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The strictly jammed (mechanically stable even as a finite system) regular sphere packing with the lowest known density is a diluted ("tunneled") fcc crystal with a density of only Ο β 2 /9 β 0.49365. [6] The loosest known regular jammed packing has a density of approximately 0.0555. [7]
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume [2] =. The altitude h satisfies [3] = + +. The area of the base is given by [4] =. The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure Ο /2 steradians, one eighth of the surface area of a unit sphere.
If the regular tetrahedron has edge length π = 2, its characteristic tetrahedron's six edges have lengths , , around its exterior right-triangle face (the edges opposite the characteristic angles π, π, π), [a] plus , , (edges that are the characteristic radii of the regular tetrahedron).
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
where N particle is the number of particles in the unit cell, V particle is the volume of each particle, and V unit cell is the volume occupied by the unit cell. It can be proven mathematically that for one-component structures, the most dense arrangement of atoms has an APF of about 0.74 (see Kepler conjecture ), obtained by the close-packed ...
The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. [1] [2] Highest density is known only for 1, 2, 3, 8, and 24 dimensions. [3]
The empty space between spheres varies depending on the type of packing. The amount of empty space is measured in the packing density, which is defined as the ratio of the volume of the spheres to the volume of the total convex hull. The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of ...