Topological Mesh Modeling is an umbrella term that covers all our work based on extensions the theory of graph rotation systems.
It includes (1) Orientable 2manifol mesh modeling using graph rotation systems and its computer graphics applications,
(2) Knot modeling with immersions of nonorientable manifold meshes and (3) Topological constructions that is based on geometric and physical
constraints with graph rotation systems. We recently started to work on
immersions of 3manifolds as a representation to develop shape modeling systems. This webpage provides links to approximately
50 papers that are roughly organized approximately 10 categories. Click links below to directly go to the the categories that provides
related papers and manuscripts.
 Orientable Mesh Modeling: We have
provided a solid foundation for orientable 2manifold mesh modeling using graph rotation systems.
Based on this theory, we have developed TopMod ,
which is is an orientable 2manifold mesh modeling system. TopMod provides
a wide vriety of High Genus Modeling tools, Remeshings & Subdivisions, and
Extrusions & Replacements.
Using TopMod, one can find a wide variety of ways to create high genus shapes;
almost all subdivision algorithms, wide variety of ways to remeshing shapes and
new extrusions. These tools are also useful for Architectural applications, Design and Sculpting.
We also hav additional tools for Surface Parameterization and Texturing and Tiling.
 Knots Modeling: We have developed provided
a solid foundation for knot, link and cyclic woven object modeling using extended graph rotation systems.
If we twist an arbitrary subset of edges of a mesh on an
orientable surface, we can obtain nonorientable surfaces. The resulting extended graph rotation system can be used
to induce a cyclic weaving on the original surface, that corresponds a 3space immedding of a nonorientable surface.
 Topological Constructions: Discrete GaussianBonnet theorem and Gaussian curvatures related
mesh topologic concepts to geometry. Using this relationship, we have developed methods to phsyically construct shapes.
 Immersions of 3Manifolds: Using an extension of graph rotation systems it is possible
to represent 3space immersions of 3manifolds by employing a topological graph theory concept called 3D thickening.
This work partially supported by the National Science
Foundation under Grant No. NSFCCF0917288.

Knot Modeling and Cyclic Woven Objects: Modeling with Immersions of NonOrientable Manifold Meshes 
If we twist an arbitrary subset of edges of a mesh on an orientable
surface, we can obtain nonorientable surfaces. The resulting extended graph rotation system
(EGRS) can be used to induce a cyclic weaving on the original
surface, that corresponds a 3space immedding of a nonorientable surface.
In extended graph rotation systems, an edge is
viewed as a paper strip that can be twisted. The sides
of the paper strips provide “two strands” to construct
weaving structures. Either these strands are “parallel” to
the mesh edge for an “untwisted edge”, or they both cross
over the edge and over each other for a “twisted edge”. If
an arbitrary subset of edges of a mesh on an orientable
surface is twisted in the same helical sense, then the
EGRS induces a cyclic plainweaving on the surface, which
consists of cycles that cross other cycles (or themselves) by
alternatingly going over and under. For theoretical treatment see this manuscript.


E. Akleman, J. Chen, Q. Xing and J. Gross, "Cyclic plainweaving on polygonal mesh surfaces with graph rotation systems",
ACM SIGGRAPH, Transactions on Graphics (TOG), Volume 28, Issue 3, pp 78.178.8, August 2009.
(Paper) (Video)
Description: We showed how to create plainweaving over an arbitrary surface. To create a plainweaving on a surface,
one need to create cycles that cross other cycles (or themselves) by alternatingly going over and under.
We use the fact that it is possible to create such cycles, starting from any given manifoldmesh surface by
simply twisting every edge of the manifold mesh. We have developed a new method that converts plainweaving
cycles to 3D thread structures. Using this method, it is possible to cover a surface without large gaps
between threads by controlling the sizes of the gaps. We have developed a system that converts any manifold
mesh to a plainwoven object, by interactively controlling the shapes of the threads with a set of parameters.
We have demonstrated that by using this system,
we can create a wide variety of plainweaving patterns, some of which may not have been seen before.


Q. Xing, E. Akleman, J. Chen, and J. Gross, "SingleCycle PlainWoven Objects",
Proceedings of Shape Modeling International, 2010,
(Paper)
Description:
In this paper, we show that it is always possible to create
a singlecycle plainweaving starting from a mesh on an
arbitrary surface, by selecting an appropriate subset of
edges to be twisted. We also demonstrate how, starting
from a mesh, to construct a large number of singlecycle
plainwoven objects. Interestingly, the singlecycle solutions
with a minimal number of edge twists correspond to plainwoven
objects that are visually similar to Celtic knots.
For converting plainweaving cycles to 3D thread structures,
we extend the original projection method, which
previously worked only when all mesh edges are twisted.
With the extension described here, our projection method
can also be used to handle untwisted edges. We have
developed a system that converts any manifold mesh into
singlecycle plainwoven objects, by interactively controlling
the proportion of edges that are twisted. The system
also allows us to change the shapes of the threads with a set
of parameters, interactively in realtime. We demonstrate
here that by using this system, we can create a wide variety
of singlecycle plainwoven objects.


E. Akleman, J. Chen, YL Chen, Q. Xing and J. Gross, "Cyclic Twill Woven Objects", Computers & Graphics 35 (2011) 623–631.
(Paper) (Video)
Description: Classical (or biaxial) twill is a textile weave in which the weft threads pass over and under two or more warp threads,
with an offset between adjacent weft threads to give an appearance of diagonal lines.
We ideveloped a theoretical framework for constructing twillwoven objects, i.e.,
cyclic twillweavings on arbitrary surfaces. We also presented methods to convert
polygonal meshes into twillwoven objects. We also identified a general technique
to obtain exact triaxialwoven objects from an arbitrary polygonal mesh surface.


