📐Discrete Geometry Unit 4 – Lattices and Packings

Introduction to Lattices

• Lattices are regular arrangements of points in space that form a repeating pattern
• Can be defined in any number of dimensions, but most commonly studied in 2D and 3D
• Consist of a set of basis vectors that span the space and define the repeating unit cell
• Play a crucial role in various fields such as crystallography, coding theory, and discrete optimization
• Provide a framework for studying the geometric and algebraic properties of point configurations
• Enable the analysis of symmetry, density, and efficiency in packing problems
• Serve as a bridge between continuous and discrete geometry, allowing for the application of continuous methods to discrete problems

Fundamental Concepts of Packings

• Packings involve the arrangement of objects (spheres, polyhedra, etc.) in space without overlap
• Aim to maximize the density or minimize the empty space between objects
• Can be periodic (lattice packings) or aperiodic (non-lattice packings)
• Characterized by the packing density, which is the fraction of space occupied by the objects
  • Packing density is calculated as the volume of the objects divided by the total volume of the space
• Influenced by the shape and size of the objects being packed
  • Spheres are the most commonly studied objects in packing problems due to their symmetry and simplicity
• Optimal packings are those that achieve the highest possible density for a given object and space
• Have applications in various fields, such as materials science, logistics, and information theory

Types of Lattices

• Bravais lattices are the most fundamental types of lattices, classified by their symmetry properties
  • There are 14 Bravais lattices in 3D and 5 in 2D
• Cubic lattices are characterized by three mutually perpendicular basis vectors of equal length
  • Simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) are the three main types of cubic lattices
• Hexagonal lattices have a six-fold rotational symmetry and are common in 2D and 3D crystal structures
  • The hexagonal close-packed (HCP) lattice is a 3D example of a hexagonal lattice
• Rhombohedral lattices have a three-fold rotational symmetry and can be described by three equal-length basis vectors with equal angles between them
• Tetragonal, orthorhombic, monoclinic, and triclinic lattices are characterized by varying degrees of symmetry and basis vector relationships
• Special lattices, such as the diamond lattice and the honeycomb lattice, have unique properties and are important in specific applications

Lattice Properties and Operations

• Lattice constants define the lengths of the basis vectors and the angles between them
• Primitive cell is the smallest repeating unit that can generate the entire lattice through translations
  • Contains exactly one lattice point and has a volume equal to the determinant of the basis vector matrix
• Reciprocal lattice is a Fourier transform of the direct lattice and plays a crucial role in crystallography
  • Basis vectors of the reciprocal lattice are perpendicular to the planes of the direct lattice
• Lattice transformations, such as rotations, reflections, and shears, can be used to manipulate and analyze lattice structures
• Dual lattices are related to each other through a duality transformation, which exchanges the roles of points and planes
• Lattice reduction techniques, such as the LLL algorithm, aim to find a "good" basis for a given lattice that minimizes certain properties (e.g., basis vector lengths)

Packing Densities and Efficiency

• Packing density quantifies the fraction of space occupied by the packed objects
  • Calculated as the volume of the objects divided by the total volume of the space
• Lattice packing density depends on the arrangement of objects in the unit cell and the lattice type
  • For spheres, the FCC and HCP lattices achieve the highest packing density of $\frac{\pi}{3\sqrt{2}} \approx 0.74048$
• Packing efficiency measures how close a given packing is to the optimal packing density
  • Defined as the ratio of the actual packing density to the optimal packing density
• Kepler's conjecture states that the FCC and HCP lattices are the densest possible sphere packings in 3D
  • Proved by Thomas Hales in 1998 using a combination of mathematical arguments and computer-assisted proofs
• Irregularly shaped objects, such as polyhedra, can have different packing densities and efficiencies compared to spheres
  • The optimal packing arrangements for irregular objects are often more complex and less symmetric than those for spheres
• Jamming is a phenomenon that occurs when a packing becomes mechanically stable and cannot be further compressed without deforming the objects
  • Jamming transitions and critical packing densities are important in the study of granular materials and soft matter physics

Sphere Packing Problems

• Sphere packing problems seek to find the densest possible arrangement of equal-sized spheres in a given space
• The Kepler conjecture, proved by Thomas Hales in 1998, states that the FCC and HCP lattices are the densest possible sphere packings in 3D
  • These lattices achieve a packing density of $\frac{\pi}{3\sqrt{2}} \approx 0.74048$
• In 2D, the hexagonal lattice is the densest possible sphere packing, with a packing density of $\frac{\pi}{2\sqrt{3}} \approx 0.90690$
• The kissing number problem asks for the maximum number of spheres that can touch a central sphere without overlapping
  • In 3D, the kissing number is 12, achieved by the FCC and HCP lattices
• Sphere packing in higher dimensions has important applications in coding theory and information theory
  • The 24-dimensional Leech lattice and the 8-dimensional E8 lattice are examples of dense sphere packings in higher dimensions
• Apollonian sphere packing is a fractal arrangement of spheres that fills space more densely than any lattice packing
  • It is constructed by recursively filling the gaps between spheres with smaller spheres

Applications in Crystallography

• Crystallography is the study of the arrangement of atoms in crystalline solids
• Lattices provide a mathematical framework for describing the periodic structure of crystals
  • The unit cell of a crystal corresponds to the primitive cell of the underlying lattice
• X-ray crystallography uses the diffraction of X-rays by the lattice planes to determine the atomic structure of crystals
  • The reciprocal lattice plays a crucial role in interpreting X-ray diffraction patterns
• Crystal symmetry is described by the combination of the Bravais lattice and the basis (arrangement of atoms within the unit cell)
  • The 230 space groups classify all possible crystal symmetries in 3D
• Lattice defects, such as vacancies, interstitials, and dislocations, can significantly influence the properties of crystalline materials
  • The study of lattice defects is important for understanding the mechanical, electrical, and optical behavior of crystals
• Quasicrystals are materials with long-range order but no translational symmetry, challenging the traditional notion of lattices in crystallography
  • The discovery of quasicrystals led to a broader understanding of the possible types of order in materials

Advanced Topics and Open Problems

• Lattice-based cryptography uses the hardness of certain lattice problems, such as the shortest vector problem (SVP), to construct secure cryptographic systems
  • Examples include the learning with errors (LWE) and the ring learning with errors (RLWE) cryptographic schemes
• Sphere packing in high dimensions has connections to error-correcting codes and the design of efficient communication channels
  • The Shannon limit, which defines the maximum achievable rate for reliable communication, is related to the asymptotic density of sphere packings
• The Leech lattice is a remarkable 24-dimensional lattice that achieves the highest possible packing density among lattices in its dimension
  • It has deep connections to exceptional structures in mathematics, such as the Monster group and the Golay code
• Aperiodic tilings, such as the Penrose tiling, exhibit long-range order without translational symmetry
  • The study of aperiodic tilings has led to new insights into the nature of quasicrystals and the role of symmetry in materials
• The sphere packing problem in dimensions 8 and 24 is of particular interest due to the existence of highly symmetric and dense lattices (E8 and Leech lattice)
  • The optimality of these lattices for sphere packing remains an open problem
• The Minkowski conjecture, which states that every convex body has a lattice packing with density at least $2^{-n}$, where n is the dimension, is a long-standing open problem in geometry
  • The conjecture has been proved for certain classes of convex bodies, but the general case remains unresolved