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Figure 8. TIA material system based on tetrahedra, architectural speculations [26].

Figure 8 TIA material system based on tetrahedra, architectural speculations [26].

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Figure 1. Topological interlocking assembly (TIA) principle: (a) elements’ identical configuration and mutual arrangement; (b) different variations of alternating rotation of elements’ contact faces. The second potential engages the TIA constituent geometric relationships. A TIA : defined by an identical configuration of its elements, their mutual arrangement, and boundary constraint that holds the whole assembly together [14] (Figure la). Element configuration necessitates the alternating rotation of their contact faces to generate th interlocking principle (Figure 1b). In this way, the arrangement of elements in TIA impose ocal kinematic constraints, uniquely keeping each element locked in its position solel by the geometrical constraints of its adjacent elements. This distinctive feature, excer for the elements at the assembly’s periphery, prevents the removal of each element b either upward, downward, sideways, or rotational displacement by its neighbors [14 n short, a TIA works as a system of geometries and geometric interdependencies fror which the assembly’s form emerges. This approach parallels computational architectur which treats form as an emergent outcome of a system. Computational design processe abandon the traditional Modernist linear approach where the form and its aesthetic dictat structure and material choices. Instead, they synthesize the form from spatial, structure and material principles [15]. Thus, the focus is not on the manifested form, that whic appears, but on the material system from which the form emerges. The material syster is an interaction between internal rules that embed the form and external morphogeneti pressures activated by contextual feedback [16]. The computational architectural theoretic discourse lacks recognition of the link between TIA and the material system, representin the second research gap that this paper addresses.
Figure 1. Topological interlocking assembly (TIA) principle: (a) elements’ identical configuration and mutual arrangement; (b) different variations of alternating rotation of elements’ contact faces. The second potential engages the TIA constituent geometric relationships. A TIA : defined by an identical configuration of its elements, their mutual arrangement, and boundary constraint that holds the whole assembly together [14] (Figure la). Element configuration necessitates the alternating rotation of their contact faces to generate th interlocking principle (Figure 1b). In this way, the arrangement of elements in TIA impose ocal kinematic constraints, uniquely keeping each element locked in its position solel by the geometrical constraints of its adjacent elements. This distinctive feature, excer for the elements at the assembly’s periphery, prevents the removal of each element b either upward, downward, sideways, or rotational displacement by its neighbors [14 n short, a TIA works as a system of geometries and geometric interdependencies fror which the assembly’s form emerges. This approach parallels computational architectur which treats form as an emergent outcome of a system. Computational design processe abandon the traditional Modernist linear approach where the form and its aesthetic dictat structure and material choices. Instead, they synthesize the form from spatial, structure and material principles [15]. Thus, the focus is not on the manifested form, that whic appears, but on the material system from which the form emerges. The material syster is an interaction between internal rules that embed the form and external morphogeneti pressures activated by contextual feedback [16]. The computational architectural theoretic discourse lacks recognition of the link between TIA and the material system, representin the second research gap that this paper addresses.
In the first step, students use physical scale models to expand their understanding of the geometries of individual elements, the overall assembly, and their interdependencies (Figure 2). Here, the TIA is approached as a spatial arrangement that is concurrently a whole subdivided into parts and propagation of parts that generate a whole. Students, thus, have to concurrently understand and define geometric rules that govern the relationship between individual elements and the procedural rationality of the assembly as a whole.
In the first step, students use physical scale models to expand their understanding of the geometries of individual elements, the overall assembly, and their interdependencies (Figure 2). Here, the TIA is approached as a spatial arrangement that is concurrently a whole subdivided into parts and propagation of parts that generate a whole. Students, thus, have to concurrently understand and define geometric rules that govern the relationship between individual elements and the procedural rationality of the assembly as a whole.
Figure 3. Architectural speculations of TIA material systems based on (a) tetrahedra [26], (b) octahe- dra [27].
Figure 3. Architectural speculations of TIA material systems based on (a) tetrahedra [26], (b) octahe- dra [27].
The specific input variations investigated included the surface curvature (planar, single curvature, and double curvature) (Figure 4), the angle of the contact face rotation, and trimming planes’ positions. Students recorded outputs through cataloguing to assess the emergent forms and the material system’s versatility and limitations. Figure 4. TIA material system of octahedra catalogue: variations of the surface curvature inputs [27]. 4.1. Case Study 01: TIA of Tetrahedra
The specific input variations investigated included the surface curvature (planar, single curvature, and double curvature) (Figure 4), the angle of the contact face rotation, and trimming planes’ positions. Students recorded outputs through cataloguing to assess the emergent forms and the material system’s versatility and limitations. Figure 4. TIA material system of octahedra catalogue: variations of the surface curvature inputs [27]. 4.1. Case Study 01: TIA of Tetrahedra
Figure 5. TIA of tetrahedra: (a) the Abeille flat vault patent; (b) TIA of regular tetrahedra.