Shiyu Hu, Qing Xing, Ergun Akleman, Jianer Chen, Jonathan L. Gross, "Pattern Mapping with QuadPatternCoverable QuadMeshes",
Computers & Graphics 36 (2012) 455465.
(Paper) (Video)
Description: We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texturemapped
with locally translated copies of a single squaretexture pattern. This implies, for instance, that every positivegenus
surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify su
cient conditions for meshes to be classified as “quadpatterncoverable”, and we present several methods to construct
such meshes. Moreover, we identify some mesh operations that preserve the quadpatterncoverability property. For
instance, since vertex insertion remeshing, which is the remeshing operation behind CatmullClark subdivision, preserves
quadpatterncoverability, it is possible to cover any surface of positive genus with iteratively finer versions of
the same texture.


Ergun Akleman, Qing Xing, Pradeep Garigipati, Gabriel Taubin, Jianer Chen, Shiyu Hu, "Hamiltonian Cycle Art:
Surface Covering Wire Sculptures and Duotone Surfaces",
Computers & Graphics 36 (2012) 455465.
(Paper) (Video1) (Video2)
Description: In this work, we present the concept of ”Hamiltonian Cycle Art” that is based on the fact that any mesh surface
can be converted to a single closed 3D curve. These curves are constructed by connecting the centers of every two
neighboring triangles in the Hamiltonian triangle strips. We call these curves surface covering since they follow the
shape of the mesh surface by meandering over it like a river. We show that these curves can be used to create wire
sculptures and duotone (twocolor painted) surfaces.
To obtain surface covering wire sculptures we have developed two methods to construct corresponding 3D wires from
surface covering curves. The first method constructs equal diameter wires. The second method creates wires with
varying diameter and can produce wires that densely cover the mesh surface.
For duotone surfaces, we have developed a method to obtain surface covering curves that can divide any given mesh
surface into two regions that can be painted two di
erent colors. These curves serve as a boundary that define two
visually interlocked regions in the surface. We have implemented this method by mapping appropriate textures to
each face of the initial mesh. The resulting textured surfaces look aesthetically pleasing since they closely resemble
planar TSP (traveling salesmen problem) art and Truchetlike curves.

Topological constructions is based on the relation between topology and geometry
through GaussBonnet theorem and Euler characteristics. In this work, we turn
data structures that is used to represent 2manifolds into physical data structures.
The fundamental
HeffterEdmunds theorem of GRS asserts
that there is a bijective correspondence between the set of
pure rotation systems of a graph and the set of equivalence
classes of embeddings of the graph in the orientable surfaces.
As a direct consequence of the theorem, to assemble a
structure all construction workers have to do is to attach
the corresponding phsyical components. Once
all the components are attached to each other, the whole
structure is guaranteed to be correctly assembled.
GaussBonnet theorem, moreover, asserts that the total
Gaussian curvature of a surface is the Euler characteristics
times 2pi. Since we use only developable
components, Gaussian curvature is zero everywhere on the
solid parts. The Gaussian curvature happens only in empty
regions and that are determined uniquely. Since, we cor
rectly form Gaussian curvature of holes, the structures always be raise and form in 3space.


Edwin Alexander Peraza Hernandez, Shiyu Hu, Han Wei Kung, Darren Hartl, Ergun Akleman,
"Towards building smart selffolding structures". Computers & Graphics 37(6): 730742 (2013)
(paper)
Description: We report our initial progress on synthesizing complex structures from programmable selffolding active materials,
which we call Smart MultiUse Reconfigurable Forms. We have developed a method to unfold a given convex polygonal
mesh into a onepiece planar surface. We analyze the behavior of this surface as if it were constructed from
realistic active materials such as shape memory alloys (SMAs), in which sharp creases and folds are not feasible.
These active materials can change their shapes when they are heated and have been applied to medical, aerospace, and
automotive applications in the engineering realm. We demonstrate via material constitutive modeling and utilization
of finite element analysis (FEA) that by appropriately heating the unfolded planar surface it is possible to recover the
3D shape of the original polygonal mesh. We have simulated the process and our finite element analysis simulations
demonstrate that these active materials can be raised against gravity, formed, and reconfigured automatically in three
dimensions with appropriate heating in a manner that extends previous work in the area of programmable matter.
Based on our results, we believe that it is possible to use active materials to develop reprogrammable selffolding
complex structures.


Qing Xing, Gabriel Esquivel, Ergun Akleman, Jianer Chen, Jonathan L. Gross, "Band Decomposition of 2Manifold Meshes
For Physical Construction of Large Structures",
Siggraph
'2011, Posters & Talks (2011).
(extended abstract)
Description: In this work, we introduce an approach to automatically create such easily assembled developable components from any
given manifold mesh. Our approach is based on classical
Graph Rotation Systems (GRS). Each developable component, which we call vertex component, is a physical equiva
lent of a rotation at the vertex v of a graph G. Each vertex
component is a star shaped polygon that physically corresponds to the cyclic permutation of the edgeends incident
on v (See Figure 2(a)). We engrave edgenumbers with laser
cutters directly on edgeends of vertex components to simplify nding corresponding edge ends. When we print edge numbers, we actually define a collection of rotations, one for
each vertex in G. This is formally called a pure rotation
system of a graph. Using this approach, Architecture students have constructed a
large version of Stanford Bunny (see Figure 1) in a design
and fabrication course in College of Architecture.


