S2 explained: cube-based projection

How can you accurately aggregate and compare point-based data from different parts of the world? When analyzing crime rates, population, or environmental factors, how do you divide the entire globe into equal, comparable units for analysis?   For data scientists and geospatial analysts, these are fundamental challenges. The solution lies in a powerful class of tools called Discrete Global Grid Systems (DGGS). These systems provide a consistent framework for partitioning the Earth's surface into a hierarchy of cells, each with a unique identifier. The most well-known systems, Google's S2 and Uber's H3, have become industry standards for everything from database optimization to logistics.   However, these systems come with inherent trade-offs. Now, a new DGGS called A5 has been developed to solve some of the critical limitations of its predecessors, particularly concerning area distortion and analytical accuracy.   Why Gridding the Globe is Harder Than It Looks The core mathematical challenge of any DGGS is simple to state but difficult to solve: it is impossible to perfectly flatten a sphere onto a 2D grid without introducing some form of distortion. Think of trying to apply a perfect chessboard or honeycomb pattern to the surface of a ball; the shapes will inevitably have to stretch or warp to fit together without gaps.   All DGGS work by starting with a simple 3D shape, a polyhedron, and projecting its flat faces onto the Earth's surface. The choice of this initial shape and the specific projection method used are what determine the system's final characteristics. As a simple analogy, consider which object you’d rather be hit on the head with: a smooth ball or a spiky cube? The ball is a better approximation of a sphere. When you "inflate" a spiky polyhedron to the size of the Earth, the regions nearest the sharp vertices get stretched out the most, creating the greatest distortion.   A Quick Look at the Incumbents: S2 and H3   To understand what makes A5 different, it's essential to have some context on the most popular existing systems.   Google's S2: The Cube-Based Grid The S2 system is based on projecting a cube onto the sphere. On each face of this conceptual cube, a grid like a chessboard is applied. This approach is relatively simple but introduces significant distortion at the cube’s vertices, or "spikes." As the grid is projected onto the sphere, the cells near these vertices become stretched into diamond shapes instead of remaining square. S2 is widely used under the hood for optimizing geospatial queries in database systems like Google BigQuery.   Uber's H3: The Hexagonal Standard Uber's H3 system starts with an icosahedron—a 20-sided shape made of triangles. Because an icosahedron is a less "spiky" shape than a cube, H3 suffers from far less angular distortion. Its hexagonal cells look more consistent across the globe, making it popular for visualization. H3's immense success is also due to its excellent and user-friendly ecosystem of tools and libraries, making it easy for developers to adopt. However, H3 has one critical limitation for data analysis: it is not an equal-area system. This was a deliberate trade-off, not a flaw; H3 was built by a ride-sharing company trying to match drivers to riders, a use case where exact equal area doesn't particularly matter. To wrap a sphere in hexagons, you must also include exactly 12 pentagons—just like on a soccer ball. If you look closely at a football, you'll see the pentagonal panels are slightly smaller than the hexagonal ones. This same principle causes H3 cells to vary in size. The largest and smallest hexagons at a given resolution can differ in area by a factor of two, meaning that comparing raw counts in different cells is like comparing distances in miles and kilometers without conversion. For example, cells near Buenos Aires are smaller because of their proximity to one of the system's core pentagons, creating a potential source of error if not properly normalized.   Introducing A5: A New System Built for Accuracy A5 is a new DGGS designed from the ground up to prioritize analytical accuracy. It is based on a dodecahedron, a 12-sided shape with pentagonal faces that is, in the words of its creator, "even less spiky" than H3's icosahedron.   The motivation for A5 came from a moment of discovery. Its creator, Felix Palmer, stumbled upon a unique 2D tiling pattern made of irregular pentagons. This led to a key question: could this pattern be extended to cover the entire globe? The answer was yes, and it felt like uncovering something "very, very fundamental." This sense of intellectual curiosity, rather than a narrow business need, is the foundation upon which A5 is built. A5's single most important feature is that it is a true equal-area system. Using a specific mathematical projection, A5 ensures that every single cell at a given resolution level has the exact same area. This guarantee even accounts for the Earth's true shape as a slightly flattened ellipsoid, not a perfect sphere.   This is a game-changer for analysis. By providing cells of identical size, A5 eliminates the need for analysts to perform complex area-based normalization. This prevents a common source of error and dramatically simplifies workflows when calculating metrics like population density, risk exposure, or any other value that depends on a consistent spatial unit.   A5 vs. H3 vs. S2: A Head-to-Head Comparison The choice of base polyhedron and projection method results in significant differences between the major DGGS. Here is a direct comparison of their key technical characteristics. Metric A5 H3 S2 Base Polyhedron Dodecahedron (12 pentagonal faces) Icosahedron (20 triangular faces) Cube (6 square faces) Equal-Area Cells Yes (Exact) No (Up to 2x area variation) No Max Resolution ~30 square millimeters ~1 square meter ~1 square centimeter Global Hierarchy Yes (Single top-level world cell) No (122 top-level cells) Yes (6 top-level cells)   The A5 Ecosystem and its "Polyglot Mirroring" Approach   The success of H3 proves that a powerful mathematical system is not enough; it needs a rich ecosystem of accessible tools to gain adoption. A5 is being built with this principle in mind, but with a novel development strategy.   This approach is called "polyglot mirroring." Instead of building a single core library in C and creating language bindings, A5 maintains separate, complete, and equivalent codebases in multiple languages, including TypeScript, Python, and Rust. To keep these distinct codebases synchronized, Large Language Models (LLMs) are used to port changes and new features from one language to another. This strategy makes the system more accessible and maintainable for developers within each language's native community.   The power of this approach was proven in a true "wow moment" during A5's development. The creator, having never written a single line of Rust, fed the existing TypeScript and Python versions and a comprehensive test suite to an LLM. After about a week of guided iteration, the model produced a complete, working, high-performance Rust library. This demonstrates how modern tools can enable a single developer to build and maintain a truly multi-lingual ecosystem, something that would have been impossible just a few years ago.   Conclusion: When Should You Choose A5?   A5 offers a powerful and precise alternative to existing global grid systems. Its primary advantages make it the ideal choice for specific, demanding use cases.   • Statistical Validity: Any analysis where equal-area cells are paramount for accuracy is a prime candidate for A5. This includes density mapping, demographic studies, environmental modeling, and financial risk assessment. • Extreme Resolution: For applications requiring precision beyond what H3 or S2 can offer, A5's ability to index down to cells of approximately 30 square millimeters provides unmatched granularity. • Efficient Global Hierarchy: Workflows that need to query data at a global scale benefit from A5's simple hierarchy, which starts from a single cell representing the entire world. In contrast, loading global data with H3's 122 top-level cells could require 122 separate requests, creating unnecessary complexity and inefficiency. To explore the A5 system, see detailed visualizations, and understand the technical comparisons in more depth, visit the official website at a5geo.org.