"To cover" and, more strictly, subpaving is a set of nonoverlapping "boxes" of R⁺. A subset X of R² can be approximated by two subpavings X⁻ and X⁺ such that
X⁻ ⊂ X ⊂ X⁺.
Illustrating bellow X as black line, X⁻ the red cover and X⁺ as the union of red and yellow.
In R², and a multi-scale regular tesselations, the "generic boxes" (also named "grid cells") can be quadrilateral shapes (square, rectangles or diamonds), triangular shapes, or hexagonal shapes.
In other words, the subpaving is a cover using a system of regular grids, where each cell can be split into a refined grid with same shapes. The subpaving is a set without holes or overlaps.
Another important restrictions:
the area of the parent-cell is exactly the sum of the areas of all its child cells.
only one parent per cell.
(repeating subpaving definition) covering without holes or overlaps.
Finite satisfactory range of levels: need for 2, 3 or 4 hierarchical levels, not minus (2 as minimal) and no more for "satisfactory approximation" of X (by X⁻ or X⁺).
... Seems subpaving is impossible with hexagons.
It is?
Is there a mathematical proof that it is impossible?
Motivation and illustration
The cell (and its refiniment) is a central concept of the DGGS standard (ISO 19170-1 or OGC Topic 21), and, as a regular tile system, the only possible shapes are triangular, hexagonal or quadrilateral.
... But, when the task is subpaving, that is "to cover with multiple grid-scales and without holes or overlaps", seems it is impossible with hexagons. It is? Is there a mathematical proof that it is impossible?
The map on the left (California) was covered by hexagons of four different sizes and the hexagons overlapped. The map on the right (Florida) was covered by quadrilateral shapes (diamonds) of four different sizes, with no holes and no overlaps. Below is a detail in California, highlighting the holes (yellow) and overlaps (red) between the larger cells.
Notes
A cell-refinement process illustration, with hierarchical labeling schema and non-overlaping cells.
Bellow all are non-valid cell-refinement of hexagon:
The cases a and b are invalid by restrictions 1 and 2. The case c by restriction 3.
PS: about “to split a DGGS cell”, for an exact definition, see DGGS standards or this animation. About "Finite satisfatory limit" (restriction 4) see National and Global applications of DGGS, por example to solve the Land Ownership by approximating X⁻ zones.
In a regular hexagon, all angles are $120$ degrees. If you have two regular hexagons of different sizes share part of a side, you create an angle of $180 - 120 = 60$ degrees. There is no way to pave that $60$ degree sector with a finite number of regular hexagons.