E. Akleman, J. Chen and J. Gross, "PaperStrip Sculptures", Proceedings of Shape Modleing International'2010.
(Paper)
Description: This paper introduces paperstrip sculptures,
a physical mesh datastructure used to represent 2
manifold mesh surfaces for understanding topological and
geometrical aspects of shape modeling with visual and
tactual examples. With paper strips it is possible to
construct simple paper sculptures that can convincingly
illustrate a variety of ideas in shape modeling — such as 2
manifold mesh surfaces, discrete Gaussian curvature, and
the GaussBonnet theorem — with handson experiments.
Such sculptures can also represent links, knots and weaving.
Paperstrip sculptures are also useful to represent and
understand nonorientable surfaces such as the projective
plane and the Klein bottle.

 E. Akleman and J. Chen,
"Practical Polygonal Mesh Modeling with Discrete GaussianBonnet Theorem", Proceedings of Geometry, Modeling and Processing 2006, Pittsburg.
(Paper)
Description: In this paper, we introduce a practical modeling approach to improve
the quality of polygonal mesh structures. Our approach is
based on a discrete version of GaussianBonnet theorem on piecewise
planar manifold meshes and vertex angle deflections that determines
local geometric behavior. Based on discrete Gaussian
Bonnet theorem, summation of angle defects of all vertices is
independent of mesh structure and it depends on only the topology
of the mesh surface. Based on this result, it can be possible to
improve organization of mesh structure of a shape according to its
intended geometric structure.

Using an extension of graph rotation systems it is possible
to represent 3space immersions of 3manifolds by employing a topological graph theory concept called 3D thickening.
(joint work Jianer Chen and Jonathan Gross)


E. Akleman, "Extended Graph Rotation Systems and Its Applications to Modeling 2Manifolds, Woven Surfaces and 3Manifolds",
GD/SPM'2013, Presentation at the Minisymposium: Shaping Surfaces.
(Presentation)
Description: In this talk, I demonstrated that extended graph rotation systems and 3D thickenings has a potential
to describe 3manifolds that can help our understanding and modeling 3manifold structures.
Such a generalized 3manifold mesh representations can be used in modeling solids,
architectural shapes, highgenus surfaces, knots and links. For 3manifolds, I started with prisms that represents
3D thickened edges of 3manifold meshes and discussed what kind of models can be constructed using those prisms.
I also introduced the concepts of chambers and blocks. Using boundary walk I demonstrated the faces of
3manifolds can be both one and twosided. If we want duality,
this suggests that 3D thickened edges should also be one or twosided and 3D thickened vertex boundaries can be any 2manifold.


O. Gonen and E. Akleman, "Sketch Based 3D Modeling with
Curvature Classification", Computers & Graphics, 2012, 36(5), 521525.
(extended Abstract)
Description: In this paper, we introduce a simple approach for sketching 3D models in arbitrary topology. Using this approach,
we have developed a system to convert silhouette sketches to 3D meshes that mostly consists of quadrilaterals and
4valent vertices. Because of their regular structures, these 3D meshes can e
ectively be smoothed using Catmull
Clark subdivision. Our approach is based on the identification of corresponding points on a set of curves. Using the
structure of correspondences on the curves, we partition curves into junction, cap and tubular regions and construct
mostly quadrilateral meshes using these partitions.


O. Gonen and E. Akleman, "Sketching Knots", Siggraph 2012, Posters.
(paper)
Description: We present an unexpectedly easy to use interface to create
knots by using sketch based modeling. In our interface the
only thing the users need to do for creating knots and links is
to draw a set of curves. These curves serves the medial axis
of the knots to be constructed. To construct knots we rst
estimate of the depth {z{ value for every point on the medial
axis curve. The depth estimation turns 2D medial axis curve
into a 3D medial axis. We then extrude a polygon along the
3D medial axis curve to obtain the tread that forms the
physical knot. If the medial axis consists of closed curves,
the result is a mathematical knot.

Theoretical Framework Theoretical Framework for Orientable 2Manifold Modeling:
Topological Graph Embeddings and Its Computer Graphics Applications

Topologically Robust Mesh Modeling:
Concepts, Data Structures and Operations
We extend the theory of graph rotation systems and
provide a solid foundation for orientable 2manifold mesh modeling.
Based on this theory, we identify a group of simple validity
rules, and show that the validity of 2manifold structures
can be tested very efficiently on all existing data structures
for mesh modeling. Moreover, the theory enables us
to develop very efficient implementations for manifold preserving
operations, which are essential for a robust interactive
modeling system. For theoretical treatment see this manuscript.

 E. Akleman, J. Chen, "Guaranteeing 2Manifold
Property for Meshes by Using Doubly Linked Face
List", International Journal of Shape Modeling,
Volume 5, No. 2, pp. 149177, 2000.
(Paper)
Description: Meshes, which generalize polyhedra by using nonplanar faces,
are the most commonly used objects in computer graphics.
Modeling 2dimensional manifold meshes with a simple user interface
is an important problem in computer graphics and computer aided
geometric design. In this paper, we propose a conceptual
framework to model meshes. Our framework guarantees topologically
correct 2dimensional manifolds and provides a new user interface
paradigm for mesh modeling systems.

