the whole and did not address the assembly as a whole subdivided into parts. The actualization of the resultant material system was limited to a planar surface and regular square gird. Figure 4: TIA material system of octahedra catalogue: variations of the surface curvature inputs [27]. The second procedural logic derived tetrahedra from a rectangular grid on a planar surface using a checkered pattern and rotated lines. The checkered pattern defined two types of tetrahedra, A and B. Each side of every rectangular grid cell had a line rotated around its midpoint. For type A tetrahedra, lines on opposite sides of the rectangle were rotated by an angle \(\alpha\) in one direction and \(-\alpha\) in the other. Conversely, for the type B tetrahedra, this rotation was reversed (Figure 7). Following the checkered pattern, type A and B tetrahedra arrayed alternately in the X and Y directions on the planar surface. The procedural logic and algorithm were derived and validated through hands-on investigations using physical working models at different scales. The TIA principle was retained for planar or single-curvature surfaces but not for double-curved ones. Finally, students developed several proposals that speculate the architectural applications of the TIA material system (Figure 8). Figure 5: TIA of tetrahedra: (**a**) the Abeille flat vault patent; (**b**) TIA of regular tetrahedra. Figure 6: Diagram of TIA material system based on tetrahedra, the first procedural logic based on the diagonals of a cube’s top and bottom faces. Figure 7: TIA material system based on tetrahedra, the second procedural logic diagram. The third procedural logic defined the tetrahedron based on planar curves derived from the contouring of a planar or single-curvature surface. The resultant curves, curve c, were approximated with equal linear segments, line a. For each even curve in the array, segmenting was shifted by half the length of line a. Next, each line a was rotated in the curve plane around its end and start points for angle \(\alpha\) and \(-\alpha\), respectively. The intersection of the rotated lines, line a\({}^{\prime}\) and line a\({}^{\prime\prime}\), is Point A. Adjacent Point As defined lines b.