Maryna Viazovska and sphere packing in 8 and 24 dimensions

The Genesis of Sphere Packing

The sphere packing problem is a foundational inquiry in Euclidean geometry, asking for the arrangement of non-overlapping, congruent spheres that occupies the maximum possible proportion of an $n$-dimensional space. The fraction of space filled by the spheres is referred to as the packing density. As a local density measurement in an infinite space can vary depending on the chosen volume, the mathematical objective is generally to maximize the asymptotic or average density across the entire coordinate space ¹².

Early Dimensions and the Kepler Conjecture

In the lowest dimensions, the optimal arrangements are historically well-understood and intuitively derived. In one-dimensional space, a "sphere" is merely a line segment; packing line segments end-to-end fills the entire linear universe, resulting in a trivial packing density of exactly 1 ¹³.

In two dimensions, the problem translates to the packing of circles on a plane. The optimal arrangement is the hexagonal lattice, which produces a daisylike pattern where each circle is surrounded by exactly six others ⁴. This configuration covers a density fraction of $\pi / \sqrt{12}$, or approximately 90.69% of the area. A rigorous mathematical proof for the optimality of the hexagonal packing was provided by Axel Thue in 1892, with further definitive refinements by László Fejes Tóth in 1940 ³⁵⁶.

The three-dimensional case introduces significant complexity. The densest arrangement corresponds to the familiar pyramidal piling of spherical objects, such as oranges in a grocery store or cannonballs on a battlefield ¹⁷⁸. This structure, known as the face-centered cubic (FCC) lattice or cubic close packing, attains a density of $\pi / \sqrt{18}$, covering roughly 74.048% of the available volume ¹²⁹. Johannes Kepler conjectured in 1611 that this arrangement was the absolute maximum possible density for both regular and irregular configurations ¹⁷. While Carl Friedrich Gauss proved in 1831 that the FCC lattice is the densest among all possible lattice (regular) packings, the overarching Kepler conjecture for all possible amorphous arrangements remained unproven for centuries ¹²¹⁰. It was not until 1998 that mathematician Thomas Callister Hales announced a proof of the Kepler conjecture. Hales' proof utilized an exhaustion method, checking thousands of individual cases via massive computer calculations spanning over 250 pages and gigabytes of code, an approach that required years of subsequent formal computer verification to be fully accepted by the mathematical community ¹⁷¹¹.

High-Dimensional Geometry and Information Theory

As dimensions increase beyond the familiar three, human spatial intuition fails, and the geometric properties of spheres change dramatically. In higher dimensions, the volume of a sphere becomes increasingly concentrated near its boundary. Consequently, when multi-dimensional spheres are packed together, the interstitial spaces - or "deep holes" - between them grow exponentially larger ³⁷.

While packing spheres in 10 or 24 dimensions may seem like an abstract geometric exercise, it holds profound practical importance in information theory and electrical engineering. In 1948, Claude Shannon established that analog communication codes and digital error-correcting codes can be modeled geometrically ³¹². A discretized electromagnetic signal with finite bandwidth and average power can be represented as a vector in an $n$-dimensional space ¹². When a signal is transmitted over a noisy continuous communication channel - such as radio waves, optical fibers, or signals to deep space probes - the noise perturbs the signal vector. By placing allowable signal vectors at the centers of $n$-dimensional spheres, engineers ensure that if the noise does not push the signal outside its surrounding sphere, the original message can be correctly decoded (a process utilizing Voronoi cells) ⁷¹². Finding the densest possible sphere packing in high dimensions directly translates to finding the most efficient way to transmit data without errors ⁸¹²¹³.

Asymptotic Bounds and Kissing Numbers

In the vast landscape of higher dimensions, the exact optimal packing arrangements remain largely unknown. The discipline relies heavily on establishing absolute upper and lower bounds for packing densities.

Theoretical Limits of Packing Density

For an arbitrary dimension $n$, the packing density $\Delta_n$ is constrained by both constructive lower bounds and theoretical upper bounds. Hermann Minkowski established an early lower bound demonstrating that lattice packings exist with a density of at least $c n 2^{-n}$, indicating that the optimal density drops exponentially as the dimension increases. Akshay Venkatesh improved this asymptotic lower bound for specific infinite sets of dimensions to $c n (\log \log n) 2^{-n}$ ⁹¹³.

Conversely, establishing an upper bound - the absolute ceiling of density that no packing can exceed - has proven difficult. The Kabatiansky-Levenshtein bound, established in 1977, provides an asymptotic upper limit of $\Delta_n \le 2^{-(0.5990\dots + o(1))n}$ ⁹¹². The gap between the best known lower bounds (actual packings) and the upper bounds grows exponentially with $n$. For instance, in dimension 36, the upper bound and the known packings differ by a multiplicative factor of 52 ³.

