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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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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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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.
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Johannes Kepler was a pivotal figure in the scientific revolution, best known for his laws of planetary motion which laid the groundwork for Newton's theory of universal gravitation. His work bridged the gap between Copernican heliocentrism and the physics of motion, fundamentally altering our understanding of the cosmos.
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Formal verification is a mathematical approach to prove or disprove the correctness of algorithms underlying a system with respect to a certain formal specification or property. It is crucial in ensuring the reliability and safety of critical systems where failure is not an option, such as in aerospace, medical devices, and cryptographic protocols.
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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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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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