 E. Akleman, J. Chen, V. Srinivasan and
F. Eryoldas, "A New Corner Cutting Scheme
with Tension and HandleFace Reconstruction",
International Journal of Shape Modeling,
Volume 7, No. 2, pp. 111121, 2001.
(Paper)
Description: A recently developed topological mesh modeling approach allows users to change topol
ogy of orientable 2manifold meshes and to create unusual faces. Handlefaces are one of
such faces that are commonly created during topology changes. This paper shows that
vertex insertion and corner cutting subdivision schemes can effectively be used to recon
struct handlefaces. These reconstructions effectively show the structure of these unusual
faces. The paper has three contributions. First, we develop a new corner cutting scheme,
which provides a tension parameter to control the shape of subdivided surface. Second,
we develop careful and e±cient remeshing algorithms for our corner cutting scheme that
use only the basic operations provided by our topological mesh modeling approach. This
implementation ensures that our new corner cutting scheme preserves topological robust
ness. Finally, a comparative study shows that the corner cutting schemes create better
handles and holes than the wellknown CatmullClark scheme.

 E. Akleman, J. Chen and V. Srinivasan,
"A minimal and complete set of operators for the
development of robust manifold mesh modelers",
Graphical Models, Volume 65, Issue
5, pp. 286304, September 2003.
(Paper)
Description: In this paper, we identify a minimal and complete set of fundamental operators,
which is necessary and su±cient for performing all homeomorphic and topological
operations on 2manifold mesh structures. E±cient algorithms are developed for
the implementation of these operators. We also developed a set of powerful, user
friendly, and effective operators at the level of userinterface. Using these operators,
we have developed a prototype system for robust, interactive and user friendly mod
eling of orientable 2manifold meshes. Users of our system can perform a large set
of homeomorphic and topological changes with these userinterface level operators.
Our system is topologically robust in the sense that users will never create invalid
2manifold mesh structure with these operators.
In our system, the homeomorphic and topological surgery operations can be ap
plied alternatively on 2manifold meshes. With our system,users can blend surfaces,
construct rinds and open holes on these rind shapes. With our system, the shapes
that look like solid, nonmanifold, or 2manifold with boundary can be manipulated.
The system also provides automatic texture mapping during topology changes.

 E. Akleman, J. Chen, V. Srinivasan,
"A New Paradigm for Changing Topology During Subdivision Modeling",
Proceedings of Pacific Graphics 2000,
Hong Kong, China, pp. 192201, October 2000.
(Paper)
Description: In this paper, we present a new paradigm that allows dynamically
changing the topology of 2manifold polygonal
meshes. Our new paradigm always guarantees topological
consistency of polygonal meshes. Based on our paradigm,
by simply adding and deleting edges, handles can be created
and deleted, holes can be opened or closed, polygonal
meshes can be connected or disconnected.
These edge insertion and edge deletion operations are
highly consistent with subdivision algorithms. In particular,
these operations can be easily included into a subdivision
modeling system such that the topological changes
and subdivision operations can be performed alternatively
during model construction.
We demonstrate practical examples of topology changes
based on this new paradigm and show that the new
paradigm is convenient, effective, efficient, and friendly to
subdivision surfaces.

Regular Meshes A Family of Meshes Includes Regular Maps

 E. Akleman and J. Chen,
"Regular Meshes",
Proceedings of Solid and Physical Modeling 2005, Boston, June 2005.
(Paper)
Description: This paper presents our preliminary results on regular meshes in
which all faces have the same size and all vertices have the same
valence. A regular mesh is denoted by (n,m,g) where n is the number
of the sides of faces, m is the valence of vertices and g is the
genus of the mesh. For g = 0, regular meshes include regular platonic
solids, all two sided polygons. For g = 1 regular meshes include
regular tilings of infinite plane. Our work shows that there
exist infinitely many regular meshes for g > 1. Moreover, we have
constructive proofs that describe how to create high genus regular
meshes that consist of triangles and quadrilaterals (3,m,g) and
(4,m,g).

 E. Akleman and J. Chen,
"Regular Meshes Construction Algorithms Using Regular Handles", Proceedings of Shape Modeling International 2006, Matsushima, japan
(Paper)
Description: We introduce a new concept called regular handles. Using
regular handles it is possible to increase genus without
increasing the number of vertices.
We develop a general procedure based on
regular handles. Our procedure allows us to greatly extend
”regular mesh families”. We provide 14 regular mesh
families that includes all genus2 primary regular meshes:
(3,7,2), (3,8,2), (3,9,2), (3,10,2), (3,12,2), (3,18,2),
(4,5,2), (4,6,2),(4,8,2), (4,12,2) and (5,5,2), (5,10,2),
(6,6,2) and (8,8,2). Our regular mesh families are constructed
by adding the regular handles to an initial regular
mesh M0. By using the same procedure iteratively we construct
a series of regular meshes M0,M1,M2, . . .Mn.

 E. Akleman, V. Srinivasan and J. Chen,
"Interactive Rind Modeling", Proceedings of Shape Modeling
International 2003, Seoul, Korea, May 2003.
(Paper)
Description: In this paper, we describe a technique, with roots in topological
graph theory, that we call rind modeling. It provides
for the easy creation of surfaces resembling peeled
and punctured rinds. We show how the method’s two main
steps of 1) creation of a shell or crust like the rind of an
orange, and 2) opening holes in the crust by punching or
peeling can be encapsulated into a real time semiautomatic
interactive algorithm. We include a number of worked examples,
some by students in a first modeling course, that
demonstrate the ease with which a large variety of intricate
rind shapes can be created.