The Kissing Number Problem

A corollary to sphere packing is the kissing number problem, which seeks the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere of the same radius ¹⁴¹⁵. For a lattice packing, this number is uniform for every sphere in the arrangement and corresponds directly to the number of minimal-length vectors in the lattice ¹⁴¹⁶.

In one dimension, the kissing number is trivially 2. In two dimensions, it is exactly 6 ⁴¹⁵. In three dimensions, a famous dispute arose in 1694 between Isaac Newton, who believed the maximum was 12, and David Gregory, who hypothesized that the extra empty space around the central sphere could accommodate a 13th sphere. The debate was not settled until 1953 when a correct proof confirmed Newton's assertion that 12 is the strict maximum ⁴⁵.

The problem was solved for four dimensions in 2003 by Oleg Musin, who modified Delsarte's method to prove the kissing number is exactly 24 ⁵¹⁵. Strikingly, prior to breakthroughs in packing densities, the exact maximal kissing numbers were rigorously established for exactly two high dimensions: 8 and 24. Vladimir Levenshtein in Russia proved that the kissing number is 240 in 8 dimensions, and 196,560 in 24 dimensions ¹⁴¹⁵. These precise resolutions were only possible because of the existence of uniquely symmetric lattices in these spaces.

The Geometry of Eight and Twenty-Four Dimensions

While most dimensions exhibit a chaotic and poorly understood packing landscape, dimensions 8 and 24 host geometric configurations of exceptional symmetry. These dimensions allow for the construction of specific lattices where the rapidly expanding "deep holes" present in lower-dimensional analogs are suddenly large enough to accommodate additional spheres, which then lock tightly into place ⁵⁷.

The E8 Root Lattice

A standard method for generating packings is the $D_n$ checkerboard lattice, which consists of points $(x_1, \dots, x_n) \in \mathbb{Z}^n$ where the sum of the coordinates is an even integer. The $D_3$ lattice forms the optimal face-centered cubic packing in three dimensions. However, as $n$ grows, the empty spaces between the spheres in the $D_n$ lattice expand. By dimension 8, the gaps in the $D_8$ lattice become precisely large enough to hold a secondary, shifted copy of the $D_8$ lattice ⁷⁸.

When these spheres are added, the structure solidifies into the $E_8$ root lattice. The $E_8$ lattice is an even, unimodular lattice, meaning the dot product of any two vectors is an even integer, and it has a covolume of 1 ¹⁶¹⁷. It is highly symmetrical, with 240 minimal vectors corresponding to its 240 kissing spheres ¹⁴¹⁶. The exact packing density fraction of the $E_8$ lattice is $\pi^4 / 384$, filling roughly 25.36% of the 8-dimensional volume ¹⁸⁹. The lattice is also the root lattice of the exceptional Lie algebra $E_8$ and relates to the optimal 8-dimensional Hamming binary error-correcting code ¹⁷.

The Leech Lattice

In 24-dimensional space, an analogous but far more complex object exists: the Leech lattice ($\Lambda_{24}$). The Leech lattice is the unique even unimodular lattice in $\mathbb{R}^{24}$ that possesses no vectors of length $\sqrt{2}$ (no roots) ¹⁶¹⁸.

The lattice was discovered in 1964 by John Leech during his study of the extended binary Golay code. The Golay code is a highly efficient error-correcting code that uses 24 bits to transmit 12 bits of data, capable of fixing up to 3-bit errors. Leech realized that when mapping this code to a 24-dimensional space, the resulting packing had gaps large enough to fit an identical set of spheres, doubling the density and creating the Leech lattice ⁴⁵.

The packing density of the Leech lattice is precisely $\pi^{12} / 12!$, which is roughly 0.193% of the 24-dimensional space. While this represents a tiny fraction of the total volume, it is profoundly denser than any other candidate arrangement in that dimension ¹⁹¹⁹. Like $E_8$, the Leech lattice connects disparate areas of mathematics; compactifying bosonic string theory on a torus defined by the Leech lattice yields the Griess algebra, an object whose automorphism group is the famous "Monster group," the largest sporadic finite simple group ¹⁶¹⁷.

The Cohn-Elkies Linear Programming Bound

While mathematicians heavily suspected that $E_8$ and the Leech lattice were the optimal packings in their respective dimensions, proving this mathematically proved extraordinarily difficult. The framework that eventually enabled the proof was formulated in 2003 by Henry Cohn and Noam Elkies ⁷¹⁸.

