If you’ve used H3 or S2 it should be familiar, the major difference (apart from the fact it uses pentagons) is that the cell areas are practically uniform, whereas alternative systems have a variance of around 2 between the largest and smallest cells, making them less useful for aggregation. The site has many visual demos, e.g. https://a5geo.org/examples/area
The code is open source: https://github.com/felixpalmer/a5
Is it essential that the cells be the same shape?
Also where does the name "A5" come from exactly? I get that 5 is because it has five sides, but why A?
However the symmetry of H3’s hexagonal cells lends itself well to flow analysis, or routing - which is no surprise as it was developed at Uber.
As for the name, it follows the convention of S2 and H3, which come from group theory and refer (loosely) to the symmetry groups of the various systems
H3 is based on a dodecahedron it is it the reason the cell areas range so much, the same is true of S2 - but this is based on a cube.
The shapes look a bit wonky when projected onto a map, though, and it may not be as intuitive to reason about as the hexagons that would (mostly) result from subdividing an icosahedron. With a subdivided icosahedron you end up with a regular lattice of shapes that is easier to reason about. I think an icosahedron might be a better fit for an indexing scheme for that reason, despite it's higher mathematical error in approximating the sphere at a given resolution.
I explored a similar idea four or five years ago, without being aware of H3. My goal was to find a compact multi-resolution geospatial height map format. My idea was closer to H3 than to yours, it seems.