# Topological Origami Models of Non-Convex

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Topological Origami Models of Non-Convex

Topological Origami Models of Non-Convex Polyhedra Andrea Hawksley∗ Abstract. We present a series of ‘topological’ modular origami models of star polyhedra, as well as a method for creating such models and a theoretical enumeration of which models are possible to create with our methodology. 1 Introduction and Previous Work. Modular origami is a standard technique for creating polyhedral models, but most existing. Most current origami models of nonconvex polyhedra obscure the underlying connectedness of their faces. We present a series of ‘topological’ origami modular constructions that emphasize the internal connectedness of self-intersecting polyhedra. The models bear some resemblance to other modular origami Lang’s ‘polypolyhedra’, as well as to the work by Hideaki Kawashima, in that they show underlying intersecting geometries; however their work is focused on creating edge skeletons of the shapes, while the current work focuses on representing the polyhedral faces [3, 4]. Each model is not of intersecting compounds, but a single self-intersecting star polyhedron. Sculptures similar to those created with our technique have, however, appeared in non-origami sculpture [1, 2, 5, 6] ter discarding those with vertices at infinity and those with self-intersecting faces. The underlying geometries of these ‘topological’ models are constructed by removing parts of each face of the nonconvex polyhedra symmetrically such that the remaining portions of each face no longer intersect. The modules and the resulting models are necessarily chiral. Figure 3 shows an example of a face of a great rhombic triacontahedron. In order to maintain connectedness while avoiding intersection with other units, each face must be represented by a rotationally symmetric unit that passes through all of the green points (where the units connect and the center point) but avoids all of the red points (where more than two faces intersect in the 3: Face of original polyhedron). These con- Figure the great rhombic straints still allow room for much triacontahedron, artistic interpretation, so two mod- with intersection els designed with this technique and points marked based on the same polyhedron may still be very visually distinct. 3 Future Work. Future work in the area of topological origami models include models of non-isohedral polyhedral by using non-identical modules or by not representing every face, models with self-intersecting faces with modules that are not strict subsets of the underlying faces, and models of polyhedra with nonconvex vertex figures by connecting faces along edges rather than at vertices or by connecting faces in a non-convex manner, perhaps by removing parts of each face at the vertices. References. Figure 1: Origami models of the great rhombic triacontahedron, the medial rhombic triacontahedron, and the small triambic icosahedron 2 Constraints. We use the constraint that models must be made out of identical units or modules that represent faces rather than edges. We choose to connect the modules at their vertices, a choice that best shows the underlying connectedness of the faces. This requirement makes it difficult to model polyhedra with nonconvex vertex figures. With the constraint that all vertices be represented in the figure and that each module represent a strict subset of a face, none of the KeplerPoinsot polyhedra are possible. Allowing models where not all vertices are represented enables us to theoretically model 33 dual nonconvex uniform polyhedra, af- [1] Vladimir Bulatov. Using polyhedral stellations for creation of organic geometric sculptures. In Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture, pages 193–198, 2009. [2] George Hart. Frabjous. http://georgehart. com/sculpture/frabjous.html. [3] Hideaki Kawashima. Works of hideaki kawashima. http://kawacho.seesaa.net/. [4] Robert J Lang. Polypolyhedra in origami. In Origami3: Proceedings of the 3rd International Meeting of Origami Science, Math, and Education, 2: pages 153–168, 2002. Figure A single module of the great rhombic triacontahedron model ∗ Communications Design Group, SAP Labs, San Francisco, CA, USA, [email protected] [5] Eve Torrence. A workshop on stellation inspired sculpture. In Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture, pages 665–670, 2011. [6] Robert Webb. Topological models. http://www. software3d.com/Misc.php#topo.