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Lattice

The term lattice has multiple meanings across various fields. In mathematics, a lattice is a partially ordered set (poset) in which every pair of elements has a unique least upper bound (join, denoted ∨) and a unique greatest lower bound (meet, denoted ∧).^[1] It can also refer to a discrete subgroup of full rank in Euclidean space or \mathbb{R}^n (lattice in group theory), or structures in discrete mathematics and algorithms. For details, see the Mathematics section. In physics and chemistry, a crystal lattice is a symmetrical, three-dimensional arrangement of atoms, ions, or molecules in a crystalline solid, forming a repeating pattern known as a unit cell.^[2] Lattice models approximate continuous systems on discrete grids, and lattice energy describes the strength of ionic bonds in crystals. In computer science and engineering, lattices underpin discrete algorithms for optimization and form the basis of lattice-based cryptography, which relies on the hardness of problems like shortest vector in lattices. In arts and design, latticework is an openwork framework of crossed strips, typically wood or metal, used in architecture for screens, fences, or decorative elements.^[3] Organizations named Lattice include Lattice Semiconductor Corporation, a U.S.-based company specializing in low-power field-programmable gate arrays (FPGAs) and related semiconductors (as of 2025).^[4]

Mathematics

Lattice (order theory)

In order theory, a lattice is a partially ordered set (poset) in which every pair of elements possesses a least upper bound, known as the join and denoted by ∨, and a greatest lower bound, known as the meet and denoted by ∧.^[5] Formally, for elements a and b in the lattice L, the join is a \vee b = \sup\{a, b\} and the meet is a \wedge b = \inf\{a, b\}, ensuring these operations are uniquely determined by the partial order.^[6] This structure generalizes both total orders and more complex partially ordered systems, providing a framework for algebraic manipulations within ordered sets.^[5] Key properties distinguish various classes of lattices. A bounded lattice includes a least element, denoted ⊥ (bottom), and a greatest element, denoted ⊤ (top), such that ⊥ ≤ x ≤ ⊤ for all x in the lattice.^[5] Distributive lattices satisfy the identities a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c) and its dual a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c), enabling the operations to distribute over each other like addition and multiplication in rings.^[5] Modular lattices weaken this to the condition that if a \leq c, then a \vee (b \wedge c) = (a \vee b) \wedge c, preserving a form of associativity without full distributivity.^[5] Representative examples illustrate these concepts. The power set of a set S, ordered by inclusion, forms a distributive lattice where the join of subsets is their union and the meet is their intersection, with ∅ as ⊥ and S as ⊤.^[5] The set of positive divisors of an integer n, ordered by divisibility, yields a distributive lattice with greatest common divisor as meet and least common multiple as join.^[5] Boolean algebras emerge as complemented distributive lattices, where every element a has a complement a' satisfying a \vee a' = \top and a \wedge a' = \perp, underpinning classical propositional logic.^[5] Garrett Birkhoff formalized lattice theory in the 1930s and 1940s, with his 1948 book Lattice Theory (revised edition) synthesizing earlier work by Dedekind and others into a comprehensive treatment, including applications to logic and the foundations of algebra.^[5] These structures extend to subtypes like complete lattices, where every subset—not just pairs—has a join and meet, facilitating analysis of infinite systems.^[6] Heyting algebras, as bounded distributive lattices equipped with an implication operation a \to b = \sup\{x \mid a \wedge x \leq b\}, model intuitionistic logic and constructive mathematics.^[7] Birkhoff's framework has influenced computer science, notably in domain theory for denotational semantics of programming languages.^[8]

Lattice (group theory)

