Imagining and designing complicated topologies, such as multi-scale lacunar objects, is challenging. Generally, designers generate these structures from ad hoc algorithms with limited options for parametrization. Some models, like Boundary Controlled Iterated Function Systems (BC-IFS), provide a complete formalism for describing and controlling both their topology and geometry. However, encoding the topology involves abstract knowledge and can be tedious to do manually. On the other hand, polyhedral crystallographic circle packings provide a complete construction of fractal shapes arising from polyhedra. The geometry of such packing relies on circles, but the simple input polyhedron entirely determines their topology. This article shows how the BC-IFS model can encode the topology of fractals obtained from polyhedral crystallographic circle packings. Our method automatically deduces the topological BC-IFS constraints directly from the polyhedron’s structure. As a result, polyhedra serve as an intuitive topological editor for creating multi-scale lacunar polygons. Finally, BC-IFS allows the end-user to modulate the global aspect using control points and refine local details with subdivision points. • Design of a new class of 2D fractal structures with an editable shape. • Polyhedra can encode the topology of the fractal structures. • Polyhedra are intuitive tools for constructing complex fractal shapes automatically. • Simple rules ensure valid connections in the assembly of the fractal structures.

Abstract. In this paper, we investigate representations of links that are either centrally symmetric in [Formula: see text] or antipodally symmetric in [Formula: see text]. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [L. Montejano, J. L. Ramírez Alfonsín, and I. Rasskin, SIAM J. Discrete Math., 36 (2022), pp. 1551–1566], we are able to present sufficient combinatorial conditions for a link [Formula: see text] to admit such representations. The latter naturally provide sufficient conditions for [Formula: see text] to be amphichiral. We also introduce another (closely related) method yielding again sufficient conditions for [Formula: see text] to be amphichiral. We finally prove that a link [Formula: see text], associated to a map [Formula: see text], is amphichiral if the self-dual pairing of [Formula: see text] is not one of 6 specific cases among the classification of the 24 self-dual pairing [Formula: see text].

A self-dual map $G$ is said to be antipodally self-dual if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be antipodally self-dual. In particular, we present a combinatorial characterization for map $G$ to be antipodally self-dual in terms of certain involutive labelings. The latter lead us to obtain necessary conditions for a map to be strongly involutive (a notion relevant for its connection with convex geometric problems). We also investigate the relation of antipodally self-dual maps and the notion of antipodally symmetric maps. It turns out that the latter is a very helpful tool to study questions concerning the symmetry as well as the amphicheirality of links.