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 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 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 « in one direction and —« in the other. Conversely, for the type B tetrahedra, this rotation was reversed (Figure 7).
Figure 7. TIA material system based on tetrahedra, the second procedural logic diagram. 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 « in one direction and —« in the other. Conversely, for the type B tetrahedra, this rotation was reversed (Figure 7).
Figure 9. TIA material system based on tetrahedra, the third procedural logic diagram. The third procedural logic defined the tetrahedron based on planar curves derivec 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 « and —«, respectively. The intersectior of the rotated lines, line a’ and line a”, is Point A. Adjacent Point As defined lines b. Line « was also rotated by 90° around its mid-point in the plane normal to the plane of curve « The rotated line, line a’”, was translated to line start and end points as lines c. Line c wa: also translated to point A as line d. Lines a and d make the tetrahedron type A (green) and lines b and c make the tetrahedron type B (violet) (Figure 9). The TIA principle wa: retained for planar or single-curvature input surfaces but not for double-curvature ones.
Figure 9. TIA material system based on tetrahedra, the third procedural logic diagram. The third procedural logic defined the tetrahedron based on planar curves derivec 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 « and —«, respectively. The intersectior of the rotated lines, line a’ and line a”, is Point A. Adjacent Point As defined lines b. Line « was also rotated by 90° around its mid-point in the plane normal to the plane of curve « The rotated line, line a’”, was translated to line start and end points as lines c. Line c wa: also translated to point A as line d. Lines a and d make the tetrahedron type A (green) and lines b and c make the tetrahedron type B (violet) (Figure 9). The TIA principle wa: retained for planar or single-curvature input surfaces but not for double-curvature ones.
Figure 10. TIA material system based on tetrahedra as a geodesic dome, parametric variations [31].
Figure 10. TIA material system based on tetrahedra as a geodesic dome, parametric variations [31].
Figure 11. TIA material system based on tetrahedra as a geodesic dome, physical working model [31]
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 12. TIA material system based on tetrahedra as a geodesic dome, conceptual proposals [31].
Figure 13. (a) Truchet flat vault; (b) TIA of intersecting cones, truncated top and bottom views. 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 13. (a) Truchet flat vault; (b) TIA of intersecting cones, truncated top and bottom views. 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].
= The first procedural logic started with a grid of intersecting cones of radius r anc height H (Figure 14a). The grid checkered pattern defined cone types A (light green) and I (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 se 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 materia system retained the TIA principle for planar or single-curvature input surfaces but not fo: double-curved ones (Figure 15a). As the final step, students developed design proposal: speculating architectural applications of the material system (Figure 15b).
= The first procedural logic started with a grid of intersecting cones of radius r anc height H (Figure 14a). The grid checkered pattern defined cone types A (light green) and I (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 se 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 materia system retained the TIA principle for planar or single-curvature input surfaces but not fo: double-curved ones (Figure 15a). As the final step, students developed design proposal: speculating architectural applications of the material system (Figure 15b).
Figure 15. TIA material system based on intersected cones, the first procedural logic: (a) single-curved surface [33]; (b) architectural speculations [33].
Figure 15. TIA material system based on intersected cones, the first procedural logic: (a) single-curved surface [33]; (b) architectural speculations [33].
Figure 16. TIA material system based on intersected cones, the second procedural logic: (a) grid cells defining the cone base as regular and irregular polygons [33]; (b) an irregular grid [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).
Figure 16. TIA material system based on intersected cones, the second procedural logic: (a) grid cells defining the cone base as regular and irregular polygons [33]; (b) an irregular grid [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).
Figure 17. TIA material system based on intersected cones, the second procedural logic: (a) on a planar surface [33]; (b) on a double-curved surface [33].
Figure 17. TIA material system based on intersected cones, the second procedural logic: (a) on a planar surface [33]; (b) on a double-curved surface [33].
Figure 18. TIA material system based on octahedra, the first procedural logic based on two fields of parallel triangle diagrams.
