# Multistep Evolution Method to Generate Topological Interlocking Assemblies
## 1 Introduction  
The topological interlocking (TI) principle states that two adjacent blocks interlock if their respective top and bottom evolution sections (cross-sections) degenerate into a line segment or a point. [PERSON] et al. defined this principle [1, 2], and [PERSON] et al. later formalized it [3]. The Platonic solids and uniform antiprisms satisfy the TI principle since we can find polygons that describe their mid-sections with respective evolutions as cross-sections. There are relevant instances of TI assemblies throughout architectural history. [PERSON] discussed the case of topological variation in the flat vault (examples in Figure 1) and its relevance in history [4, 5]. In the author's words, _\"The historical problem is to find a building solution to cover a space with a flat floor constituted by discrete elements: in other words, to build a vault with zero radius stone-ashlar\"_[4]. [PERSON] discussed the flat vault designs from [PERSON] (in 1699) and [PERSON] (in 1704), both considered examples of topological interlocking assemblies centuries before such a term was coined [6]. To generate a TI assembly, we start with a surface tessellation composed of convex tiles with an even number of sides each. A generation method designs an interlocking block per tile. The traditional generation method [3] requires angle parameters that describe the tilted faces of the blocks. A recently proposed method named Height-Bisection [7] introduces height parameters to generate valid, block-aligned TI assemblies even for non-planar surface tessellations. Figure 2 shows the generation methods for generating an octahedron using either the traditional method or the Height-Bisection method. However, both methods can only generate uniform antiprisms and three of the Platonic solids: the tetrahedron, the cube, and the octahedron. We require the post-processing of the blocks to generate other known TI block shapes or clip overlapping, misaligned blocks. For example, to generate a truncated tetrahedron, we need to generate a traditional tetrahedron and then truncate its vertices adequately. Designers and engineers must modify the blocks either manually or computationally. Recent work on the performance of TI assemblies shows that block shapes play a significant role in the integrity of a structure. [PERSON] et al. considered the shapes of the interlocking pieces in their study of the strength and toughness of TI materials [8; 9]. Multiple assembly designs (based on square and hexagon tiles) were 3D-printed and tested. The authors found that increasing the tilting angles of the contact surfaces improved the energy absorption of the TI. [PERSON] et al. considered the relation between the geometry of the blocks and the structural performance of the assembly [10]. The authors considered a variety of planar TI assemblies based on different interlocking geometries. Every configuration was subjected to indentation tests that measured the displacement of the pieces after applying loads. Their results showed that a higher number of tiles and block faces with lower tilting angles reduced the structure's strength. Finally, [PERSON] and [PERSON] described how the geometry of the blocks affected the paths traversing a TI assembly under loads [11]. In this article, we report a generalization of the mid-section evolution concept. It allows for the design of any block with TI properties. We approach the concept by including evolution steps for a given polygon such that its evolution along a direction vector produces the corresponding block vertices. Each step requires a collection of angle and distance parameters that reshape and translate a polygon from one evolution step to another. The evolution parameters are similar, in nature and purpose, to the parameters of the Tilting Angles and Height-Bisection methods. We begin with a discussion of the current work on the TI principle, focusing on the geometry of the blocks. Next, we describe the polygon evolution process, which passes through several evolution steps.