Harmonic Analysis and Poisson Summation

Cohn and Elkies adapted the linear programming bounds originally developed by Philippe Delsarte for error-correcting codes, translating them into continuous Euclidean space using Fourier analysis ¹⁵¹⁸²⁰. They proved that one could establish a strict upper limit on the density of any sphere packing in $\mathbb{R}^n$ by identifying a suitable "auxiliary" or "magic" function $f : \mathbb{R}^n \to \mathbb{R}$.

To serve as a valid bound, this function must be a radial Schwartz function - a smooth, rapidly decaying function whose value depends solely on the distance from the origin - that satisfies three rigid conditions: 1. Both the function and its Fourier transform must evaluate to 1 at the origin: $f(0) = \hat{f}(0) = 1$. 2. The Fourier transform must be everywhere non-negative: $\hat{f}(y) \ge 0$ for all $y \in \mathbb{R}^n$. 3. The function must be non-positive outside the central sphere of the packing: $f(x) \le 0$ for all $|x| \ge r$, where $r$ is the minimal distance between sphere centers ¹²¹²².

If such a function exists, the Poisson summation formula, which intimately connects a lattice sum to its dual lattice sum, dictates that the density of any sphere packing in that dimension can never exceed the volume of a sphere of radius $r/2$ multiplied by $f(0)$ ⁸²¹²³.

Cohn and Abhinav Kumar ran extensive computational optimizations to find functions that satisfied these constraints. For dimensions 8 and 24, the numerical results were staggering. The computer-generated auxiliary functions yielded upper density limits that were within a factor of 1.000001 of the $E_8$ lattice density and 1.0007071 of the Leech lattice density ¹⁸²⁴. This microscopic margin strongly implied that the lattices were indeed optimal and that perfectly sharp bounds could be achieved if an exact, analytic magic function could be formulated.

To achieve a perfectly tight bound proving a specific lattice is optimal, the roots of the Cohn-Elkies auxiliary function $f(x)$ must align perfectly with the specific distances between spheres in that lattice. For $E_8$, the theoretical magic function must drop below zero at the sphere boundary and then undulate to touch the x-axis from below exactly at the lengths of the lattice vectors (e.g., distances of $\sqrt{2}, \sqrt{4}, \sqrt{6}$) without ever crossing into positive territory. Simultaneously, its Fourier transform must remain strictly non-negative everywhere, touching the axis at corresponding discrete intervals ^1182124.

The Obstruction of Multiples of Four

The Cohn-Elkies numerical experiments revealed that the linear programming bound was exceptionally tight only in dimensions 1, 2, 8, and 24 ⁹²⁶. In most other dimensions, the best available upper bounds and the known lower bounds remained far apart, leading researchers to question whether the linear programming method simply required better optimization or if fundamental mathematical barriers existed ²⁵.

Cohn and Ngoc Mai Tran, and subsequently Cohn and Nicholas Triantafillou, addressed this by computing dual linear programming bounds. They proved the existence of strict theoretical obstructions to the Cohn-Elkies bound in specific dimensions. By formulating modular forms to calculate limits on how effective a primal linear program could be, Triantafillou demonstrated that for dimensions that are multiples of 4 - specifically 12, 16, 20, 28, and 32 - the LP method is mathematically incapable of providing an upper bound that matches the known best packings ²⁵²⁶. In spaces like 9D, where continuous families of non-lattice "fluid diamond packings" exist, or 10D, dominated by the non-lattice Best packing, the gap remains similarly unbridgeable via current LP techniques ¹²¹³²⁴. Thus, 8 and 24 dimensions are extraordinary anomalies where the geometric alignment of the lattices perfectly matches the analytic requirements of Fourier eigenfunction bounds.

Viazovska's Modular Form Solution in Eight Dimensions

Despite the overwhelming numerical evidence in 8 and 24 dimensions, deriving the exact, explicit auxiliary function to satisfy the Cohn-Elkies inequalities proved impossible for thirteen years. The challenge required finding a function with highly erratic properties: strict roots at lattice vector lengths, rapid decay, and simultaneous positivity constraints on its Fourier transform. Mathematician Thomas Hales noted that the community suspected the function existed but felt "it would take a Ramanujan to find it," referencing the legendary mathematician known for pulling deep algebraic identities from thin air ⁷²⁷.

On March 14, 2016, Maryna Viazovska, a postdoctoral researcher at the Berlin Mathematical School, published a 23-page paper titled The Sphere Packing Problem in Dimension 8. She provided the exact analytic construction of the elusive magic function, definitively proving that no packing of unit balls in $\mathbb{R}^8$ has a density greater than the $E_8$ lattice ⁷²⁸²⁹.