In group theory, a lattice in the Euclidean space \mathbb{R}^n is defined as a discrete additive subgroup generated by n linearly independent basis vectors that span \mathbb{R}^n.^[9] This structure ensures the lattice is full rank, meaning it has the same dimension as the ambient space, and discrete, with a positive minimum distance \epsilon > 0 between any two distinct points.^[9] Lattices arise naturally as the set of integer linear combinations of these basis vectors, denoted \Lambda = \{ B \mathbf{z} \mid \mathbf{z} \in \mathbb{Z}^n \}, where B is the n \times n basis matrix with linearly independent columns.^[10] Key properties of lattices include the covolume, which measures the "density" of the lattice and equals the absolute value of the determinant of the basis matrix, \det(\Lambda) = |\det(B)|; this represents the volume of the fundamental parallelepiped spanned by the basis vectors.^[10] The Voronoi cell of a lattice is the convex polytope consisting of all points in \mathbb{R}^n closer to the origin than to any other lattice point, serving as a fundamental domain whose volume equals the covolume.^[11] Primitive vectors in a lattice context refer to a basis of linearly independent vectors that generate the entire lattice via integer coefficients, ensuring no proper sublattice is missed.^[12] A fundamental result is Minkowski's theorem, which states that any centrally symmetric convex set in \mathbb{R}^n with volume greater than $2^n \det(\Lambda) must contain a non-zero point of the lattice.^[10] Representative examples include the integer lattice \mathbb{Z}^n, generated by the standard orthonormal basis vectors e_1, \dots, e_n, with covolume 1.^[9] In two dimensions, the hexagonal lattice is generated by vectors (1, 0) and \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right), achieving the densest packing of circles among 2D lattices.^[13] In three dimensions, the body-centered cubic lattice consists of all points with integer coordinates plus those shifted by \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right), notable for its role in efficient sphere packings.^[12] Lattices find significant applications in number theory, particularly in reduction theory, where algorithms seek short, nearly orthogonal bases to approximate solutions to Diophantine equations.^[10] The Lenstra–Lenstra–Lovász (LLL) algorithm, introduced in 1982, provides a polynomial-time method for lattice basis reduction, enabling practical solutions to problems like integer relation detection and polynomial factoring over the rationals.^[14] These techniques trace back to Diophantine approximation, where lattices model simultaneous approximations of real numbers by rationals.^[15] Historically, lattice theory emerged in the 19th century through Charles Hermite's work on reducing positive definite quadratic forms in the 1850s, which laid foundations for analyzing lattice minima, and was formalized by Hermann Minkowski in his 1896 monograph Geometrie der Zahlen, establishing deep connections between lattices and quadratic forms in the geometry of numbers.^[16]^[17]

Other mathematical lattices

Lattice multiplication is a visual arithmetic technique for multiplying multi-digit numbers by organizing partial products in a grid, where diagonals are summed to obtain the result. This method, which breaks down multiplication into smaller steps using a lattice-like diagram, facilitates error checking and is particularly useful for manual calculations. It is also referred to as the Chinese multiplication method due to its adoption and variation in Chinese mathematical texts, and as a Vedic math technique in modern interpretations, though its roots predate the formalized Vedic mathematics system. The technique originated in India around the 10th to 11th century and was later introduced to Europe by Fibonacci in the 13th century under the name gelosia, meaning "jealousy" in Italian, referring to the crisscross pattern resembling a latticed window.^[18]^[19] In combinatorics, lattice paths refer to sequences of moves, typically right (east) and up (north), on a grid from the origin (0,0) to a point (m,n), often with restrictions such as avoiding certain lines or obstacles. The unrestricted number of such paths equals the binomial coefficient \binom{m+n}{m} = \frac{(m+n)!}{m! \, n!}, which counts the ways to choose m right steps out of m+n total steps. For paths that stay above the diagonal (Dyck paths when m=n), the count is given by the Catalan number C_n = \frac{1}{n+1} \binom{2n}{n}, appearing in numerous enumerative problems. Restricted variants, such as those never falling below a line, are quantified by the Ballot theorem, which states that if candidate A receives a votes and B receives b votes with a > b, the probability that A is always ahead is \frac{a - b}{a + b}; generalizations apply to lattice paths from (0,0) to (m,n) staying above y = (n/m)x. These concepts underpin modern combinatorial enumerations developed since the late 19th century.^[20]^[21]^[22] In representation theory, Gel'fand-Tsetlin patterns serve as lattice structures indexing bases for irreducible representations of the general linear group GL(n), consisting of triangular arrays of integers satisfying interlacing inequalities that ensure non-negativity and decreasing rows. These patterns correspond to lattice points within the Gel'fand-Tsetlin polytope, whose vertices determine the highest-weight vectors. Similarly, Young tableaux act as lattice diagrams in the study of symmetric group representations, where standard Young tableaux of shape λ enumerate the dimension of the Specht module via the hook-length formula, providing a combinatorial basis linked to semi-standard fillings for GL(n) weights.^[23]^[24] Lattice polygons, simple closed paths on the integer grid, have areas computed by Pick's theorem: A = I + \frac{B}{2} - 1, where I is the number of interior lattice points and B is the number on the boundary; this relates geometric measure to discrete counts, foundational in Ehrhart theory. In geometry, modular lattices model incidence structures like subspace lattices of vector spaces, where the modular law x \leq z implies x \vee (y \wedge z) = (x \vee y) \wedge z holds, and geometric lattices—atomic and semimodular—capture matroid geometries with flats as closed sets.^[25]^[26]

