A surprise to be on HN!

Tech notes for the curious:

I started the page in 2013 based on notes I had been collecting for ~20 years before that. I update this reference page as I come up with better visualizations and explanations. The last major update was in 2018 when I rewrote it from using d3.js to using vue.js. It greatly reduced the amount of code I was writing, and also let me make the page more interactive. I then added lots more (doubled coordinates, links to papers, axes legends, diagonal directions, side by side comparisons) and improved most of the existing sections. In 2019 the changes were minor, rewriting the map storage section and improving performance. In 2020 I improved the cube/hex animation, the pixel-to-hex section, and added Rust sample code. There's lots more I could add to the page[1] but I don't have any specific plans for when I'll get around to it.

The page is svg- and js-heavy. I prerender the diagrams on the server (try loading it without js!). Then using IntersectionObserver on each section of the page, I swap the static diagrams and text with interactive versions. This improves load times somewhat, because almost no JS has to run on page load.

For printability I process the page with xsltproc, extract all the links, and turn them into footnotes. Try print preview on the page to see these. Unfortunately the diagrams are broken when printing in Chrome (it works in Firefox) so I need to investigate.

[1]: https://www.notion.so/redblobgames/f8bc2f44fba94607afa9c0671...

I love this page. I've used this page as a reference, it saved me a lot of time and left me with a stronger understanding as well.

It reminds me of old school web. When google first came on the scene I was amazed at how many informational pages from either hobbyists or professionals existed (but which I wasn't finding with older search engines), where someone just decided to put everything they know about some topic into a page in glorious detail.

FWIW, I strongly prefer square grids for tabletop games.

Hexagonal grids give you precise measurements in exactly 6 directions.

If you approximate sqrt(2) as 1.5 then square grids give you pretty darn accurate measurements in 8 directions, including 2 pairs of directions that are orthogonal, which is good for rectangular features like buildings.

Furthermore the distance between two points can be quickly calculated with (long-side + 1/2 short-side), which is doable in your head, unlike the distance calculation on a hex grid.

Lastly, if you divide the circle into a large number of slices and calculate the average error for path distance using hex or squares, among those slices, the average error (compared to pythogorean) is smaller with squares (obviously for small multiples of 6 and small multiples of 8 respectively, one or the other will be preferred).

For a semi-goofy take (CGPGrey) on hexagons:

https://youtu.be/thOifuHs6eY

In case anyone is doing signal processing on a hexagonal grid, the https://en.wikipedia.org/wiki/Hexagonal_fast_Fourier_transfo... is pretty useful. I work on consumer phased arrays with hexagonal grids, so the HFFT is useful for computing the far-field beam pattern.

Surprised they didn’t cover hexagonal ring data structures. Here[0] is a visualization along with the calculations needed to find nearest neighbors.

[0] https://www.flickr.com/photos/jeremyjohnstone/3979747216/

CGP Grey did a video about hexagons recently. I mean, there's only so much you can say about hexagons really, but he manages to cover a lot of ground.

https://www.youtube.com/watch?v=thOifuHs6eY

If curious see also

2019 https://news.ycombinator.com/item?id=19184412

2017 https://news.ycombinator.com/item?id=14627711

2015 https://news.ycombinator.com/item?id=9537009

2015 https://news.ycombinator.com/item?id=8941588

Discussed at the time: https://news.ycombinator.com/item?id=5809724

This is from 2013, so I guess it comes up every so often (2019: https://news.ycombinator.com/item?id=19184412). But, I've had it bookmarked since the first time I saw it come up. It's a great article for anyone who's ever played a hexagonal game or has thought about making a hexagonal game.

The source site also has a lot of great stuff on applications of computer science to less-thought-of aspects of game development (by which I mean people usually think of pixel-blasting 3D engines as "game development", but there's a lot more to it)

The guides on that website are fantastic. I wrote multiple games using them as reference. Well written, explained clearly and a wealth of information.

Ah, I wish I had come across this two weeks ago!

I've been doodling with Processing and was working on hexagonal Truchet tiles. (basically random tiles that have paths connecting to any random tiles next to it, with hexagons as tiles instead of squares)

I ended up having to figure out some of that stuff again, in particular the coordinate system inside each tile so that I could deal with integer coordinates and translate them into the right fraction of the radius.

It was fun to figure out what I needed, so I don't regret it :)

And I am also planning on doing something with cubes, so this page will come in handy.

Thanks for sharing! I love how it covers all aspects of the problem.

Result of the tiling: https://twitter.com/_timothee/status/1330315929641570304

Diagram of my coordinate system: https://twitter.com/_timothee/status/1327859098642046976

See also several past discussions, though none recent:

https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...

Every time I see a post of redblobgames, I realize what a treasure this website is.

I've been on this website so often, and pointed other people so many times to it... I hope this never gets lost and is archived.

If Amit would write a book about gamedev, I'd definitely buy it and add it to my collection.

I almost never use bookmarks any more, but I’ve had this page bookmarked for the better part of a decade now and still refer to it at least once a year, and I don’t even work in game development.

