Cube | EPFL Graph Search

En géométrie euclidienne, un cube est un prisme droit dont toutes les faces sont carrées donc égales et superposables. Le cube figure parmi les solides les plus remarquables de l'espace. C'est le seul des cinq solides de Platon ayant exactement 6 faces, 12 arêtes et 8 sommets. Son autre nom est « hexaèdre régulier ». Le cube est un zonoèdre à trois générateurs. Comme il a quatre sommets par face et trois faces par sommet, son symbole de Schläfli est {4,3}. L'étymologie du mot cube est grecque ; cube provient de kubos, le dé. Le terme de cube, appliqué à un nombre, désigne la valeur obtenue en multipliant ce nombre par lui-même et en remultipliant le résultat par le nombre initial. Cette expression s'est imposée durant la période où l'algèbre géométrique était omniprésente, le carré d'un nombre était vu comme la surface d'un carré de côté le nombre initial et le cube d'un nombre comme le volume d'un cube de côté le nombre initial. L'expression « a3 » peut se lire « a au cube » et «

Multi-physical characterization of cellular ceramics for high-temperature applications

Ehsan Rezaei

The use of cellular ceramics in enhancing the performance of a high-temperature latent heat thermal energy storage unit was investigated. A detailed design methodology is presented, which consists of a combined analytical-numerical analysis followed by a multi-objective optimization. This optimization indicated that within the selected design space, effectiveness values as large as 0.95 and energy densities as large as 810 MJ/m3 could be achieved. Motivated by the results of this study, new porous structures were investigated. As the classical computer aided design tools are not optimized for quick and efficient design of cellular structures with large number of geometrical features, new design approaches were presented: two methods to design structured and unstructured lattices and a Voronoi-based design approach to create structures consisting of different unit-cells combined together. We then used a combined experimental-numerical approach to investigate the effect of the cell morphology on the heat and mass transport behavior of the porous structures. Different morphologies, namely tetrakaidecahedron, Weaire-Phelan, rotated cube and random foam, were investigated. These structures were designed in cylindrical forms, 3D printed and then manufactured in SiSiC via replica technique followed by silicon reactive infiltration. Permeability and Forchheimer coefficients of the structures were experimentally measured by pressure drop tests at room temperature. The volumetric convective heat transfer coefficients were estimated using temperature measurements and fitting a thermal non-equilibrium heat and fluid flow model to these experiments. It was observed that for the same porosity and cell density the cubic lattice and the random foam exhibited lower pressure drops but also lower heat transfer rates. Undesirable manufacturing anomalies such as pore clogging, was observed for tetrakaidecahedron and Weaire-Phelan structures, which led to a tortuosity larger than calculated, causing additional pressure drop. Finally, the mechanical and degradation behavior of five SiSiC cellular structures, namely simple cube, rotated cube, tetrakaidecahedron, modified octet-truss and random foam, was experimentally investigated in early stage oxidation conditions at 1400 °C. The samples were oxidized in two different environments: in a radiant burner and inside an electric furnace. The results revealed different mechanisms, namely silicon alloy bead formation and H2O/CO2-based corrosion, simultaniously degrading the specimens. It is shown that different lattice architectures led to different oxidation behavior on the struts resulting from the changing gas flow paths inside each ceramic architecture. The effect of the morphology on the elastic behavior of lattice structures was studied in more detail by adapting a numerical approach consisting of a unit-cell model with periodic boundaries. The elastic anisotropies of the lattices were explored by calculating the elastic modulus in different directions. The results revealed that all the studied lattices, and in particular the cubic lattice, have an anisotropic elastic behavior. A new strategy is presented to obtain unit-cells with high elastic modulus and controled anisotropy.

EPFL2020

Decomposition of a Cube into Nearly Equal Smaller Cubes

János Pach

Let d be a fixed positive integer and let epsilon > 0. It is shown that for every sufficiently large n >= n(0)( d, e), the d-dimensional unit cube can be decomposed into exactly n smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most 1 +epsilon. Moreover, for every n >= n(0), there is a decomposition with the required properties, using cubes of at most d + 2 different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of three different sizes.

Mathematical Assoc Amer2017

Planck 2013 results. XXVI. Background geometry and topology of the Universe

Fabio Finelli, Julien Lesgourgues, Jason Douglas McEwen

The new cosmic microwave background (CMB) temperature maps from Planck provide the highest-quality full-sky view of the surface of last scattering available to date. This allows us to detect possible departures from the standard model of a globally homogeneous and isotropic cosmology on the largest scales. We search for correlations induced by a possible non-trivial topology with a fundamental domain intersecting, or nearly intersecting, the last scattering surface (at comoving distance chi(rec)), both via a direct search for matched circular patterns at the intersections and by an optimal likelihood search for specific topologies. For the latter we consider flat spaces with cubic toroidal (T3), equal-sided chimney (T2) and slab (T1) topologies, three multi-connected spaces of constant positive curvature (dodecahedral, truncated cube and octahedral) and two compact negative-curvature spaces. These searches yield no detection of the compact topology with the scale below the diameter of the last scattering surface. For most compact topologies studied the likelihood maximized over the orientation of the space relative to the observed map shows some preference for multi-connected models just larger than the diameter of the last scattering surface. Since this effect is also present in simulated realizations of isotropic maps, we interpret it as the inevitable alignment of mild anisotropic correlations with chance features in a single sky realization; such a feature can also be present, in milder form, when the likelihood is marginalized over orientations. Thus marginalized, the limits on the radius R-i of the largest sphere inscribed in topological domain (at log-likelihood-ratio Delta ln L > -5 relative to a simply-connected flat Planck best-fit model) are: in a flat Universe, R-i > 0.92 chi(rec) for the T3 cubic torus; R-i > 0.71 chi(rec) for the T2 chimney; R-i > 0.50 chi(rec) for the T1 slab; and in a positively curved Universe, R-i > 1.03 chi(rec) for the dodecahedral space; R-i > 1.0 chi(rec) for the truncated cube; and R-i > 0.89 chi(rec) for the octahedral space. The limit for a wider class of topologies, i. e., those predicting matching pairs of back-to-back circles, among them tori and the three spherical cases listed above, coming from the matched-circles search, is R-i > 0.94 chi(rec) at 99% confidence level. Similar limits apply to a wide, although not exhaustive, range of topologies. We also perform a Bayesian search for an anisotropic global Bianchi VIIh geometry. In the non-physical setting where the Bianchi cosmology is decoupled from the standard cosmology, Planck data favour the inclusion of a Bianchi component with a Bayes factor of at least 1.5 units of log-evidence. Indeed, the Bianchi pattern is quite efficient at accounting for some of the large-scale anomalies found in Planck data. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. In the physically motivated setting where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VIIh cosmology and constrain the vorticity of such models to (omega/H)(0) < 8.1 x 10(-10) (95% confidence level).