**Snyder equal-area projection** is used in the * ISEA (Icosahedral Snyder Equal Area) discrete global grids *. The first projection studies was conducted by John P. Snyder in the 1990s.^{ [1] }

It is a modified Lambert azimuthal equal-area projection, most adequate to the polyhedral globe, a truncated icosahedron with 32 same-area faces (20 hexagons and 12 pentagons).^{ [2] }^{ [3] }

For non-exact approximations (to equal-area) it can be replaced by Gnomonic projection, as in *H3 Uber*.^{ [4] }^{ [5] }

As stated by Carr at al. article^{ [3] }, page 32:

*The S in ISEA refers to John P. Snyder. He came out of retirement specifically to address projection problems with the original EMAP grid (see Snyder, 1992). He developed the equal area projection that underlies the gridding system.**ISEA grids are simple in concept. Begin with a Snyder Equal Area projection to a regular icosahedron (...) inscribed in a sphere. In each of the 20 equilateral triangle faces of the icosahedron inscribe a hexagon by dividing each triangle edge into thirds (...). Then project the hexagon back onto the sphere using the Inverse Snyder Icosahedral equal area projection. This yields a coarse-resolution equal area grid called the resolution 1 grid. It consists of 20 hexagons on the surface of the sphere and 12 pentagons centered on the 12 vertices of the icosahedron.*

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most sides.

In geometry, an **icosidodecahedron** is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a **Kepler–Poinsot polyhedron** is any of four regular star polyhedra.

In three-dimensional space, a **Platonic solid** is a regular, convex polyhedron. It is constructed by congruent, regular, polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:

In geometry, the **truncated icosahedron** is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons.

In geometry, the **truncated icosidodecahedron** is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

In geometry, the **truncated dodecahedron** is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

In geometry, the **rhombic triacontahedron**, sometimes simply called the **triacontahedron** as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

In geometry, the **triakis icosahedron** is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

The **icosahedral honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra, {3,5}, around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral, {5,3}, vertex figure.

The **chamfered dodecahedron** is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

A **regular dodecahedron** or **pentagonal dodecahedron** is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

A **geodesic grid** is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.

The **pentakis icosidodecahedron** or **subdivided icosahedron** is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the *truncated rhombic triacontahedron*.

In mathematics, and more specifically in polyhedral combinatorics, a **Goldberg polyhedron** is a convex polyhedron made from hexagons and pentagons. They were first described by Michael Goldberg (1902–1990) in 1937. They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. *GP*(5,3) and *GP*(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.

The **rectified truncated icosahedron** is a polyhedron, constructed as a rectified truncated icosahedron. It has 92 faces: 60 isosceles triangles, 12 regular pentagons, and 20 regular hexagons. It is constructed as a rectified truncated icosahedron, rectification truncating vertices down to mid-edges.

In geometry, a **hexecontahedron** is a polyhedron with 60 faces. There are many symmetric forms, and the ones with highest symmetry have icosahedral symmetry:

A **Discrete Global Grid** (**DGG**) is a mosaic which covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called a **cell**.

A **geodesic polyhedron** is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra with mostly hexagonal faces.

The **hexapentakis truncated icosahedron** is a convex polyhedron constructed as an augmented truncated icosahedron. It is geodesic polyhedron {3,5+}_{3,0}, with pentavalent vertices separated by an edge-direct distance of 3 steps.

- ↑ Snyder, J. P. (1992), “An Equal-Area Map Projection for Polyhedral Globes”, Cartographica, 29(1), 10-21. urn:doi:10.3138/27H7-8K88-4882-1752.
- ↑ PROJ guide's "Icosahedral Snyder Equal Area", proj.org/operations/projections/isea.html
- 1 2 D. Carr
*et al.*(1997), "ISEA discrete global grids"; in "Statistical Computing and Statistical Graphics Newsletter" vol. 8. - ↑ github.com/uber/h3 Overview
- ↑ github.com/uber/h3/issues/237

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