 V. Srinivasan, E. Akleman and J. Chen,
"Interactive Construction of MultiSegment
Curved Handles", Proceedings of Pacific Graphics 2002,
Beijing, China, October 2002.
(Paper)
Description: In this paper, we present a method to interactively create
multisegment, curved handles between two starshaped
faces of an orientable 2manifold mesh or to connect two
2manifold meshes along such faces. The presented algorithm
combines a very simple 2D morping algorithm with
a Hermite interpolation to construct the handle. Based on
the method, we have developed a user interface tool that allows
users to simply and easily create multisegment curved
handles. The
method can be used for handle creation (i.e., adding a handle
to a surface) and for surface blending (i.e. connecting
two distinct surfaces). Both applications of the algorithm
are useful to designers for creating manifolds of high genus.
A handle is not an extrusion (or lofting) and handle creation
is not simply an extrusion method. Handle creation
is a topological operation and it requires topological consistency.
Therefore, handle creation methods are different
than extrusion methods since they are required to guarantee
topological consistency. Our method not only guarantees 2
manifold property of final mesh, in every stage of our handle
creation costructed meshes continue to be 2manifold.

 V. Srinivasan and E. Akleman,
"Connected and Monifold Sierpinsky Polyhedra",
Proceedings of Solid Modeling 2004, Genoa, Italy, June 2004.
(Paper)
Description: In this paper, we present a subdivisioninspired scheme to construct generalized Sierpinski polyhedron. Unlike usual Sierpinski
polyhedra construction schemes, which create either an infinite set of disconnected tetrahedra or a nonmanifold polyhedron,
our robust construction scheme creates one connected and manifold polyhedron. Moreover, unlike the original schemes, this
new scheme can be applied to any manifold polyhedral mesh and based on the shape of this initial polyhedra a large variety
of Sierpinski polyhedra can be obtained.Our basic scheme can be viewed as applying simplest subdivision scheme [23] to an
input polyhedron, but retaining old vertices. The porous structure is then obtained by removing the refined facets of the simplest
subdivision.

 V. Srinivasan, E. Mandal and E. Akleman,
Solidifying Frames, Bridges:
Mathematical Connections in Art, Music, and Science
2004, Banf, Alberta, Canada, August 2005.
(Paper)
Description: In this paper we present a method to convert a wireframe mesh into a 2manifold mesh consisting
of cylindrical pipes in place of the edges and joints in place of the vertices in the original mesh. Our
method allows users to create unique artistic depictions of common objects and structures. The resulting
mesh is also more effective at conveying the overall 3D structure and any internal elements of a model
when compared to regular wireframe or boundary representations. The input wireframe mesh can be
any collection of linear edges; they do not have to form a manifold surface or even be connected to
each other. The result is always an orientable 2manifold surface. Our algorithm replaces every edge in
the wireframe mesh with a cylindrical 3D pipe. The pipes are connected to each other using 3D joints
created at the vertices in the wireframe where the edges meet. Our method has been implemented as part
of a polygonal mesh modeling system and has been used to create artistic models of popular architectural
structures as well as to create conceptual sketches for virtual environments.

 E. Mandal, E. Akleman and V. Srinivasan,
"Wire Modeling",
Visual Proceedings of ACM SIGGRAPH'2003 (Siggraph Sketch),
San Diego,
California, July 2003.
(Sketch)
Description: We present a method that will let the users create
extremely high genus manifold meshes with minimal human interaction and time.
Our method replaces each edge of a given mesh with a {\em ``3D pipe}'' by
creating a wired look. Our method guarantees that the pipes are connected
and the resulting shapes can be physically constructed. We have implemented
this method as an extension to an existing modeling system.
Our system creates a complex high genus mesh from an input polygonal mesh by
converting the edges of the input polygon to 3D pipes which looks like
wires or matchsticks. Since the quality final model completely depends the
mesh structure of the initial polygonal mesh, we have developed a set of
subdivision based approaches to create a wide variety of mesh structures.

 E. Akleman, V. Srinivasan, and E. Mandal,
"Remeshing Schemes for SemiRegular Tilings",
Proceedings of Shape Modeling International 2005, Boston, June 2005.
(Paper)
Description: Most frequently used subdivision schemes such as
CatmullClark create regular regions after several application.
This paper shows that all semiregular regions can be
created by subdivision schemes and each semiregular region
type can be created with one application of a
particular subdivision scheme to a particular regular region.
Using this property of subdivision schemes it is easy
to cover any given surface with semiregular tiles by applying
one semiregularity creating subdivision after
several applications of a regularity creating subdivision

 E. Akleman, V. Srinivasan, Z. Melek and P. Edmundson,
Semiregular Pentagonal Subdivisions, Shape Modeling
International 2004, Genoa, Italy, June 2004.
(Paper)
Description: Triangular and quadrilateral meshes are commonly used
in computer graphics applications. In this paper, we analyze
the topological existence of meshes that consist of n
sided faces where n is greater than 4 such as pentagonal
and hexagonal meshes. We show that it is possible to represent
any 2manifold with a mesh that is made up of only
pentagons. We also show that the meshes that consist of
only polygons with more than five sides cannot represent all
2manifolds.
We present a pentagonalization (or pentagonal conversion)
scheme that can create a pentagonal mesh from any
arbitrary mesh structure. We also introduce a pentagonal
preservation scheme that can create a pentagonal mesh
from any pentagonal mesh.

