# Multistep Evolution Method to Generate Topological Interlocking Assemblies
## 4 General Mid-Section Evolution
### Fundamental TI Generation Requirement  
Two neighboring polyhedra must have a common planar interface between them. Such an interface is the intersection of the respective faces from both polyhedra that are mutually coplanar. Let \(f_{i},f_{j}\in F\) be two neighboring tiles in a surface tessellation with shared edges represented as half-edges \(ab,ba\) for the respective tile. The tilting angles \(\theta_{ab}^{+}\in\Theta_{0}^{+}\) and \(\theta_{ba}^{-}\in\Theta_{0}^{-}\) (only for the first step from both evolution sequences) must be  
Figure 12: Uniform and non-uniform evolution. Seed polygons are in red. Evolution polygons are in blue. (**a**) uniform evolution (overview), (**b**) uniform evolution (lateral), (**c**) non-uniform evolution (overview), and (**d**) non-uniform evolution (lateral).  
\(\theta_{ab}^{+}=-\theta_{ba}^{-}\). The tilting angles rotate the normal vector associated with the shared edge such that it defines the same tilted plane \(P_{a,b}\) that contains the edge.  
This requirement satisfies the cross-section criteria [2] for TI arrangements. For a planar tessellation composed of square tiles, each tile is the mid-section of the respective block in the assembly. A tile evolves into rectangles as it moves upward toward its normal vector. Eventually, the rectangles collapse into a line segment when the evolving tile reaches the topmost section of the block. Simultaneously, the tile also evolves into rectangles as it moves downwards in the opposite direction of its normal vector, and it will collapse into a line segment when the rectangles reach the bottom-most section of the block. The rectangles moving towards the top of the block are rotated \(\frac{\pi}{2}\) angles concerning the rectangles moving towards the bottom of the block. Figure 13 shows an example of squares evolving into rectangles in an assembly of interlocked tetrahedra. [PERSON] et al. introduced the TI criterion using such an evolution principle [3].  
The polyhedron section contained between its first evolved \(n\)-polytopes, along with both positive and negative directions, contributes to the TI behavior of the resultant block. Such a section contains the faces in contact between the block and its neighbors. Its geometry is an antiprism without both top and bottom faces. This geometry guarantees the cross-section criteria of the block in the resultant assembly. The polyhedron sections defined after the first evolution steps (along with both positive and negative directions) can contribute to the TI behavior if their respective tilting angles are the same as those from the first evolution steps.  
Figure 12a,c show two different polyhedra evolved from the same seed polygon. Both positive evolution sequences are identical. The negative evolution sequences differ in the last step, where the evolving rectangle collapses into a line segment. This change alters the common interface with a neighboring block in the assembly (if any). Furthermore, it breaks the uniformity of the block without sacrificing the TI property. Moreover,the volumes of the polyhedra become different (assuming uniform density). Changes in the evolution step parameters have implications for the distribution of forces required to maintain the static equilibrium of an assembly.  
Figure 13: Squares evolving into rectangles while generating tetrahedra. (**Left**): assembly overview. (**Right**): top view showing the rectangles as cross-sections.