Physics and chemistry

Crystal lattice

A crystal lattice is the periodic, three-dimensional arrangement of atoms, ions, or molecules in a crystalline solid, modeled as a Bravais lattice—an infinite array of discrete points generated by integer combinations of three basis vectors—with a motif or basis of atoms attached to each lattice point to describe the full atomic periodicity. This structure underpins the long-range order in solids, distinguishing crystals from amorphous materials. The concept was formalized in 1850 by French physicist Auguste Bravais, who demonstrated that only 14 unique lattice types are possible in three dimensions, classified by their translational symmetry and belonging to seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.^[27] These lattices provide the geometric framework for understanding crystal symmetry and properties. The 14 Bravais lattices vary in the positioning of lattice points within the unit cell, including primitive (one lattice point per cell) and non-primitive forms like base-centered, body-centered, and face-centered. Representative examples include the simple cubic lattice, with points only at cube corners; the face-centered cubic (FCC), with additional points at face centers; and the body-centered cubic (BCC), with a point at the body center alongside corners.^[28] Classification arises from space group symmetries, ensuring no two distinct lattices are equivalent under rotation or translation. The unit cell represents the smallest repeating volume, where the primitive unit cell encloses exactly one lattice point and has the minimal volume, while conventional unit cells—often chosen for higher symmetry—may contain multiple lattice points, such as four in the FCC conventional cell.^[29] Miller indices (hkl) provide a standardized notation for specifying lattice planes and directions in a crystal, defined as the reciprocals of the fractional intercepts of the plane with the crystallographic axes, reduced to the smallest integers.^[30] For instance, the (100) plane intersects the a-axis at one unit length and is parallel to the b- and c-axes. These indices facilitate the description of crystal orientations, cleavage planes, and diffraction patterns. Notable examples of crystal structures include the diamond cubic lattice in carbon (diamond), which uses an FCC Bravais lattice with a two-atom basis, where each carbon atom forms tetrahedral bonds with four neighbors, yielding a coordination number of 4 and exceptional hardness.^[31] Similarly, sodium chloride (NaCl) exhibits the rock salt structure, an FCC Bravais lattice with alternating Na^+ and Cl^- ions in octahedral coordination, resulting in six nearest neighbors per ion and a 1:1 stoichiometry. The reciprocal lattice, essential for analyzing wave interactions with crystals, is defined such that its vectors \mathbf{b}_i satisfy \mathbf{b}_i \cdot \mathbf{v}_j = 2\pi \delta_{ij} for direct lattice vectors \mathbf{v}_j, explicitly given by

\mathbf{b}_i = 2\pi \frac{\mathbf{v}_j \times \mathbf{v}_k}{\mathbf{v}_i \cdot (\mathbf{v}_j \times \mathbf{v}_k)},

where i, j, k are cyclic permutations of 1, 2, 3. This construction transforms the real-space periodicity into momentum space, aiding in the interpretation of diffraction. X-ray diffraction from lattice planes occurs constructively when satisfying Bragg's law,

n\lambda = 2d \sin\theta,

where n is an integer, \lambda the X-ray wavelength, d the interplanar spacing, and \theta the incidence angle; this relation was derived in 1913 by William Lawrence Bragg to explain Laue's 1912 observations of X-ray scattering by crystals.^[32] The Braggs' work in the 1910s, building on Bravais's foundations, established X-ray crystallography as a tool for atomic-scale structure determination.^[33] In modern applications, crystal lattices are pivotal in semiconductor design, where structures like the diamond cubic in silicon and germanium dictate band gaps, carrier mobility, and device performance in transistors and photovoltaics.^[34] For example, silicon's FCC-based lattice enables precise doping and fabrication of integrated circuits, underpinning electronics technology.

Lattice models in physics

Lattice models in physics discretize continuous space into a regular grid, or lattice, to approximate the behavior of particles and fields, facilitating the study of complex interactions in statistical mechanics and quantum field theory. These models assign physical degrees of freedom, such as spins or particle densities, to lattice sites, with interactions limited to nearest neighbors to capture local correlations while simplifying computations. By replacing continuum equations with finite-difference analogs, lattice models enable exact solutions in low dimensions and numerical simulations in higher dimensions, revealing emergent phenomena like ordering and criticality.^[35] A foundational example is the Ising model, introduced by Wilhelm Lenz in 1920 as a discrete representation of ferromagnetism and exactly solved in one dimension by Ernst Ising in 1925. In this model, each lattice site hosts a spin \sigma_i = \pm 1, interacting via the Hamiltonian

