Names like S2, H3, and A5 - these are not branding choices or GIS shortcuts. They come directly from mathematics and encode how geometry and symmetry are used to model the Earth in a rigorous and globally consistent way.
S2: The Geometry of the Earth’s Surface
S2 comes from standard mathematical notation. The letter S stands for sphere, and the number indicates dimension. For example, S¹ represents a circle, S² represents the surface of a sphere, and S³ represents a higher dimensional sphere.
Although the Earth exists in three dimensional space, its surface has only two degrees of freedom: latitude and longitude. Intrinsically, the surface is two dimensional. For this reason, when the Earth is treated as a geometric object for analysis, it is modeled as S2, the two dimensional sphere.
In this context, S2 describes the space where geospatial data lives.
H3: Icosahedral Symmetry in Three Dimensions
H3 comes from Coxeter group notation, which classifies symmetry groups generated by reflections. The letter H denotes a special family of symmetry groups that include five fold symmetry. Five fold symmetry cannot tile flat space, but it appears naturally on spherical surfaces. The number 3 indicates that the symmetry acts in three dimensional space.
The icosahedron and dodecahedron are three dimensional solids whose full symmetry, including reflections, is described by the Coxeter group H3. When a DGGS uses an icosahedral or dodecahedral scaffold to distribute cells evenly over the globe, it is relying on this three dimensional symmetry structure.
H3 describes the full geometric symmetry of the polyhedral framework used to discretize the Earth.
A5: Rotation Only Symmetry of the Icosahedron and Dodecahedron
A5 comes from group theory rather than Coxeter theory. Here, A refers to the alternating group, and the number 5 refers to permutations of five abstract elements. A5 consists of all even permutations of five objects, contains 60 elements, and is the smallest non abelian simple group.
Most importantly for geospatial applications, A5 is mathematically identical to the rotation only symmetry group of the icosahedron and dodecahedron. Unlike H3, it excludes reflections. This distinction matters because reflections reverse orientation, which is usually not physically meaningful when modeling the Earth.
A5 captures the physically meaningful rotational symmetry of the grid.
How These Concepts Fit Together
These names describe different layers of the same system. S2 represents the Earth as a two dimensional spherical surface. H3 represents the full symmetry of the three dimensional polyhedral structure used to discretize that surface. A5 represents the rotation only symmetry that preserves orientation and physical meaning.
A precise way to describe this relationship is that the Earth is modeled as S2, discretized using polyhedral symmetry from H3, with physically relevant symmetry captured by A5.
Using these mathematical symmetry groups ensures that a DGGS is globally uniform, rotation invariant, and free from special cases such as poles or seams. The names S2, H3, and A5 encode these guarantees directly, making the design philosophy explicit rather than implicit.
They are not just labels. They are compact statements of geometry and symmetry that explain why DGGS works the way it does.