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The PYXIS indexing is a hierarchical indexing for the ISEA3H using 3-bit-digit to encode 7 child cells to its parent.  

The PYXIS Tile

The square root three tiling used in the ISEA3H is a series of base 3 refinements.  Hexagons can not be divided into smaller hexagons nor aggregated to form larger hexagons as is common congruent tessellations of square and triangular tiles (quad-trees). Congruency provides an efficient basis for the hierarchical division of space, but fails to retain the important mathematical property of monotonic convergence ( to the Reals) required in a complete numbering system.

The PYXIS innovation encodes both a hierarchy and 
convergence.  Two types of child cells can be identified in the square root three subdivision.  A child cell that shares the centroid of the parent cell called a Centroid Child and a child cell whose centroid is located at the vertex of a parent cell called a Vertex Child.  The PYXIS indexing allocates a digit to each of the children of a parent cell.  The parent index is refined by adding a  zero (0) placeholder to index the Centroid Child. As in the example on the left, the parent cell "0" spawns a Centroid Child "00". This can continue infinitely for all cells that are located at the center of their parent - so "00" is the parent to the vertex cell "000".  

Children share a vertex with 3 parents. This raises a problem similar to that encountered when rounding in other numbering systems such as rounding the number 23.465 to 2 decimal places - up or down?  A rule is required to overcome this dilemma.  Parent cells that were themselves Centroid Children (centroid parent) are used to form the index of the Vertex Children.  The parent index is refined by adding a digit "1,2,3...6) to index the Vertex Child.  In the illustration shown "002". This  indexing generates the elegant PYXIS Tile shown on the right.

The PYXIS DGGS SDK include operations to access, convert, and iterate over PYXIS indices.

Traversing the Globe

Coverage of PYXIS tiles over the Earth forms a complete global spatial reference system.  A full PYXIS tile is positioned on each of the 20 faces of the icosahedron and a modified 5/6th PYXIS tile, with one branch removed from the root/first parent and it subsequent Centroid Children, is positioned on each of the 12 vertices of the icosahedron as shown in the unfolded version.  Each of these 32, 20 plus 12, tiles is given a unique label.  The general form of the PYXIS indexing is shown on the right.

The PYXIS DGGS SDK includes algebraic expressions and operations to efficiently iterate/traverse over the icosahedron referenced cells covering the globe.