H = -J \sum_{\langle i,j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i,

where J > 0 denotes the ferromagnetic coupling between nearest neighbors \langle i,j \rangle, and h is an external magnetic field; the partition function is then Z = \sum_{\{\sigma\}} \exp(-\beta H), with \beta = 1/(k_B T). This model exhibits phase transitions in dimensions greater than one, illustrating spontaneous magnetization below a critical temperature. Another key example is the Hubbard model, proposed by John Hubbard in 1963 to describe strongly correlated electrons in solids, incorporating on-site repulsion U alongside hopping terms on a lattice, which captures phenomena like Mott insulation and high-temperature superconductivity. These models find broad applications in analyzing phase transitions and critical phenomena, where lattice discreteness aids in applying renormalization group theory, pioneered by Kenneth Wilson in the 1970s to explain scaling behaviors near criticality. For instance, Wilson's approach on lattice models unified short- and long-distance physics, earning the 1982 Nobel Prize and enabling predictions of critical exponents for the Ising universality class. Percolation theory, developed by Simon Broadbent and John Hammersley in 1957, models connectivity on lattices by randomly occupying sites or bonds, yielding thresholds for infinite clusters that underpin transport in disordered media. Advanced techniques include Monte Carlo simulations, first applied to lattice models like Ising by Nicholas Metropolis and colleagues in 1953 to sample equilibrium configurations via Markov chains, allowing estimation of thermodynamic averages in complex systems. Additionally, lattice Boltzmann methods, evolving from lattice gas automata in the 1970s and formalized in the 1980s, simulate fluid dynamics by propagating particle distribution functions on lattices according to a discrete Boltzmann equation, offering efficient solutions for multiphase flows and turbulence. Such models often employ underlying grids akin to crystal lattices but emphasize dynamic probabilistic evolution over static geometry. Lattice energy, denoted as U, is the energy required to separate one mole of an ionic compound into its constituent gaseous ions, a positive quantity representing the strength of the ionic bonds in the lattice. This energy is a key determinant of the stability of ionic compounds, as higher lattice energies indicate stronger electrostatic attractions between oppositely charged ions. Lattice energy depends primarily on the charges of the ions and the distance between them; for ions with the same charges, smaller ionic radii lead to shorter interionic distances and thus higher lattice energies, while higher charges amplify the attractive forces proportionally to the product of the charges. Experimental determination of lattice energy is challenging due to the hypothetical nature of gaseous ions, so it is typically calculated using the Born-Haber cycle, a thermochemical cycle based on Hess's law that combines measurable quantities such as standard enthalpies of formation, sublimation, ionization energies, and electron affinities.^[36] For theoretical estimation in ionic crystals, the Born-Landé equation provides a model incorporating Coulombic attraction and short-range repulsion:

U = \frac{N_A M |z_+ z_-| e^2}{4\pi \epsilon_0 r_0} \left(1 - \frac{1}{n}\right)

where N_A is Avogadro's constant, M is the Madelung constant specific to the crystal structure, z_+ and z_- are the absolute values of the ion charges, e is the elementary charge, \epsilon_0 is the vacuum permittivity, r_0 is the nearest-neighbor distance, and n is the Born repulsion exponent (typically 7–12 depending on the ions). An empirical alternative, the Kapustinskii equation, approximates lattice energy without detailed structural data:

U = \frac{1202.5 \nu |z_+ z_-|}{r_+ + r_-} \left(1 - \frac{34.5}{r_+ + r_-}\right) \quad (\text{in kJ/mol, with radii in pm})

where \nu is the number of ions per formula unit, and r_+, r_- are the ionic radii; this method yields estimates accurate to within 5% for many alkali halides. A representative example is sodium chloride (NaCl), with a lattice energy of approximately 787 kJ/mol, reflecting the +1/-1 charges and relatively small ionic radii (Na⁺: 102 pm, Cl⁻: 181 pm). In contrast, magnesium oxide (MgO) exhibits a much higher value of about 3791 kJ/mol due to the +2/-2 charges and smaller ions, underscoring how charge and size dominate energetic stability.^[37] Related concepts include lattice enthalpy, which is synonymous with lattice energy under constant pressure conditions, and cohesion energy, the total binding energy per atom required to separate the lattice into neutral atoms (distinct from the ion-based lattice energy). Lattice defects, such as Schottky defects (paired cation-anion vacancies maintaining charge neutrality) and Frenkel defects (an ion displaced to an interstitial site, creating a vacancy), introduce local disruptions that slightly lower the overall lattice energy and influence ionic conductivity and mechanical properties.^[38] In modern quantum chemistry, density functional theory (DFT) enables accurate computation of lattice energies for complex ionic and molecular lattices by solving the Kohn-Sham equations to obtain total electronic energies, often achieving errors below 10 kJ/mol compared to experimental values after correction for basis set superposition.^[39] These methods are particularly relevant to battery materials, where post-2020 advancements in DFT have optimized solid-state electrolytes like lithium garnets (e.g., Li₇La₃Zr₂O₁₂) by predicting lattice stabilities and ion migration barriers, enhancing energy density and safety in all-solid-state lithium-ion batteries. Such computations tie lattice energy to the structural basis of crystal lattices, where geometric arrangements determine the Madelung constants used in energy models.