 E. Akleman, P. Edmundson and O. Ozener,
A Vertex Truncation Subdivision Scheme to Create Intriguing Polyhedra, Bridges: Mathematical Connections in Art, Music, and Science 2004, Winfield, Kansas, August 2004.
(Paper)
Description: In this paper, we present a new class of semiregular polyhedra. All the faces of these polyhedra
are bounded by smooth (quadratic Bspline) curves and the face boundaries are C1 discontinues everywhere.
These semiregular polyhedral shapes are limit surfaces of a simple vertex truncation subdivision
scheme. We obtain an approximation of these smooth fractal polyhedra by iteratively applying a new
vertex truncation scheme to an initial manifold mesh. Our vertex truncation scheme is based on Chaikin’s
construction. If the initial manifold mesh is a polyhedra only with planar faces and 3valence vertices, in
each iteration we construct polyhedral meshes in which all faces are planar and every vertex is 3valence.

 E. Akleman and V. Srinivasan, "Honeycomb Subdivision", Proceedings of ISCIS'02, 17th International Symposium on Computer and
Information Sciences, pp. 137141, November 2002, Orlando, Florida.
(Paper)
Description: In this paper, we introduce a new subdivision scheme
which we call honeycomb subdivision. After one iteration
of the scheme each vertex becomes exactly 3valent and
with consecutive applications regular regions strongly resembles
a honeycomb. This scheme can be considered as a
dual for triangle schemes. The major advantage of the new
scheme is that it creates a natural looking mesh structure.
We call this scheme honeycomb since the resulting
meshes strongly resemble honeycombs, which is defined as a structure of hexagonal, thinwalled
cells constructed from beeswax by honeybees to hold honey
and larvae or something resembling this structure in configuration
or pattern.

 E. Landreneau, E. Akleman and V. Srinivasan,
"Local Mesh Operators: Extrusions Revisited", Proceedings of Shape Modeling
International 2004, 2005, Boston, June 2005.
(Paper)
Description: In this paper, we present a set of generalized “local”
mesh operators. Local operators are those that operate on
a single face without affecting the rest of the mesh. Boundary
edges of the chosen face also stay the same. We have
identified two types of local operators: (1) Extrusions that
create generalized pipes in which bottom and top polygons
have the same number of sides, and (2) Stellations that create
generalized pyramids, where there is a top vertex instead
of top polygon. Our operators can create extrusions
that are regular polyhedra including dodecahedron, icosahedron,
octahedron and tetrahedron. The tetrahedron is created
using the stellation operator, which is also useful to
create generalized versions of Kepler and Poinsot solids.
Using these extrusions unusual shapes can be
created without changing the genus. The paper also shows how to create
nontriangular planar meshes with extrusions.

 E. Landreneau, E. Akleman and J. Keyser,
"Iterative Face Replacements for Modeling Detailed Shapes", Proceedings of Geometry, Modeling and Processing 2006, Pittsburg.
(Paper)
Description: In this paper, we present a method that allows novice
users to interactively create partially selfsimilar manifold
surfaces without relying on shape grammars or fractal
methods. Moreover, the surfaces created using our method
are connected. The modelers that are based on traditional
fractal methods or shape grammars usually create disconnected
surfaces and restrict the creative freedom of users. In
most cases, the shapes are defined by hardcoded schemes
that provide only a few parameters that can be adjusted by
the users. We present a new approach for modeling such
shapes. With this approach, novice users can interactively
create a variety of unusual and interesting partially selfsimilar
manifold surfaces.

 E. Akleman, A. Kaur and L. Green, Tiled Textures: What if Miro Painted a Sphere, ISAMA'2008,
(Paper)
Description:
We present a simple and practical technique for seamlessly texturing quadrilateral meshes. Using our
technique any image can be converted to an isotropic texture that can be mapped to any quadrilateral
mesh without any discontinuity or singularity. Using our technique, we can make any abstract painter
like Miro to seamlessly paint any smooth manifold surface. The surface can have any number of holes
or handles.
Our texturing method is to organize a set of tiles that satisfy specific boundary conditions into one
texture image file which is called a tiled texture. We have also developed an algorithm to create tiled textures
from any image with a simple user interface that allows the users to specify the boundaries. Based
on tiled textures, we have developed an extremely simple texture mapping algorithm that assigns one tile
to every quadrilateral in any given quadrilateral mesh. Our mapping algorithm provides aperiodicity on
the surface of the mesh and yields singularity free textures regardless of the singularities existing in the
quadrilateral mesh


E. Akleman, J. Chen, B. Meric, "Symmetric Tile Design", pp. 283292 Proceedings of ACADIA 2000, Washington, DC., October 2000.
(Paper)
Description: This paper presents a new approach for intuitive and effective design of periodic symmetric tiles. We
observe that planar graphs can effectively represent symmetric tiles and graph drawing provides an
intuitive paradigm for designing symmetric tiles. Moreover, based on our theoretical work to represent
hexagonal symmetry by rectangular symmetry, we are able to present all symmetric tiles as graphs
embedded on a torus and based on simple modulo operations. This approach enables us to develop
a simple and efficient algorithm, which has been implemented in Java. By using this software, designers,
architects and artists can create interesting symmetric tiles directly on the web. We also have
designed a few examples of symmetric tiles to show the effectiveness of the approach.


E. Akleman, "Twirling Sculptures", Journal of Mathematics and Arts, vol. 3, no. 1, pp. 110, 2009.
(Paper)
Description: In this paper, I outline a method for constructing aesthetically pleasing sculptures containing spiral shapes. Since every face of an
orientable manifold mesh can be given a consistent edge rotation ordering, if one applies an extrusion operator to each face of an
orientable manifold mesh using the same rotation and scaling factors, each edge in the original mesh will be converted to an Sshaped
region that consists of two spiral arms. The twirling nature of my sculptures results from these Sshaped regions. The nal sculptures are
obtained by smoothing the resulting shapes with a subdivision scheme. I discuss several methods for visually emphasizing the twirling
nature of Sshaped regions. All the models and virtual sculptures in this paper are created using the Topological Mesh Modeling system
TopMod.


