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Natural gas hydrate
Published in Jon Steinar Gudmundsson, Flow Assurance Solids in Oil and Gas Production, 2017
Crystals are three-dimensional structures. Cubes have six sides (also called faces) and triangular pyramids four sides. The cube form is also called hexahedron and the pyramid form tetrahedron. Collectively, the various forms are called polyhedra (singular, polyhedron). A polyhedron with 12-faces is called dodecahedron. Each of the 12 faces is shaped like a pentagon. One of the crystal cages that can trap small gas molecules is shaped like a dodecahedron, shown in Figure 5.1.
Shape effect of grain in a granular flow
Published in Y. Kishino, Powders and Grains 2001, 2020
D. Petit, F. Pradel, G. Ferrer, Y. Meimon
To stress out the effect of the morphology factor, special polyhedra are selected to replace the spherical grain. It has been decided to select the regular polyhedra which are inscribed into the spherical grain they are replacing. In a tridimensional space, there is only five polyhedra called the Platonic solids and reported on figure 3: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
Introduction to Random Signals
Published in Shaila Dinkar Apte, Random Signal Processing, 2017
A dodecahedron is a solid object with 12 equal faces. It is frequently used as a calendar paperweight with a month name placed on each face. If such a calendar is randomly placed on a desk, the outcome is taken to be the month of the upper face.
Reimagining the history of GIS
Published in Annals of GIS, 2018
Rather than lay a quadtree on a flattened Earth, DGGs are hierarchical structures on the curved surface of the Earth itself, thus avoiding all of the distortions inherent in map projections. Unfortunately only five ways exist to create a 3D solid using pieces of equal size and shape. These are the five Platonic solids, and have been known since classical times: the 4 triangles of a tetrahedron, the 6 squares of a cube, the 8 triangles of an octahedron, the 12 pentagons of a dodecahedron and the 20 triangles of an icosahedron. Instead, DGGs begin with one of the Platonic solids or a simple modification, and then use a hierarchical scheme to subdivide each of its faces. Since triangles are the favoured display element of 3D graphics systems, it is desirable that the basic elements at any level in the hierarchy be triangles. Triangles are also convenient since it is desirable that basic elements at each level nest within the corresponding element at the next higher level, and it is easy to create four nesting triangles from a larger triangle by connecting the midpoints of its edges. On the other hand the pentagons of the dodecahedron do not lead to hierarchies with simple and desirable properties.
Geometry in Our Three-Dimensional World
Published in Technometrics, 2023
Chapter 6 is a joy to read and discusses, among other things, Platonic and Archimedean solids. These solids are polyhedral and are the natural spatial equivalents of plane objects such as triangles, squares, and other polygons. The fact that there are only five Platonic solids can be easily established thanks to Euler’s characteristics formula. Using the concept of duality, it is shown that a dodecahedron (with 20 faces and 12 vertices) is the dual of an icosahedron (with 12 faces and 20 vertices). A hexahedron (aka a cube) is shown as the dual of an octahedron. A tetrahedron, however, is self-dual and is without a Platonic partner. Somewhat less popular but equally interesting Kepler-Poinsot solids have also been discussed in considerable detail.
Conjugate heat transfer analysis within in lattice-filled heat exchanger for additive manufacturing
Published in Mechanics of Advanced Materials and Structures, 2022
Nadhir Lebaal, Abdelhakim SettaR, Sebastien Roth, Samuel Gomes
The model of lattice pattern used in this study is called the Dodecahedron network [25]. The geometry of the dodecahedron is a polyhedron with twelve flat faces as shown in Figure 1. The best-known dodecahedron is the regular dodecahedron, which is a platonic solid.