Computer science and engineering

Lattices in discrete mathematics and algorithms

In discrete mathematics and algorithms, lattices serve as foundational structures in two distinct but interconnected ways: as partially ordered sets (posets) in which every pair of elements possesses a unique supremum (least upper bound) and infimum (greatest lower bound), and as geometric objects defined as full-rank discrete additive subgroups of Euclidean space \mathbb{R}^n, generated by integer linear combinations of basis vectors.^[40]^[41] The poset formulation, rooted in order theory, enables abstract reasoning about hierarchies and approximations, while the geometric view supports computational problems involving point sets and distances in high dimensions.^[42] These dual perspectives facilitate applications in search, optimization, and formal verification, where lattices model discrete constraints or approximate continuous spaces efficiently.^[10] A key application of lattices as posets lies in abstract interpretation for programming language analysis, where program states and properties form a lattice, and dataflow analysis computes fixed points of monotone functions over this structure to approximate reachable states statically.^[40] In this framework, the lattice's join and meet operations propagate information across control flow graphs, enabling optimizations like dead code elimination while ensuring soundness through Galois connections between concrete and abstract domains.^[43] Lattices also underpin data structures for version control in distributed systems, where they model concurrent updates via conflict-free replicated data types (CRDTs); for instance, semilattices enforce monotonic merges to resolve branching histories without conflicts, supporting eventual consistency in tools like Git for tracking code evolutions.^[44] In geometric lattices, core problems include the shortest vector problem (SVP) and closest vector problem (CVP), which drive algorithms for optimization and search. SVP seeks the minimal Euclidean norm of a nonzero lattice vector:

\lambda_1(L) = \min \{ \|v\|_2 : v \in L \setminus \{0\} \},

where L is the lattice and \|\cdot\|_2 denotes the \ell_2-norm; solving it exactly is NP-hard under randomized reductions, but approximations inform practical heuristics.^[45]^[46] CVP, conversely, given a target vector t \in \mathbb{R}^n, aims to find v \in L minimizing \|t - v\|_2, with applications in error correction and pattern matching.^[45] An example arises in integer factorization, where lattice reduction constructs bases encoding polynomial relations modulo the target number, yielding short vectors that reveal factors when norms are sufficiently small.^[47] Prominent algorithms address these problems through basis reduction. The Gaussian lattice reduction, originally for two dimensions, iteratively subtracts integer multiples of one basis vector from another to minimize their lengths and angle, analogous to the Euclidean algorithm for gcd; it produces a reduced basis where the shortest vector is at most \sqrt{2} times the true minimum.^[48] For higher dimensions and CVP, Babai's nearest plane method provides a polynomial-time approximation by successively projecting the target onto orthogonal subspaces spanned by the basis vectors, rounding coefficients to the nearest integer to enumerate candidate lattice points; it achieves an exponential approximation factor of $2^{n/2} in the worst case but performs well empirically with reduced bases.^[49] Historically, lattice algorithms trace to computational number theory in the 1980s, with the Lenstra-Lenstra-Lovász (LLL) algorithm marking a breakthrough by enabling polynomial-time basis reduction for arbitrary dimensions, building on earlier two-dimensional methods to solve Diophantine approximation and factorization subproblems efficiently.^[10] In the 2020s, lattices have extended to AI for constraint satisfaction, particularly in motion planning, where lattice-based planners discretize state spaces into graphs of feasible actions, using search algorithms like A* to satisfy spatiotemporal constraints in robotics while optimizing paths.

