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.