https://www.redblobgames.com/blog/2025-03-12-hexagon-spiral-coordinates/

Hexagon spiral coordinates

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Blog post: 12 Mar 2025

My guide to hexagonal grids^[1] is one of the most popular pages on
my site. I keep a list of things I want to add to that page^[2]. One
of them has been spiral coordinate systems. I had thought I would
wait until I actually use them in a real project, so that I would
have real world experience with the thing I'm writing about. I'm
afraid of writing about things I'm unsure about, or information
that's incomplete. But I haven't used them yet.

I decided to stop waiting.

[2025-hexag]Spiral coordinates on a hexagonal grid

I decided to make some diagrams showing the parts I do understand.
This also led me to try to understand more. I gave some unoptimized
sample code. Part of me wanted to wait until I have the best sample
code to present. But ... I've already been waiting for so many years. I
decided to publish the unoptimized code for now.

There are also lots of variants I could have covered: 0-based vs
1-based, outside-in vs inside-out, ring-based vs path-based, uniform
direction vs alternating direction, single spiral vs recursive
spirals, and probably more. But if I wait until I have figured out
all of these, it will take even longer. So I published just one
variant for now.

Take a look at the new section^[3]. Please let me know what you'd
like to see changed or improved!

While working on this, I also added an accidental discovery: running
atan2(r,q) on axial coordinates instead of atan2(y,x) on cartesian
coordinates produces something angle-like that can be used for
sorting by angle, while being slightly cheaper than actually
calculating the angles:

[hexagon-ps]Something angle-like that can be used for sorting by
angle

That's not a topic I wanted for the main page, so I put it on a
separate page^[4].

Email me redblobgames@gmail.com, or comment here:

Links

 1. [1]: https://www.redblobgames.com/grids/hexagons/
 2. [2]: https://redblobgames.notion.site/
    Hexagonal-Grids-7d2d4d624bc5483dafbe615d75ab3902
 3. [3]: https://www.redblobgames.com/grids/hexagons/#
    rings-spiral-coordinates
 4. [4]: https://www.redblobgames.com/grids/hexagons/directions.html#
    angles

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 Created 12 Mar 2025;  Last modified: 12 Mar 2025