Lattice-based cryptography

Lattice-based cryptography is a prominent paradigm in post-quantum cryptography, deriving its security from the conjectured intractability of certain computational problems over integer lattices. These problems include the Shortest Vector Problem (SVP), which requires finding the shortest nonzero vector in a lattice; the Closest Vector Problem (CVP), which involves identifying the lattice vector closest to a given target point; and the Learning With Errors (LWE) problem, which posits that recovering a secret from noisy linear equations modulo a prime q is hard.^[50] The LWE problem is formally defined as follows: given a uniformly random matrix A \in \mathbb{Z}_q^{m \times n} and vector \mathbf{b} = A \mathbf{s} + \mathbf{e} \pmod{q}, where \mathbf{s} \in \mathbb{Z}_q^n is the secret vector and \mathbf{e} is a small error vector sampled from a discrete Gaussian or bounded distribution, recover \mathbf{s}. The decision variant of LWE asks to distinguish such (A, \mathbf{b}) pairs from uniformly random ones, and its hardness underpins many lattice-based schemes through worst-case to average-case reductions.^[51] The field traces its origins to the mid-1990s with the introduction of the NTRU public-key cryptosystem by Hoffstein, Pipher, and Silverman, which relies on the hardness of finding short vectors in a lattice defined over polynomial rings.^[52] A surge in research followed in the 2000s, driven by threats from Shor's algorithm—which efficiently factors integers and solves discrete logarithms on quantum computers, breaking RSA and ECC—and foundational contributions from Micciancio and Regev. Their 2004 work established efficient worst-case to average-case reductions for lattice problems using Gaussian measures, enabling provably secure constructions, while Regev's 2005 LWE formulation provided a versatile average-case assumption resistant to known quantum attacks.^[51] Key lattice-based schemes include CRYSTALS-Kyber, a key encapsulation mechanism (KEM) selected by NIST in 2022 and standardized as FIPS 203 (ML-KEM) in 2024, which uses Module-LWE for efficient key exchange; and CRYSTALS-Dilithium, a digital signature scheme standardized as FIPS 204 (ML-DSA), relying on Module-LWE and Short Integer Solution (SIS) problems for forgery resistance. Another lattice-based signature scheme, FN-DSA (based on Falcon), reached draft standard status (FIPS 206) submitted for approval in August 2025, intended as a backup to ML-DSA.^[53]^[54] For enhanced efficiency, many schemes employ Ring-LWE variants, where computations occur over rings like \mathbb{Z}_q/(x^n + 1) to reduce dimension and key sizes compared to plain LWE, as in Kyber's module variant over cyclotomic rings. Lattice-based constructions also enable advanced functionalities, such as fully homomorphic encryption, pioneered by Gentry's 2009 scheme over ideal lattices, which allows arbitrary computations on encrypted data without decryption by "bootstrapping" noisy ciphertexts.^[55] These schemes offer significant advantages, including resistance to quantum adversaries—since no efficient quantum algorithms are known for core lattice problems like approximate SVP or LWE—and often superior performance over alternatives like code-based cryptography in terms of key sizes and computation speed for equivalent security levels.^[56] NIST's 2024 standardization of Kyber and Dilithium marks a pivotal step, with implementations integrating into protocols like TLS by 2025 to migrate real-world systems to quantum-safe security; as of October 2025, providers like Cloudflare have enabled post-quantum encryption for over half of their human-initiated traffic using hybrid modes in TLS 1.3.^[53]^[57]

Arts and design

Latticework and architecture

Latticework, also known as trellis, refers to an openwork framework formed by intersecting strips of material, typically arranged in a crisscross or diagonal pattern to create geometric openings. This structure serves both ornamental and functional purposes in architecture, providing support for climbing plants, screening for privacy, or decorative elements that allow light and air to pass through while obscuring direct views.^[3] Historically, latticework appeared in Roman architecture as transennae, decorative lattice screens made of stone, metal, or wood, used in windows and railings to form repeating geometric patterns. These elements, seen in structures like the 5th-century Basilica of Santa Sabina in Rome, influenced early Christian designs and provided both aesthetic appeal and practical division of space. In Gothic architecture, tracery evolved as a sophisticated form of latticework, consisting of slender stone bars that divided windows into intricate patterns, emerging around 1240 and becoming increasingly complex to support expansive stained-glass areas in cathedrals. Victorian-era garden trellises, often featuring wooden lattice panels, were integral to veranda designs, offering shade and support for vines while blending indoor and outdoor spaces, as illustrated in 19th-century American landscape architecture.^[58]^[59]^[60] Traditional materials for latticework include wood for its workability in intricate carvings, metal for durability in structural applications, and stone for monumental permanence, with techniques involving interlacing strips, turning on lathes, or chiseling patterns. Modern advancements incorporate fiberglass for weather-resistant garden screens and 3D-printed polymers for customizable, lightweight components, enabling complex geometries unattainable by conventional methods.^[61]^[62] Notable examples include Japanese shoji screens, where translucent rice paper is stretched over wooden lattice frames to diffuse light and partition rooms, a technique dating to the Heian period (794–1185) and central to traditional interiors. The Eiffel Tower (1889), designed by Gustave Eiffel, exemplifies engineering latticework through its wrought-iron girders forming a curved, open framework that balances strength and elegance, using 18,000 prefabricated pieces connected by 2.5 million rivets.^[63] In engineering, latticework underpins load-bearing designs like the Warren truss, patented in 1848, which uses equilateral triangular diagonals in steel or iron to efficiently distribute forces across bridges, as seen in early railroad spans like the 1852 Newark Dyke Bridge. Post-2010 additive manufacturing has enabled lattice structures in aerospace, creating ultra-lightweight components with high strength-to-weight ratios for aircraft parts, mimicking natural efficiencies.^[64]^[65] Culturally, Islamic geometric lattices, such as girih tiles—sets of five decorated polygons—hold profound significance, symbolizing infinite order and divine unity in medieval architecture from the 12th century onward, as used in mosques and madrasas across the Islamic world to evoke spiritual contemplation without figurative imagery.^[66]

