3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments

Abstract

Concave polyhedra of the second sort (abbreviated: C-II-n) constitute a group of polyhedra formed over regular n-sided base polygons and having a deltahedral lateral surfaces. This group includes: concave cupolae, concave pyramids and concave antiprisms of the second sort (CC-II-n, CP-II-n and CA-II-n, respectively). The common feature of these solids is that their lateral surfaces consist of a double row of equilateral triangles which can be assembled in two ways, making two different solids’ heights: major (C-II-nM) and minor (C-II-nm). The geometrical regularities and a high level of symmetry that characterizes these polyhedra, makes them suitable for joining and combining, so they can be arrayed infinitely in space, in x, y and z direction forming 3D lattice structures. For some representatives of these solids, the congruity of their lateral deltahedral surfaces occurs, so 3D tessellations are formed. In this paper, we focus on a single "layer" of such a structure, a panel-like 3D lattice. It is generated by multiplication of the chosen unit cell – the selected C-II-n representative – along the x-y directions. In the z direction the lateral surfaces form a deltahedral structure which makes the thickness of the panel. The method we used is based on the continuous connection of the edges of the two adjacent units, by joining relevant vertices. When we remove the base polygons, the unit cells become hollow, so they can create a honeycombed structure, more desirable for the purpose of application. Then, observed in 2D, by applying symmetry transformations, we form patterns similarly to the formation of wallpaper groups. In this way, we get visually interesting patterns in 2D, which transform into 3D lattice depending on the viewing angle. The thickness of the panel can be halved in some cases, so we get a thinner structure with "face" and "back", having different tessellations of polygons appearing on them. As an artistic intervention, these panels can be modified by joining deltahedral surfaces of other C-II-n onto the compatible faces, whereby we add another layer of patterns to the resulting structure. 3D patterns and lattices are currently experiencing real boom in the design and industry, thanks to the 3D printing capabilities. As for architecture, they can be applied not only as an element of ornamentation, but also as a functional component of the project, especially concerning climate responsive facades. Due to the simplicity of the geometry of C-II-n, such 3D structures are feasible and easy to perform in terms of production and assembly. They are achievable not only with 3D printing, but can also be manually assembled or folded like origami, which allows the use of a much wider range of materials.