In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space.
Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%. [1] Tetrahedra do not tile space, [2] and an upper bound below 100% (namely, 1 − (2.6...)·10−25) has been reported. [3]
Aristotle claimed that tetrahedra could fill space completely. [4] [5]
In 2006, Conway and Torquato showed that a packing fraction about 72% can be obtained by constructing a non-Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing (with one particle per repeating unit such that each particle has a common orientation). [6] These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman. [7] In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%. [8] In 2008, Chen was the first to propose a packing of hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%. [9] [10] A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%. [11]
In mid-2009 Haji-Akbari et al. showed, using MC simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to 83.24%. They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%. [12]
In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. [13] These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, [14] and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%. [1] The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. Surprisingly, the square-triangle tiling [12] packs denser than this double lattice of triangular bipyramids when tetrahedra are slightly rounded (the Minkowski sum of a tetrahedron and a sphere), making the 82-tetrahedron crystal the largest unit cell for a densest packing of identical particles to date. [15]
Because the earliest lower bound known for packings of tetrahedra was less than that of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent spheres is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions or to non-Euclidean spaces such as hyperbolic space.
In geometry, a rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.
In physics, an entropic force acting in a system is an emergent phenomenon resulting from the entire system's statistical tendency to increase its entropy, rather than from a particular underlying force on the atomic scale.
László Fejes Tóth was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane. He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer.
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container. In other words, shaking increases the density of packed objects. But shaking cannot increase the density indefinitely, a limit is reached, and if this is reached without obvious packing into an ordered structure, such as a regular crystal lattice, this is the empirical random close-packed density for this particular procedure of packing. The random close packing is the highest possible volume fraction out of all possible packing procedures.
In geometry, circle packing is the study of the arrangement of circles on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.
The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes. It was also independently discovered by Kurt Mahler in 1947. It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.
Sharon C. Glotzer is an American scientist and "digital alchemist", the Anthony C. Lembke Department Chair of Chemical Engineering, the John Werner Cahn Distinguished University Professor of Engineering and the Stuart W. Churchill Collegiate Professor of Chemical Engineering at the University of Michigan, where she is also professor of materials science and engineering, professor of physics, professor of macromolecular science and engineering, and professor of applied physics. She is recognized for her contributions to the fields of soft matter and computational science, most notably on problems in assembly science and engineering, nanoscience, and the glass transition, for which the elucidation of the nature of dynamical heterogeneity in glassy liquids is of particular significance. She is a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences.
Salvatore Torquato is an American theoretical scientist born in Falerna, Italy. His research work has impacted a variety of fields, including physics, chemistry, applied and pure mathematics, materials science, engineering, and biological physics. He is the Lewis Bernard Professor of Natural Sciences in the department of chemistry and Princeton Institute for the Science and Technology of Materials at Princeton University. He has been a senior faculty fellow in the Princeton Center for Theoretical Science, an enterprise dedicated to exploring frontiers across the theoretical natural sciences. He is also an associated faculty member in three departments or programs at Princeton University: physics, applied and computational mathematics, and mechanical and aerospace engineering. On multiple occasions, he was a member of the schools of mathematics and natural sciences at the Institute for Advanced Study, Princeton, New Jersey.
Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says that the optimal density for packing congruent spheres is smaller than that for any other convex body. That is, according to the conjecture, the ball is the convex solid which forces the largest fraction of space to remain empty in its optimal packing structure. This conjecture is therefore related to the Kepler conjecture about sphere packing. Since the solution to the Kepler conjecture establishes that identical balls must leave ≈25.95% of the space empty, Ulam's conjecture is equivalent to the statement that no other convex solid forces that much space to be left empty.
In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of any point under the action of a double lattice is a union of two Bravais lattices, related to each other by a point reflection. A double lattice in two dimensions is a p2 wallpaper group. In three dimensions, a double lattice is a space group of the type 1, as denoted by international notation.
Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called columnar structures.
In geometry, ellipsoid packing is the problem of arranging identical ellipsoid throughout three-dimensional space to fill the maximum possible fraction of space.