Lattice patterns in other arts

In culinary arts, lattice patterns manifest as interwoven strips of pastry dough placed atop pies to facilitate steam ventilation during baking while providing an aesthetic crisscross design. This technique, common in fruit-based pies, originated with 15th-century Dutch bakers who perfected the lattice-style crust for apple pies, allowing juices to evaporate without sogginess. By the 18th and 19th centuries, American cooks adapted this method using local ingredients, making the lattice-topped apple pie a staple of holiday traditions and symbolizing national comfort food.^[67]^[68] Lattice motifs in visual arts draw inspiration from geometric lattices in mathematics, creating repetitive grids that evoke infinity and harmony. In traditional textiles, such patterns appear in Islamic carpets, where lattice designs formed by superimposed vine grids or floral compartments symbolize interconnectedness and were prevalent in classical Persian and Mamluk weaves from the 16th century onward. In modern digital art, fractal-like lattices emerge through algorithmic generation, producing self-similar structures that mimic natural complexity, as seen in software-rendered pieces exploring symmetry and recursion since the early 2000s.^[69]^[70] Techniques for creating lattice patterns extend to weaving in basketry, where lattice twining involves crossing weft strands over paired warps to form open grids, a method used in traditional crafts for durable, ventilated containers. In graphic design, software tools enable precise lattice generation; Adobe Illustrator, for instance, allows artists to define custom patterns from grid-based artwork since its pattern-making features were enhanced in the 2000s, facilitating seamless repeats for textiles and digital media.^[71]^[72] In music, lattice patterns serve as grid-based models for tuning systems, mapping pitch ratios in just intonation to visualize harmonic relationships and facilitate microtonal composition. Composer Harry Partch pioneered a 43-tone lattice in the 1940s, a just intonation scale derived from 11-limit intervals projected onto a multidimensional grid to expand beyond equal temperament for his instrumental works.^[73]^[74] Culturally, lattice appears in heraldry as a bordure of perpendicular and horizontal bars crossed at right angles, symbolizing unity or enclosure without interlacing, as defined in traditional armorial bearings. In the 2020s, lattice algorithms have influenced NFT art through generative processes, where code produces unique, blockchain-minted visuals featuring algorithmic grids and patterns, redefining digital collectibles in projects like those on platforms emphasizing procedural complexity.^[75]^[76]

Organizations

Lattice Semiconductor

Lattice Semiconductor Corporation is a leading provider of low-power programmable logic solutions, specializing in field-programmable gate arrays (FPGAs), complex programmable logic devices (CPLDs), and related semiconductor technologies.^[77] Founded in 1983 in Oregon's Silicon Forest and headquartered in Hillsboro, Oregon, the company focuses on delivering energy-efficient chips for edge computing, communications, and industrial applications.^[78] With approximately 1,100 employees globally as of 2025, Lattice emphasizes small-form-factor, low-power devices that enable rapid prototyping and deployment in resource-constrained environments.^[79] Key products include the iCE40 family of ultra-low-power FPGAs, designed for battery-operated and mobile devices, which originated from the company's 2011 acquisition of SiliconBlue Technologies for $62 million to bolster its wireless and low-cost FPGA capabilities.^[80] Another prominent offering is the Certus-NX series, which supports high-speed interfaces up to 10 Gbps for applications in networking and synchronization. These products target sectors requiring minimal power consumption, such as consumer electronics and IoT devices, distinguishing Lattice from higher-power competitors. Major milestones include its initial public offering in 1989 on the NASDAQ under the ticker LSCC, which fueled early expansion into programmable logic markets.^[81] In the 2020s, Lattice shifted toward AI edge computing, with fiscal year 2024 revenue reaching $509.4 million, reflecting a 31% decline from 2023 due to market cyclicality but stabilization through AI-driven demand.^[82] Innovations such as the Lattice Nexus platform, introduced in 2019, leverage 28nm FD-SOI technology for 75% lower dynamic power and enhanced reliability in small FPGAs.^[83] Additionally, collaborations like the 2019 partnership with SiFive have integrated open-source RISC-V processor cores into Lattice FPGAs, enabling customizable embedded solutions.^[84] Lattice holds a strong position in automotive and industrial markets, accounting for 46% of its sales as of fiscal year 2024 through solutions for machine vision, sensor fusion, and real-time control.^[85] It competes primarily with AMD (via its Xilinx acquisition) and Intel in the FPGA space, but differentiates via low-power focus for edge applications rather than high-density data centers.^[86] Recent developments include expansions into 5G infrastructure, highlighted by the 2023 Edison Award-winning ORAN Solution Stack for open radio access networks, and machine learning accelerators through a 2023 NVIDIA collaboration for edge AI inference.^[87] In 2024 and 2025, Lattice launched updated Certus-NX models and the MachXO5-NX family for post-quantum cryptography-ready security in 5G and ML deployments.^[88]^[89]