Figure 18. TIA material system based on octahedra, the first procedural logic based on two fields of parallel triangle diagrams.
Figure 19. TIA material system based on octahedra, the second procedural logic based on two fields of parallel triangle diagrams.
Figure 19. TIA material system based on octahedra, the second procedural logic based on two fields of parallel triangle diagrams.
Figure 20. TIA material system based on octahedra, the third procedural logic based on rotated planes diagrams [27]. The third procedural logic started with a hexagonal grid defining a rotated plane field. Each plane was rotated around a grid cell side by the same angle « but in a direction opposite to the rotation of its neighboring planes. The intersection of the six rotated planes generated an octahedron (Figure 20). The resulting material system retained the TIA principle for all surface curvatures.
Figure 20. TIA material system based on octahedra, the third procedural logic based on rotated planes diagrams [27]. The third procedural logic started with a hexagonal grid defining a rotated plane field. Each plane was rotated around a grid cell side by the same angle « but in a direction opposite to the rotation of its neighboring planes. The intersection of the six rotated planes generated an octahedron (Figure 20). The resulting material system retained the TIA principle for all surface curvatures.
Figure 21. TIA of osteomorphic blocks: (a) single block and assembly; (b) translated as two element types with planar faces [25]. The osteomorphic block was invented for wall and column construction in high seismic zones [35]. Similarly to Truchet voussoizr, it actualizes the interlocking principle as multiple alternations of concave and convex surfaces on the voussoir’s horizontal contact faces. Their fabrication, thus, requires six-axis CAD/CAM tools. The running bond construction sequence enables the TIA principle. Its vertical contact faces are planar (Figure 21a). The limited funds in the academic year 2017/2018 focused the study unit investigations on affordable materials and fabrication. The materials available were sheet materials like paper, cardboard, and plastic, as well as any material that fit within the study unit’s limited budget. The available fabrication tool was a laser cutter. The assembly geometry investigated was a TIA of osteomorphic blocks.
Figure 21. TIA of osteomorphic blocks: (a) single block and assembly; (b) translated as two element types with planar faces [25]. The osteomorphic block was invented for wall and column construction in high seismic zones [35]. Similarly to Truchet voussoizr, it actualizes the interlocking principle as multiple alternations of concave and convex surfaces on the voussoir’s horizontal contact faces. Their fabrication, thus, requires six-axis CAD/CAM tools. The running bond construction sequence enables the TIA principle. Its vertical contact faces are planar (Figure 21a). The limited funds in the academic year 2017/2018 focused the study unit investigations on affordable materials and fabrication. The materials available were sheet materials like paper, cardboard, and plastic, as well as any material that fit within the study unit’s limited budget. The available fabrication tool was a laser cutter. The assembly geometry investigated was a TIA of osteomorphic blocks.
Figure 22. TIA material system of osteomorphic block fabrication: (a) laser-cut, plastic sheet moulds with folding perforations [25]; (b) laser-cut MDF frames increase moulds’ rigidity during casting [25]. plaster’s viscosity and the faces’ rotation angle, stress concentration at the sharp edges, and the length of the casting and drying processes (Figure 23).
Figure 22. TIA material system of osteomorphic block fabrication: (a) laser-cut, plastic sheet moulds with folding perforations [25]; (b) laser-cut MDF frames increase moulds’ rigidity during casting [25]. plaster’s viscosity and the faces’ rotation angle, stress concentration at the sharp edges, and the length of the casting and drying processes (Figure 23).
Figure 24. TIA material system of planar osteomorphic blocks, second procedural logic: (a) single block and assembly [25]; (b) fabricated blocks did not achieve interlocking [25]. An additional procedural logic investigated a single element type with a configuration of no planar faces to increase its degree of interlocking (Figure 24a). Although theoretically, the configuration achieved a more optimal interlocking, the added complexity reduced the precision of its fabrication. The result was unavoidable warping and imperfections of elements’ contact surfaces that prevented interlocking, the principle it was trying to optimize (Figure 24b).
Figure 24. TIA material system of planar osteomorphic blocks, second procedural logic: (a) single block and assembly [25]; (b) fabricated blocks did not achieve interlocking [25]. An additional procedural logic investigated a single element type with a configuration of no planar faces to increase its degree of interlocking (Figure 24a). Although theoretically, the configuration achieved a more optimal interlocking, the added complexity reduced the precision of its fabrication. The result was unavoidable warping and imperfections of elements’ contact surfaces that prevented interlocking, the principle it was trying to optimize (Figure 24b).
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