Three-dimensional alpha shapes

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the ``shape'' of the set. For that purpose, this paper introduces the formal notion of the family of $\alpha$-shapes of a finite point set in \Real^3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter \alpha \in \Real controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size $n$ in time $O(n^2)$, worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.Comment: 32 page

Betti number signatures of homogeneous Poisson point processes

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.Comment: Submitted to PRE. 11 pages, 10 figure

Reconstructing the triaxiality of the galaxy cluster Abell 1689: solving the X-ray and strong lensing mass discrepancy

We present the first determination of the intrinsic triaxial shapes and tree-dimensional physical parameters of both dark matter (DM) and intra-cluster medium (ICM) for the galaxy cluster Abell 1689. We exploit the novel method we recently introduced (Morandi et al. 2010) in order to infer the tree-dimensional physical properties in triaxial galaxy clusters by combining jointly X-ray and strong lensing data. We find that Abell 1689 can be modeled as a triaxial galaxy cluster with DM halo axial ratios 1.24 +/- 0.13 and 2.37 +/- 0.11 on the plane of the sky and along the line of sight, respectively. We show that accounting for the three-dimensional geometry allows to solve the discrepancy between the mass determined from X-ray and strong gravitational lensing observations. We also determined the inner slope of the DM density profile alpha: we measure alpha = 0.90 +/- 0.05 by accounting explicitly for the 3D structure for this cluster, a value which is close to the cold dark matter (CDM) predictions, while the standard spherical modeling leads to the biased value alpha = 1.16 +/- 0.04. Our findings dispel the potential inconsistencies arisen in the literature between the predictions of the CDM scenario and the observations, providing further evidences that support the CDM scenario.Comment: Accepted for publication in Ap

Unveiling the three-dimensional structure of galaxy clusters: resolving the discrepancy between X-ray and lensing masses

[Abridged] We present the first determination of the intrinsic three-dimensional shapes and the physical parameters of both dark matter (DM) and intra-cluster medium (ICM) in a triaxial galaxy cluster. While most previous studies rely on the standard spherical modeling, our approach allows to infer the properties of the non-spherical intra-cluster gas distribution sitting in hydrostatic equilibrium within triaxial DM halos by combining X-ray, weak and strong lensing observations. We present an application of our method to the galaxy cluster MACS J1423.8+2404. This source is an example of a well relaxed object with a unimodal mass distribution and we infer shape and physical properties of the ICM and the DM for this source. We found that this is a triaxial galaxy cluster with DM halo axial ratios 1.53+/-0.15 and 1.44+/-0.07 on the plane of the sky and along the line of sight, respectively. We show that accounting for the three-dimensional geometry allows to solve the long-standing discrepancy between galaxy cluster masses determined from X-ray and gravitational lensing observations. We focus also on the determination of the inner slope of the DM density profile alpha, since the cuspiness of dark-matter density profiles is one of the critical tests of the cold dark matter (CDM) paradigm for structure formation: we measure alpha=0.94+/-0.09 by accounting explicitly for the 3D structure for this cluster, a value which is close to the CDM predictions, while the standard spherical modeling leads to the biased value alpha=1.24+/-0.07. Our findings provide further evidences that support the CDM scenario and open a new window in recovering the intrinsic shapes and physical parameters of galaxy clusters in a bias-free way. This has important consequences in using galaxy clusters as cosmological probes.Comment: Accepted for publication in Ap

Energy of molecular structures in 12^{12}C, 16^{16}O, 20^{20}Ne, 24^{24}Mg, and 32^{32}S

International audienceThe energy of the 12 C, 16 O, 20 Ne, 24 Mg and 32 S 4n-nuclei has been determined within a generalized liquid drop model and assuming different planar and three-dimensional shapes of α-molecules : linear chain, triangle, square, tetrahedron, pentagon, trigonal bipyramid, square pyramid, hexagon, octahedron, octogon and cube. The potential barriers governing the entrance and decay channels via α absorption or emission as well as more symmetric binary and ternary reactions have been compared. The rms radii of the linear chains differ from the experimental rms radii of the ground states. The binding energies of the three-dimensional shapes at the contact point are higher than the ones of the planar configurations. The alpha particle plus A-4 daughter configuration leads always to the lowest potential barrier. The binding energy can be reproduced within the sum of the binding energy of n α particles plus the number of bonds multiplied by 2.4 MeV or by the sum of the binding energies of one alpha particle and the daughter nucleus plus the Coulomb energy and the proximity energy

Microstructural evolution in two-dimensional two-phase polycrystals

In two-dimensional polycrystals composed of [alpha]-phase and [beta]-phase grains the stability of [alpha][alpha][alpha], [beta][beta][beta], [alpha][alpha][beta] and [alpha][beta][beta] three-grain junctions and [alpha][beta][alpha][beta] four-grain junctions depends on the [alpha]-[alpha], [beta]-[beta] and [alpha]-[beta] interfacial energies. A computer simulation which generates thermodynamically consistent microstructures for arbitrary interfacial energies has been utilized to investigate microstructural evolution in such polycrystals when phase volume is not conserved. Since grain shapes, phase volume, and phase arrangements are dictated by interfacial energies, clustered-, alternating-, isolated-, and single-phase microstructures occur in different interfacial energy regimes. Despite great differences in microstructure, polycrystals which contain only three-grain junctions evolve with normal grain growth kinetics. In contrast, structures containing flexible four-grain junctions eventually stop evolving. We conclude that two-dimensional polycrystals continually evolve when grain junction angles are thermodynamically fixed, while grain growth ultimately ceases when grain junction angles may vary. Predictions concerning three-dimensional and phase-volume conserved systems are made.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30895/1/0000564.pd

Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $\alpha$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $\alpha$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.Comment: 42 pages, 14 figure

The Effectiveness of Computer-Assisted Instruction on Students’ Cognitive Skill to Know Geometric Shapes

Abstract: Cognitive skill, the important aspect in early childhood development, consists of three skills such as learning and problem solving, symbolic thinking and logical thinking. The activities in recognizing geometric shapes in children aged 4-5 years are by mentioning geometric shapes, showing geometric shapes, and geometric grouping shapes. The result of observation shows children are difficult to distinguish between square and rectangular shapes when asked. Thus, educational media to overcome these problems as well as to meet the industrial revolution 4.0 challenge is so needed. This study aims to determine the effect of CAI (Computer Assisted Instruction) on children aged 4-5 years in recognizing geometric shapes. This is quantitative descriptive research using a quasi-experimental design with a non-equivalent control group design. The population used were 75 children aged 4-5 years. Data were collected using observation and documentation. The assessment indicators used are to mention geometric shapes, show geometric shapes, connect 2- and 3-dimensional geometry shapes (surrounding objects), and group surrounding objects. Computer-Assisted Instruction (CAI) in recognizing geometric shapes was used 3 times. The average score before treatment was 10.77, while after treatment was12.66. Through the Mann-U Whitney test, the hypothesis was sig = 0,000 < alpha (0.05) meaning that Ho was rejected and Ha was accepted. Therefore, it can be concluded that CAI (Computer Assisted Instruction) is an effective media to recognize geometric shapes in children aged 4-5 years.