Hexes & Horizons

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YE HAVE HEARD THAT IT WAS SAID BY THEM OF OLD TIME, "Your average human in a flat area without any obstructions in view (think a becalmed sea) can see up to 3 miles" [1]. This fact is the basis of both the six-mile hex where characters cannot see past the edge of the hex, and the three-mile hex where characters can see straight to the middle of the next hex over [2] [3]. There has been some discourse about which format is better, and with which timekeeping system, but I don't want to talk about that today.

Instead, I want to talk about horizons. Note the fine print in the quote from Steamtunnel's original post: "in a flat area without any obstructions in view". Treating the horizon as 3 miles away, whether this means the edge of your current hex or the center of a neighboring hex, is intuitive and useful (the literal measurement is not as important as its game function). However, this measure is also something we can treat as variable to interesting effect.

We have an equation, given an observer's height h above sea level (in meters), the distance d from that observer to the horizon (in kilometers).

d(h) ≈ 3.57 √h

The average person stands somewhere between 1.5 and 2.0 meters, so let's say that h = 2. This is where we get the aforementioned d ≈ 5 which is 5 kilometers or 3 miles. How high up do you have to stand in order to expand your horizon? First, let's rearrange the equation so that we're solving for height above sea level rather than your distance to the horizon. I'm also going to approximate the constant to keep it simple.

h(d) = 0.0784 d 2

Now this is the fun part. A table! Ain't that fun? I think it's fun. For six-mile or ten-kilometer hexes, you will only want to use every other distance (multiples of 10 km or 6 mi) since that will indicate the center of each subsequent hex in a line.

Distance to Horizon (km) Distance to Horizon (mi) Height of Observer (m) Height of Observer (ft)
5 3 1.96 6.43
10 6 7.84 25.7
15 9 17.6 57.7
20 12 31.4 103
25 15 49.0 161
30 18 70.6 232
35 21 96.0 315
40 24 125 410
45 27 159 522
50 30 196 643
55 33 237 778
60 36 282 925
65 39 331 1086

I think having this in mind would encourage you to seek out higher ground to get a better view of your surroundings. For example, consider a two-story building (about 30 feet tall): that would allow you to see another hex away! However, it would take increasingly taller buildings to see even farther away. We love diminishing returns. I haven't included heights for mountains or anything like that, but I hope you can figure that out for yourself if you need it. I'm just the messenger.

This was brought on by me writing a certain special ability for elves, inspired by the idea that the Earth is flat but only for elves (and it's round for everyone else). Their ability is just that they can extend their horizon twice as far for an hour [4]. So then my next thought was, okay, seeing six miles instead of three is nice—but what if you were standing on a really tall tower, made specifically for you to see stupidly far? Maybe it would be less overpowering, in light of this math, not to double the distance between yourself and your horizon, but to (virtually) double your height as an observer. Anyway, that's all besides the point. Towers go brr.

Endnotes

[1] (Steamtunnel). 2009-12-03. "In Praise of the 6 Mile Hex", The Hydra's Grotto.

[2] (Steamtunnel). 2018-09-21. "The Ergonomic 3 Mile Hex", The Hydra's Grotto.

[3] Hines, Joel. 2022-10-23. "Down With The 6 Mile Hex! A Modest Proposal", Silverarm.

[4] B., Marcia. 2023-01-10. “Fivey: Oops, All Feats!”, Traverse Fantasy.

Comments

  1. emarskJanuary 17, 2023 at 8:09 AM

    There are a couple of other considerations to make.

    First. Not only if you're on a 331m tall hill you can see 65km away on the surrounding flat terrain, but the other way round works as well: on flat terrain (and no obstructions) you can see the top of any landmark taller than 331m. You can see mountains from VERY far away.

    Second. Obstructions. Good luck being in a place with NO obstacles whatsoever in ALL directions. Not impossible, but not very common either.

    All in all, the horizon argument for the size of hexes always seemed a pretty rubbish one to me.

    ReplyDelete
    Replies
    1. Marcia B.January 23, 2023 at 4:58 PM

      absolutely agree on all accounts! especially w.r.t. your last point--i think that the distance to the horizon is a very flimsy point of support, at least compared to more pertinent things like how far someone can travel in a given amount of time (and which unit of time or distance works best for one's own campaign).

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