The Theory of Modular Forms

Viazovska's breakthrough hinged on a profound connection between the discrete geometry of sphere packing and the continuous analytic theory of modular forms ²⁰³². Modular forms are highly symmetric, complex analytic functions defined on the upper half of the complex plane. They satisfy intricate functional equations related to the modular group $SL(2, \mathbb{Z})$ and exhibit an infinite number of "hidden" symmetries, often described as trigonometric functions "on steroids" ²⁰³⁰. Because of these extreme symmetric constraints, the vector spaces of modular forms of a specific "weight" are finite-dimensional, making them rigid and predictable objects ¹⁶³⁰.

Viazovska realized that the required double roots at specific lattice distances ($\sqrt{2}, \sqrt{4}, \dots$) could be systematically engineered by utilizing the structured coefficients of modular and quasimodular forms ⁸²¹.

Constructing the Magic Function

To build the Cohn-Elkies auxiliary function, Viazovska defined the target function as a linear combination of radial eigenfunctions of the Fourier transform. She explicitly constructed a $+1$ eigenfunction and a $-1$ eigenfunction ²⁰.

The components of these eigenfunctions were built using classical modular functions: 1. Eisenstein Series: $E_2$, $E_4$, and $E_6$, which are fundamental examples of modular and quasimodular forms that dictate specific growth and asymptotic properties. 2. Jacobi Thetanull Functions: $\theta_2, \theta_3$, and $\theta_4$, which are deeply connected to the combinatorial properties of lattices and the distribution of their vector lengths ³¹³².

By carefully calibrating these objects, Viazovska created two complex functions, $\phi_0(z)$ and $\psi_I(z)$. These functions were then mapped into the real domain via an integral Laplace transform. The resulting function $f(x)$ naturally possessed roots at the exact vector lengths of the $E_8$ lattice and vanished identically where required, forming an optimal "certificate" of the bound ¹²⁴³³.

Romik's Human-Verifiable Proof

While Viazovska's construction was analytically exact, verifying that the resulting function strictly obeyed the inequalities $f(x) \le 0$ (for $|x| \ge \sqrt{2}$) and $\hat{f}(y) \ge 0$ was arduous. In her original 2016 paper, she proved these conditions by utilizing interval arithmetic and computer-assisted calculations to truncate the asymptotic expansions of the functions ²²³².

Although the computer verification was robust, it obscured the conceptual elegance of the inequalities. In 2023, mathematician Dan Romik published a paper providing a direct, fully human-verifiable proof of Viazovska's modular form inequalities. Romik demonstrated that the inequalities governing $\phi_0$ and $\psi_I$ could be proven directly using elementary manipulations of standard mathematical constants, completely removing the reliance on computational truncation and further solidifying the geometric inevitability of the result ³¹³².

The Generalization to Twenty-Four Dimensions

The brevity and power of Viazovska's 8-dimensional proof - 23 pages compared to the hundreds required for Hales' 3-dimensional proof - stunned the mathematical community ⁷³³. The immediate question was whether her modular form machinery could be adapted to the Leech lattice in 24 dimensions.

Within a week of her preprint appearing online, Viazovska was contacted by Henry Cohn. She subsequently joined forces with Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko ²⁴³⁴. On March 21, 2016, just days after the 8-dimensional result, the team posted a second paper proving that the Leech lattice is the unique optimal periodic packing in $\mathbb{R}^{24}$ ²²²⁴. Both papers were subsequently published side-by-side in the Annals of Mathematics in 2017 ²²²⁸³⁵.

Weight 12 Forms and the Ramanujan Tau Function

The transition to 24 dimensions required engineering a new magic function tailored to the geometry of the Leech lattice. Because the Leech lattice contains no roots (no vectors of length $\sqrt{2}$), the minimal non-zero distance between sphere centers is exactly 2, and the required root structure of the auxiliary function shifts accordingly ¹⁶²².

Since the Leech lattice is an even, unimodular lattice in 24 dimensions, its theta series $\Theta_{24}$ is a weight 12 modular form ¹⁶. The space of such forms is exactly 2-dimensional. To build the required Fourier eigenfunctions, the researchers utilized the modular discriminant $\Delta$, a cusp form deeply tied to the Ramanujan tau function $\tau(n)$ ¹⁶. By executing an integral transform over a carefully weighted combination of these weight -8 and weight -10 forms, they produced an exact auxiliary function that proved the Leech lattice could not be surpassed in density by any other regular or irregular arrangement ¹⁶²². Furthermore, their proof established uniqueness: because the generated function possesses no extraneous roots, the Leech lattice is the unique optimal periodic packing in $\mathbb{R}^{24}$, up to scaling and isometries ²².

