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Sphere packing is the arrangement of non-overlapping spheres within a given space to maximize density, a problem that has applications in fields such as crystallography, coding theory, and discrete geometry. The densest known packing in three dimensions is the face-centered cubic lattice, proven by Thomas Hales in the Kepler conjecture resolution.
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The Kepler Conjecture posits that no arrangement of equally sized spheres filling space has a greater average density than that of the face-centered cubic packing or hexagonal close packing, both of which have a density of approximately 74.048%. This conjecture, first proposed by Johannes Kepler in 1611, was proven by Thomas Hales in 1998 using a combination of traditional mathematical proof and computer verification, marking a significant milestone in the field of discrete geometry.
Concept
A face-centered cubic (FCC) lattice is a crystal structure where atoms are located at each corner and the center of each face of the cube, resulting in a highly efficient packing arrangement. This structure is prevalent in many metals, contributing to their ductility and high packing density, which is 74% of the volume occupied by atoms.
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Density is a measure of how much mass an object or substance has in a given volume, often expressed in units like kilograms per cubic meter or grams per cubic centimeter. It is a fundamental property used to characterize materials and can affect how substances interact, float, or sink in different environments.
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Lattice packing is a mathematical arrangement of non-overlapping spheres in a regular, repeating pattern within a given space, aiming to maximize the density of the spheres. It is a fundamental problem in discrete geometry and has applications in physics, materials science, and coding theory.
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Discrete geometry is a branch of geometry that studies combinatorial properties and constructive methods of discrete geometric structures. It often focuses on the arrangements and properties of finite sets of geometric objects, such as points, lines, and polygons, and has applications in computer science, particularly in algorithms and computational geometry.
Concept
Hexagonal close packing (HCP) is a highly efficient arrangement of spheres where each sphere is surrounded by 12 others, forming a hexagonal lattice structure. This arrangement is commonly found in metals like magnesium and titanium, providing them with unique mechanical properties due to the dense packing and specific symmetry of the lattice.
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Packing fraction is a measure of the density of particles in a given volume, often used in nuclear physics to describe the stability of a nucleus. It represents the difference between the actual mass of a nucleus and its mass number, indicating how tightly nucleons are bound together.
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A Voronoi diagram is a partitioning of a plane into regions based on the distance to a specific set of points, where each region contains all points closer to one particular seed point than to any other. This geometric structure is widely used in fields like computer graphics, spatial analysis, and optimization to model natural phenomena and solve proximity problems.
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The Kissing Number Problem is a classic problem in geometry that seeks to determine the maximum number of non-overlapping unit spheres that can touch another unit sphere in n-dimensional space. The problem has been solved for dimensions 1, 2, 3, 4, 8, and 24, with the solution for three dimensions famously being 12, a result first proven by Isaac Newton and later rigorously confirmed in the 20th century.
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Crystallography is the scientific study of crystal structures and properties, primarily using X-ray diffraction to determine the atomic and molecular arrangement within a crystal. It is crucial in fields like materials science, chemistry, and biology for understanding the structural basis of material properties and biological functions.
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A sphere is a perfectly symmetrical three-dimensional geometric object where every point on its surface is equidistant from its center, making it a fundamental shape in mathematics and physics. Spheres are prevalent in natural and artificial contexts, from celestial bodies to everyday objects, and their properties are crucial in fields such as geometry, calculus, and topology.
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Packing and covering are fundamental concepts in combinatorial optimization and geometry, focusing on how to efficiently fill or cover a space with geometric shapes without overlap (packing) or with complete coverage (covering). These concepts have applications in fields such as network design, coding theory, and resource allocation, where optimal space utilization or coverage is critical.
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The Leech lattice is a highly symmetrical, 24-dimensional lattice that is the densest known sphere packing in 24 dimensions. It plays a crucial role in various areas of mathematics and theoretical physics, including string theory, coding theory, and the study of sporadic simple groups.
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Geometric discrepancy measures the deviation of a point set from an ideal distribution in a geometric space, serving as a fundamental tool in numerical analysis, computational geometry, and quasi-Monte Carlo methods. It quantifies how uniformly or non-uniformly points are distributed, impacting the efficiency and accuracy of algorithms in these fields.
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Space-filling geometry explores how shapes can completely cover a space without leaving any gaps or overlaps, significantly influencing fields such as crystallography and materials science. This concept is foundational to understanding both natural structures, like honeycombs, and engineered solutions, such as efficient packing and storage designs.
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