Q. Xing, G. Esquivel and E. Akleman, "Twisted DForms: Design and Construction of DForms with Twisted
Prismatic Handles with Developable Sides", Proceedings of Bridges 2012.
(Paper)
Description: In this work, we present shapes that
are constructed from a set of twisted papers,
which we call Twisted Dforms. These
shapes consists of twisted prismatic handles
with developable sides. We design these
handles using the handle creation tool in
TopMod. The handle creation tool allows
designing twisted handles that consists
of strips of long triangles. Using this
approach it is possible to design shapes of
high genus. This initial triangulated model let us do minor modifications in the designs using commercial
software such as Maya without destroying the developable property.
We constructed a large number of small scale prototypes using paper.


Ozgur Gonen, Ergun Akleman and Vinod Srinivasan, "Modeling DForms", Proceedings of Bridges 2008.
(Paper)
Description:
Recently, very interesting developable sculptures, called Dforms,
were invented by the London designer Tony Wills. Dforms are created by joining the edges of
a pair of sheet metal or paper with the same
perimeter. Despite its power to construct unusual shapes easily, there are two problems with physical Dform construction.
First, the physical construction is limited to only two pieces. It is hard to figure out the perimeter
relationships if we try to use more than two pieces. Second problem with Dform construction is that until
we finalize the physical construction of the shape we do not exactly know what kind of the shape to be
constructed. In this paper we introduce a computation method that provides an alternative to physical Dform construction.
Using our method, Dforms can directly be designed with our software. Our Dforms can consist
of more than two pieces. Another advantage of our method is that before physical construction
of the shape we exactly know what kind of shape to be constructed.


Jace Miller and E. Akleman, "EdgeBased Intersected Polyhedral Paper Sculptures
Constructed by Interlocking Slitted Planar Pieces", Proceedings of Bridges 2008.
(Paper)
Description:
In this paper, we generalize George Hart’s slidetogether sculptures as edgebased intersected polyhedral paper sculptures.
Edgebased intersected polyhedra are also a conceptual generalization of Kepler’s Small Stellated Dodecahedron.
These sculptures are constructed by interlocking slitted planar pieces without using glue. We present a simple
procedure to construct slitted planar pieces for any given polyhedron. These sculptures can easily be constructed by
children and can be used to teach properties of Platonic or Archimedean Solids through handson experience.


Yutu Liu, Hernan Molina and E. Akleman, "Inout Sculptures", Proceedings of Bridges 2007.
(Paper)
Description:
The people innately find a mysterious beauty in sculptures with smooth saddle regions that exists in hyperbolic sculptures.
Wellknown examples hyperbolic sculptures are Robert Longhurst’s Arabesque 29th
and Brent Collins’s hyperbolic sculptures with many smooth holes and handles. We
present a new method to create a new set of hyperbolic sculptures, which we call inout sculptures.
Our idea is simply to simultaneously show both inside
and outside of an already complicated shape that contains many holes. These sculptures are
obtained by showing both inside and outside of a shape with holes. Inout sculptures looks interesting since they allow
to simultaneously view both inside and outside of complicated shapes.


Vinod Sribnivasan, Hernan Molina and E. Akleman, "Multiple Handle Creation and Multiple Hole Opening", Proceedings of Bridges 2007.
(Paper)
Description:
In this paper, we present the concept of multiple handle operation to create complicated high genus virtual sculptures.
We have developed and implemented a simple procedure to create multiple handles that connect a set of faces in 3D.
To create multiple handles, we first create a connector, which is a convex shaped mesh surface. We then simply
connect each selected face to this connector surface with a simple one segment handle. If the connector is inside of
the original mesh and the handles goes through the inside of the objects, the result becomes multiple hole.


E. Akleman, "Designing Symmetric HighGenus Sculptures", Siggraph'2006 Art Exhibition and Presentation.
(Extended Abstract) (paper)
Description:
In this paper, we present a procedure to create a new sculptural
family with interactive topological modeling. Using this procedure
a large set of sculptures that have a similar conceptual form can easily
be created. We have tested the procedure in
a computer aided sculpting course. We observe
that, using the procedure, students can rapidly create a wide variety
of shape. Although these shapes are completely different; they
are indistinguishably belong the same family.

 E. Akleman, O. Ozener and C. Yuksel,
"Designing Symmetric HighGenus Sculptures, Proceedings of Bridges 2006, London .
(Paper)
Description: This paper introduces a design guideline to construct a family of
symmetric, connected sculptures with high number of holes and
handles. Our guideline provides users a creative flexibility.
Using this design guideline, sculptors can easily create a wide
variety of sculptures with a similar conceptual form.


E. Akleman, O. Ozener and V. Srinivasan, Rind Architecture, International Journal of Architectural Computing, 2005
(Paper)
Selected from ECAADE 2004 paper by O. Ozener, E. Mandal and E. Akleman,
"Rind Modeling for Architectural Design", Education and Research in Computer Aided Architectural Design in Europe:
EcaadE'04, Copenhagen, Denmark, September 2004.
(Paper)
Description: This paper presents a new modeling technique for architectural design. Rind modeling
provides for the easy creation of surfaces resembling peeled and punctured
rinds. We show how the method‘s two main steps of 1) creation of a shell or crust
2) opening holes in the crust by punching or peeling can be encapsulated into a
real time semiautomatic interactive algorithm. We include a number of worked
examples, some by students in a special modeling workshop that demonstrate the
ease with which a large variety of intricate rind shapes can be created.
Rind modeling method allows us developing a userfriendly tool for designers and
architects. The new tool extends the abilities of polygonal modeling and allows
designers to work on structured and consistent models for architectural design
purposes. Rind modeling gives architects and designers a processing flexibility.


