2,209 research outputs found
Coverage and Connectivity in Three-Dimensional Networks
Most wireless terrestrial networks are designed based on the assumption that
the nodes are deployed on a two-dimensional (2D) plane. However, this 2D
assumption is not valid in underwater, atmospheric, or space communications. In
fact, recent interest in underwater acoustic ad hoc and sensor networks hints
at the need to understand how to design networks in 3D. Unfortunately, the
design of 3D networks is surprisingly more difficult than the design of 2D
networks. For example, proofs of Kelvin's conjecture and Kepler's conjecture
required centuries of research to achieve breakthroughs, whereas their 2D
counterparts are trivial to solve. In this paper, we consider the coverage and
connectivity issues of 3D networks, where the goal is to find a node placement
strategy with 100% sensing coverage of a 3D space, while minimizing the number
of nodes required for surveillance. Our results indicate that the use of the
Voronoi tessellation of 3D space to create truncated octahedral cells results
in the best strategy. In this truncated octahedron placement strategy, the
transmission range must be at least 1.7889 times the sensing range in order to
maintain connectivity among nodes. If the transmission range is between 1.4142
and 1.7889 times the sensing range, then a hexagonal prism placement strategy
or a rhombic dodecahedron placement strategy should be used. Although the
required number of nodes in the hexagonal prism and the rhombic dodecahedron
placement strategies is the same, this number is 43.25% higher than the number
of nodes required by the truncated octahedron placement strategy. We verify by
simulation that our placement strategies indeed guarantee ubiquitous coverage.
We believe that our approach and our results presented in this paper could be
used for extending the processes of 2D network design to 3D networks.Comment: To appear in ACM Mobicom 200
Repetitive Delone Sets and Quasicrystals
This paper considers the problem of characterizing the simplest discrete
point sets that are aperiodic, using invariants based on topological dynamics.
A Delone set whose patch-counting function N(T), for radius T, is finite for
all T is called repetitive if there is a function M(T) such that every ball of
radius M(T)+T contains a copy of each kind of patch of radius T that occurs in
the set. This is equivalent to the minimality of an associated topological
dynamical system with R^n-action. There is a lower bound for M(T) in terms of
N(T), namely N(T) = O(M(T)^n), but no general upper bound.
The complexity of a repetitive Delone set can be measured by the growth rate
of its repetitivity function M(T). For example, M(T) is bounded if and only if
the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and
densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive
sets and densely repetitive sets have strict uniform patch frequencies, i.e.
the associated topological dynamical system is strictly ergodic. It follows
that such sets are diffractive. In the reverse direction, we construct a
repetitive Delone set in R^n which has
M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform
patch frequencies. Aperiodic linearly repetitive sets have many claims to be
the simplest class of aperiodic sets, and we propose considering them as a
notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37
pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te
Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions
Three-dimensional integer partitions provide a convenient representation of
codimension-one three-dimensional random rhombus tilings. Calculating the
entropy for such a model is a notoriously difficult problem. We apply
transition matrix Monte Carlo simulations to evaluate their entropy with high
precision. We consider both free- and fixed-boundary tilings. Our results
suggest that the ratio of free- and fixed-boundary entropies is
, and can be interpreted as the ratio of the
volumes of two simple, nested, polyhedra. This finding supports a conjecture by
Linde, Moore and Nordahl concerning the ``arctic octahedron phenomenon'' in
three-dimensional random tilings
An Overview of Rendering from Volume Data --- including Surface and Volume Rendering
Volume rendering is a title often ambiguously used in science. One meaning often quoted is: `to render any three volume dimensional data set'; however, within this categorisation `surface rendering'' is contained. Surface rendering is a technique for visualising a geometric representation of a surface from a three dimensional volume data set. A more correct definition of Volume Rendering would only incorporate the direct visualisation of volumes, without the use of intermediate surface geometry representations. Hence we state: `Volume Rendering is the Direct Visualisation of any three dimensional Volume data set; without the use of an intermediate geometric representation for isosurfaces'; `Surface Rendering is the Visualisation of a surface, from a geometric approximation of an isosurface, within a Volume data set'; where an isosurface is a surface formed from a cross connection of data points, within a volume, of equal value or density. This paper is an overview of both Surface Rendering and Volume Rendering techniques. Surface Rendering mainly consists of contouring lines over data points and triangulations between contours. Volume rendering methods consist of ray casting techniques that allow the ray to be cast from the viewing plane into the object and the transparency, opacity and colour calculated for each cell; the rays are often cast until an opaque object is `hit' or the ray exits the volume
Puzzling the 120-cell
We introduce Quintessence: a family of burr puzzles based on the geometry and
combinatorics of the 120-cell. We discuss the regular polytopes, their
symmetries, the dodecahedron as an important special case, the three-sphere,
and the quaternions. We then construct the 120-cell, giving an illustrated
survey of its geometry and combinatorics. This done, we describe the pieces out
of which Quintessence is made. The design of our puzzle pieces uses a drawing
technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout.
This is the long version of the shorter published versio
Recursive tilings and space-filling curves with little fragmentation
This paper defines the Arrwwid number of a recursive tiling (or space-filling
curve) as the smallest number w such that any ball Q can be covered by w tiles
(or curve sections) with total volume O(vol(Q)). Recursive tilings and
space-filling curves with low Arrwwid numbers can be applied to optimise disk,
memory or server access patterns when processing sets of points in
d-dimensional space. This paper presents recursive tilings and space-filling
curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube
tilings and space-filling curves cannot have optimal Arrwwid number, and we see
how to construct alternatives with better Arrwwid numbers.Comment: Manuscript accompanying abstract in EuroCG 2010, including full
proofs, 20 figures, references, discussion et
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