45 research outputs found
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
Discrete Differential Geometry
This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry
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Geometry-based structural analysis and design via discrete stress functions
This PhD thesis proposes a direct and unified method for generating global static equilibrium
for 2D and 3D reciprocal form and force diagrams based on reciprocal discrete stress
functions. This research combines and reinterprets knowledge from Maxwell’s 19th century
graphic statics, projective geometry and rigidity theory to provide an interactive design and
analysis framework through which information about designed structural performance can be
geometrically encoded in the form of the characteristics of the stress function. This method
results in novel, intuitive design and analysis freedoms.
In contrast to contemporary computational frameworks, this method is direct and analytical.
In this way, there is no need for iteration, the designer operates by default within
the equilibrium space and the mathematically elegant nature of this framework results in its
wide applicability as well as in added educational value. Moreover, it provides the designers
with the agility to start from any one of the four interlinked reciprocal objects (form diagram,
force diagram, corresponding stress functions).
This method has the potential to be applied in a wide range of case studies and fields.
Specifically, it leads to the design, analysis and load-path optimisation of tension-and compression
2D and 3D trusses, tensegrities, the exoskeletons of towers, and in conjunction
with force density, to tension-and-compression grid-shells, shells and vaults. Moreover, the
abstract nature of this method leads to wide cross-disciplinary applicability, such as 2D and
3D discrete stress fields in structural concrete and to a geometrical interpretation of yield line
theory
Incircular nets and confocal conics
We consider congruences of straight lines in a plane with the combinatorics
of the square grid, with all elementary quadrilaterals possessing an incircle.
It is shown that all the vertices of such nets (we call them incircular or
IC-nets) lie on confocal conics.
Our main new results are on checkerboard IC-nets in the plane. These are
congruences of straight lines in the plane with the combinatorics of the square
grid, combinatorially colored as a checkerboard, such that all black coordinate
quadrilaterals possess inscribed circles. We show how this larger class of
IC-nets appears quite naturally in Laguerre geometry of oriented planes and
spheres, and leads to new remarkable incidence theorems. Most of our results
are valid in hyperbolic and spherical geometries as well. We present also
generalizations in spaces of higher dimension, called checkerboard IS-nets. The
construction of these nets is based on a new 9 inspheres incidence theorem.Comment: 33 pages, 24 Figure
Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Folding Orthogonal Polyhedra
In this thesis, we study foldings of orthogonal polygons into orthogonal polyhedra. The particular problem examined here is whether a paper cutout of an orthogonal polygon with fold lines indicated folds up into a simple orthogonal polyhedron. The folds are orthogonal and the direction of the fold (upward or downward) is also given. We present a polynomial time algorithm to solve this problem. Next we consider the same problem with the exception that the direction of the folds are not given. We prove that this problem is NP-complete. Once it has been determined that a polygon does fold into a polyhedron, we consider some restrictions on the actual folding process, modelling the case when the polyhedron is constructed from a stiff material such as sheet metal. We show an example of a polygon that cannot be folded into a polyhedron if folds can only be executed one at a time. Removing this restriction, we show another polygon that cannot be folded into a polyhedron using rigid material