V. Srinivasan, O. Ozener, E. Mandal and E. Akleman, Solidfying Frames For Architecture, CAAD FUTURES 2005, Vien, Austria, June 2005.
(Paper)
Description: In this paper we present a method to convert a wireframe mesh into a 2manifold mesh consisting
of cylindrical pipes in place of the edges and joints in place of the vertices in the original mesh. Our
method allows users to create unique artistic depictions of common objects and structures. The resulting
mesh is also more effective at conveying the overall 3D structure and any internal elements of a model
when compared to regular wireframe or boundary representations. The input wireframe mesh can be
any collection of linear edges; they do not have to form a manifold surface or even be connected to
each other. The result is always an orientable 2manifold surface. Our algorithm replaces every edge in
the wireframe mesh with a cylindrical 3D pipe. The pipes are connected to each other using 3D joints
created at the vertices in the wireframe where the edges meet. Our method has been implemented as part
of a polygonal mesh modeling system and has been used to create artistic models of popular architectural
structures as well as to create conceptual sketches for virtual environments.


E. Akleman, J. Chen and V. Srinivasan, "An Interactive Shape Modeling System for Robust Design of Functional 3D Shapes", pp. 248257 Proceedings of ACADIA 2001, Buffalo, NW., October 2001.
(Paper)
Description: In Architecture, it is essential to design functional and topologically complicated
3D shapes (i.e. shapes with many holes, columns and handles). In this
paper, we present a robust and interactive system for the design of functional
and topologically complicated 3D shapes. Users of our system can easily change
topology (i.e. they can create and delete holes and handles, connect and disconnect
surfaces). Our system also provide smoothing operations (subdivision
schemes) to create smooth surfaces. Moreover, the system provides automatic
texture mapping during topology and smoothing operations.
We also present new design approaches with the new modeling system. The
new design approaches include blending surfaces, construction of crusts and
opening holes on these crusts.

Topological Repair and Simplification 
 V. Srinivasan, E. Akleman and J. Keyser,
Topological Construction of 2Manifold Meshes
from Arbitrary Polygonal Data,
Technical Report, January 2004.
(Paper)
Description: In this paper we present a simple algorithm to construct
2manifold meshes from arbitrary collections of polygons.
We form our final data structure using two very basic
manifoldpreserving operations, thus guaranteeing that the
result is a valid manifold. Our algorithm is purely topological
and does not consider the geometric properties of the
underlying shape.
The algorithm automatically and correctly creates the
missing faces of manifolds with boundaries. It also eliminates
all twogons (2sided polygons) and converts nonmanifold
meshes into one of the possible manifold interpretations.
We have implemented this algorithm and we
highlight the performance of our algorithm on a number
of sample models.

 E. Akleman and J. Chen,
Progressive Refinement with Topological Simplification, Technical Report,
January 2003.
(Paper)
Description: This paper presents a theoretical framework for progressive
refinement of manifold meshes with topological simplification.
We demonstrate that topology changes are not
intuitive and therefore a great deal of care is necessary
for handling topological simplification. We illustrate nonintutive
nature of topology changes with several examples.
We also show how to use the nonintutive nature of the
topology changes as an advantage and develop a theretical
framework for progressive refinement with topological simplification.

These two manuscripts presents TopMod3D (a.k.a. TopMod).
The concepts and algorithms behind the tools in TopMod have
been developed, implemented and published by our research group that can be seen above. Many of these
tools are unique to TopMod.

 E. Akleman, V. Srinivasan, J. Chen, David Morris, and Stuart Tett,
"TopMod3D: An Interactive Topological Mesh Modeler", Computer Graphics International 2008, pp. 1018.
(Paper)
Description: This is a description of TopMod 2.0. In August 2007, we released a new version, TopMod
2.0, with an improved user interface and scripting editor. For the interface of the new version, we switched
from FLTK to Qt. The new version also runs on Mac,
Linux and Windows platforms.We have also developed a
website to create a community around the software. This experience is a strong example of the impor
tance of creation of a community for the useability of
software. For instance, many people discovered ways to
create unusually interesting shapes and shared their ex
periences by developing video tutorials. Other users, fol
lowing video tutorials, created similar shapes. Having a
community also helped to solve portability problems. For
instance, the script editor was initially developed on the
Mac platform and we had trouble compiling the code for
Windows. One user from Italy provided a solution to this
problem. Another user from France translated the user
interface from English to French. The model is in the left is created by one of the users, Jonathan Johanson from Germany.

 E. Akleman, V. Srinivasan, E. Mandal, J. Chen, Z. Melek, and E. Lendreneau,
"Topmod: Topological Mesh Modeling System", Technical Report, Aug. 2004.
(Report)
Description: This is description of TopMod 1.0.
The initial version of the software, TopMod 1.0, has
been available as free software since 2003. Since then, several talented artists
created very interesting sculptures using TopMod 1.0.
TopMod 1.0 was implemented in C++ using OpenGL
and FLTK. It runs on Mac, Linux and Windows
platforms. This initial version, although was not userfiendly, was discovered by a few designers and
they created interesting models. The image in the left was created by Torolf Sauermann from Germany

