A5 v1.0
18 points by pheelicks
18 points by pheelicks
I don't have anything to say about this particular release, but I've heard of A5 from an episode of the Geomob podcast: 312. Felix Palmer: A5 Grid System.
I wondered how the pentagonal grid works. The introduction says the pentagons are equal area and links to some pretty pictures, but doesn’t explain. The blog post says “all the cells at all resolutions have the same pentagonal topology”, which makes me wonder if they are using “topology” in the usual stretchy abstract sense, or if they actually mean geometrically congruent.
Then I wondered how the space-filling curve works. The blog post has a magic word “teohedral” which links to a technical details section of the documentation that I looked for and failed to find in the home page burger menu. I’m surprised that the docs and homepage burger menus look the same but have different contents.
How does projecting the dodecahedral tiling onto a sphere keep the pentagons congruent?
Topology is used in the sense of a “mesh topology”, in graphics one talks of quad or triangle topology. For A5 single topology t is shorthand for “all cells have 5 edges”
If you check https://a5geo.org/examples/tiling and select “dodecahedron” you will see a tiling on congruent pentagons. Select “project” and you will get the sphere, but the cells are no longer exactly congruent. That is geometrically impossible. However they retain their shapes better than if the same thing is done with a cube (as another project, S2 does)
I’m working on a paper which will cover how the curve is constructed, at the core is an L-system you can get an intuition of how it works on https://a5geo.org/examples/lattice
can someone working in the area explain the problem with triangle meshes that all these other cell systems are trying to fix?
I'd never heard of H3 but the first thought I had: why pentagons and not hexagons?
There are already a number of hex systems (and yes you can also make them equal area with a similar projection). However the 12 pentagons any hex system must have will always be outliers and not the same area as the hex cells. Another advantage is that A5 is aperture 4 (each resolution has 4x as many cells as the one above), h3 is aperture 7 so you get fewer resolution levels to choose from. (Other apertures are possible with hexagons but do not nest as naturally)
The fact that pentagons are used might be the first visible characteristic of the system, but it isn’t somehow fundamental to making the rest of it work, it is just one if the choices made in the design as a whole.
Looks like the answer is here but...I don't get it https://a5geo.org/docs/recipes/a5-vs-h3
I didn't look until now (apparently H3 is hexagonal) but I don't see what the shape has to do with the variance. One could have a hex grid without variance in size, I guess, then it's just not this H3?
You’re right, the variance is due to the projection used. H3 chose to use the gnomonic projection, which is simple and quick but doesn’t preserve area