Energy Minimization and Universal Optimality

The classic sphere packing problem models hard spheres that interact exclusively upon contact, preventing overlapping interiors ²². However, in physical systems, particles rarely behave as impenetrable boundaries. Atoms, molecules, and electrons often repel one another over continuous distances governed by potential energy forces, such as the Coulomb force or inverse power laws ¹⁷³³. This introduces the broader mathematical problem of energy minimization: how should points be arranged in space to minimize the total pairwise repulsive energy?

In 2019, Viazovska, Cohn, Kumar, Miller, and Radchenko expanded their scope beyond hard spheres to address this continuous domain. They published a paper demonstrating that the $E_8$ and Leech lattices are "universally optimal" ³³³⁶.

Completely Monotonic Potentials

Universal optimality is a remarkably rare property. A configuration is universally optimal if, among all point arrangements of the same density, it achieves the absolute minimum Gaussian energy for every potential function that is a completely monotonic function of the squared distance ⁸¹⁷²³. This means that whether the particles are repelling each other sharply according to an inverse square law, or softly according to a Gaussian distribution, the $E_8$ lattice (in 8D) and the Leech lattice (in 24D) represent the definitive ground state ⁸¹⁷.

By contrast, three-dimensional space is chaotic regarding potential energy; changing the repulsive force changes the optimal physical structure, leading to a zoo of different ground states where universal optimality is impossible ³³. In two dimensions, strong numerical evidence suggests the hexagonal (equilateral triangle) lattice is universally optimal, but rigorous proof remains elusive ⁸³³.

Fourier Interpolation and Ground States

To prove universal optimality, the research team had to upgrade the Cohn-Elkies linear programming bounds. They formulated a robust Fourier interpolation theory capable of recovering continuous radial Schwartz functions based solely on their values and derivatives at discrete lattice points ⁸¹⁷.

By generalizing Viazovska's original "magic function" construction into an explicit Fourier interpolation formula, the team generated an infinite family of auxiliary functions. These functions provided immediate mathematical "certificates" that the $E_8$ and Leech lattices minimize energy for any completely monotonic repulsive force ⁸³³. This result established that dimensions 8 and 24 serve as unique nexus points where geometric density and continuous energy optimization align perfectly.

Formal Verification and the Fields Medal

The application of modular forms to solve centuries-old questions of discrete geometry represented a paradigm shift in modern mathematics. The elegance and profound cross-disciplinary utility of these proofs prompted immediate international recognition.

The 2022 Fields Medal Citation

On July 5, 2022, Maryna Viazovska was awarded the Fields Medal at the International Congress of Mathematicians in Helsinki, Finland ³⁷³⁸. The Fields Medal is widely considered the highest accolade in mathematics, awarded quadrennially to researchers under the age of 40 for groundbreaking existing work and the promise of future achievement ³⁴³⁸. Viazovska was only the second woman to receive the award in its 86-year history ¹¹³⁴.

The official citation from the International Mathematical Union commended Viazovska "for the proof that the $E_8$ lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis" ²³³⁸. The scientific community lauded her approach for directly engaging with the mathematical heart of the matter without relying on the exhaustive computational techniques that defined previous dimensional proofs ⁸¹¹.

AI-Assisted Formalization in Lean

While the human mathematical community accepted Viazovska's proofs swiftly, the highest standard of modern algorithmic rigor requires translating every conceptual step into a strict logical language that software can verify automatically ³²³⁹. In 2024, Viazovska and mathematician Sidharth Hariharan launched the Formalising Sphere Packing in Lean project, utilizing the widely-used interactive proof assistant Lean ³²³⁹.

The project marked a historical turning point in human-AI mathematical collaboration. The researchers utilized a specialized artificial intelligence named Gauss, developed by Math, Inc., to assist in the formalization. By completing complex intermediate steps regarding contour integration, discrete geometry, and harmonic analysis, the AI dramatically accelerated the timeline ³²³⁹.

The 8-dimensional proof was formally verified in a mere five days - a task human maintainers estimated would otherwise take six additional months of manual coding. Subsequently, Gauss autoformalized the vastly more intricate 24-dimensional proof in just two weeks, generating over 200,000 lines of rigorous code ³²³⁹. This 2026 achievement certified the absolute mathematical truth of the packing limits in dimensions 8 and 24, and stands as the first complete computer formalization of a Fields Medal-winning result from the 21st century ³²³⁹.