Other organizations named Lattice

Lattice is a San Francisco-based people management platform founded in 2015 that provides software tools for performance reviews, employee engagement, goal setting, and career development, serving more than 5,000 organizations worldwide, including companies like Cruise and Instacart, as of 2025.^[90]^[91] The company, which emerged from Y Combinator's accelerator program, has raised more than $328 million in venture funding from investors such as Thrive Capital and Workday Ventures, achieving a valuation exceeding $3 billion as of 2021.^[92] Lattice Engines, established in 2008 and headquartered in San Mateo, California, developed cloud-based predictive analytics software for B2B marketing and sales, enabling companies to identify high-value leads and personalize customer interactions using AI-driven insights.^[93] The firm served enterprise clients such as Dell and Microsoft before being acquired by Dun & Bradstreet in 2019 for approximately $130 million, integrating its technology into broader data analytics offerings.^[94]^[95] Lattice Inc., a Pennsylvania-based provider of correctional technology solutions founded over 40 years ago, specializes in inmate management systems, telecommunications, and visitation tools designed to enhance security and operational efficiency in prisons and jails across the United States and internationally.^[96] Its offerings include integrated platforms for commissary management, video visitation, and 24/7 technical support, with deployments in facilities serving more than 100,000 inmates.^[97] Lattice Automation, launched in 2017 and based in Boston, Massachusetts, develops software for synthetic biology, automating the design, simulation, and execution of genetic engineering workflows to accelerate research in biotech and pharmaceuticals.^[98] The platform supports data-driven optimization of biological constructs, serving academic labs and industry partners in fields like gene therapy and sustainable materials.^[99] The Lattice Companies, formed as a holding entity and backed by private equity firm Gridiron Capital since 2021, operates a commerce enablement platform for independent dealers in home improvement sectors such as flooring, cabinets, and appliances, providing supply chain, e-commerce, and marketing tools to over 1,200 locations nationwide.^[100] Lattice Ventures, a New York City-based early-stage venture capital firm founded in 2016, invests in seed and pre-seed technology startups, particularly in fintech, healthtech, and consumer internet, with a portfolio including companies like Alloy and Grain.^[101] The firm manages over $100 million in assets and emphasizes founder-led innovation through its network-driven approach.^[102]

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[PDF] arXiv:2206.02276v1 [math.CO] 5 Jun 2022
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Unit Cells
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[PDF] Crystal basis:
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Lattice Planes and Miller Indices (all content) - DoITPoMS
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[PDF] Lecture 3 CVP Algorithm 1 The Nearest Plane Algorithm
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[PDF] The Learning with Errors Problem
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[PDF] On Lattices, Learning with Errors, Random Linear Codes, and ...
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Mashrabiya Screen with Medallions - The Metropolitan Museum of Art
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Homemade Apple Pie with a Lattice Crust | Thrive Market
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Pieces of a Puzzle: Classical Persian Carpet Fragments
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Fragments of a Carpet with Lattice and Blossom Pattern
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[PDF] EXPLORING BASKET WEAVING TECHNIQUES FOR SPACE ...
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Pattern overview - Adobe Help Center
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Harmonic Lattice Diagrams, by Joe Monzo - Tonalsoft
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Partch 43-tone scale lattice (c) 2000 by Joe Monzo - Tonalsoft
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7 Generative Art NFTs Redefining Digital Creativity | OpenSea
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Lattice Semiconductor to Acquire SiliconBlue
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About us - Lattice
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Lattice - Crunchbase Company Profile & Funding
Lattice is a people success platform that help business leaders develop engaged, high-performing employees, and winning cultures.
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About Lattice |
For over 40 years, Lattice has been a trusted global partner to correctional facilities in providing an efficient, safe, and productive environment.
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Lattice Automation
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Lattice Automation - LinkedIn
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The Lattice Companies - Gridiron Capital
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Lattice Ventures investment portfolio | PitchBook
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