An added challenge is to efficiently construct hex grids by hand——quickly, at the expense of accuracy. I didn’t have access to a printer early in the pandemic, but wanted to put together maps for a tabletop rpg. This is a clever solution: http://www.rdwarf.com/users/wwonko/hex/irregular/ It has the added benefit of being easy to remember

Uber's H3 geospatial library uses a lot of these concepts. https://eng.uber.com/h3/

In case you want to make a hexagonal cellular automata:

https://github.com/churchofthought/HexagonalComplexAutomata

Or in WebGL:

https://github.com/churchofthought/Grautamaton/tree/complex

This just came up a bit ago in another thread. RBG is so good. Like I did a while back I'll mention that that work is the reason I could write this Ruby svg viz library https://github.com/SpeciesFileGroup/waxy and it be a fun, rather than painful process.

Red Blob Games has quite a few excellent blog posts. I highly recommend checking out the rest of their blog :)

And if you want something cool to do with this, try implementing Gliński's variant of chess.

It’s something I’m dabbling in, but no doubt an order of magnitude more difficult to implement than orthodox chess.

The idea of indexing by cubes is so obvious once you discover it, but maybe not as easy to intuit. Same goes for the data structures.

Such an invaluable resource! I spent many hours here while helping develop https://hexiconapp.com/. The axial coordinate system, in particular, makes working with hexagons much simpler.

Are you familiar with the kisrhombille grid?

https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling#...

Scaleable like a square grid.

12 directions.

I remember this page. I used it almost a decade ago when implementing a settlers of catan clone for that was going to be the main project for a class curriculum. I was a TA/RA at the time.

It's a great page. It was invaluable to me.

This is an awesome resource I keep going back to now and again for the sheer fun of it. Glad to see it on HN again (it’s well worth reposting occasionally for folk who haven’t seen it yet).

Good stuff. I’m surprised that I have not seen a voxel-based game that leverages a purely 2D render and expects the player to switch between projections to work with the area.

I like octagonal grids. Sure, you have to have your game in hyperbolic space, but you can have accurate movement in 8 directions. Your players will just have to deal with parallel lines diverging, there being unique things like pseudocycles, horocycles, circles with a circumference that grows exponentially with increasing radius, etc. Sadly such geometry isn't supported by any tabletop software I'm familiar with, and crocheting a game board of sufficient size takes a very long time.

> Hexagonal grids give you precise measurements in exactly 6 directions.

Square grids give you precise measurements in exactly 4.

> If you approximate sqrt(2) as 1.5 then square grids give you pretty darn accurate measurements in 8 directions, including 2 pairs of directions that are orthogonal

If you approximate sqrt(3) as 1.75, then hex grids give you pretty darn accurate measurements in 12 directions, including 3 pairs of directions that are orthogonal.

(If square grids have an advantage here, it's that the quantum of measurement the same on each axis of each orthogonal pair, 1 on one axis and 1.5 on the other; whereas each orthogonal pair on a hex grids has one axis with quantum 1 and one with quantum 1.75.)

> Furthermore the distance between two points can be quickly calculated with (long-side + 1/2 short-side), which is doable in your head, unlike the distance calculation on a hex grid.

The equivalent loose approximation on a hex grid would seem to be long direction minus 1/4 short direction, which doesn't seem any harder.

       ______            ______      
      /      \          /      \     
    /    B     \______/          \___
    \          /      \          /   
      \      /          \      /     
        ----      C       ----       
      /      \          /      \     
    /    A     \______/    D     \___
    \          /      \          /   
      \      /          \      /     
        ----              ----       
      /      \          /      \     
    /          \______/          \___
    \          /      \          /   
      \      /          \      /     
        ----              ----

I prefer Penrose tiling for all tabletop games. Sure, the number of directions for each tile vary, but distances are incalculable

I always found the hex grid useful in Battletech as your facing has an impact on where you get hit and having 6 sides provides more potential combinations of facing.

>FWIW, I strongly prefer square grids for tabletop games.

I agree mostly, though i prefer hex grids for war games and square grids for rpgs/roguelikes.

Hex grids allow for more units in a given encounter, plus, i'm likely biased as the first tabletop war game i ever played was battle masters which used a huge hex grid map and a card draw system for movement with some 'cool skull dice' for battles and big plastic figures on bases with flags for units. There was like 6-7 unique unit types for each side, including a special unique unit for each army with special abilities. You got to build an awesome plastic tower units could occupy, it was basically orcs vs humans like the first warcraft. My 8-9 year old self was pretty stoked.

Also, for some reason i remember there was an elven shortsword item card that had this description

'Light as a moonbeam, sharp as a tiger's claw'

That's really stuck in my head as the definitive description of elvish/elfish(for you pedantic Tolkien fans) weapons.

Just seems fitting.

On the other hand, for movement along the 8 directions the distance is 41% short, for movement along the 4 main directions, the taxicab distance is 41% longer than the actual distance.

For a hexagonal grid, the distance is approximated by counting the number of cells of the shortest path. This has <10% relative error (IIRC) from the actual distance.