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= The beauty of hexagons =

[[File:Hexagons-honey-bees.jpg|right|400px]]

Hexagons are beautiful! For me, they are little masterpieces. A simple geometric shape that somehow conveys something magical. Depending on the context, they are organic or artificial, historical or futuristic, stable or flexible, but always elegant and efficient.

Maybe my view is a bit biased. I've always associated hexagons with games of all kinds and that's why they are part of some of my fondest memories. The first strategy games I had to lose to my big brother. But hey, at least I got to play with him. And the many rounds of Settlers of Catan at the table with good friends. That's where friendships were put to the test. Some of the best games of all time rely on hexagonal boards. But why are hexagons so popular in game design? Well, there's a good reason.

== Hexagonal grids ==

[[File:Hexagons-HexMapDistance.png|left|400px]]

If you arrange squares on a game board as equilateral hexagons, then all neighboring squares are equidistant from each other. This means that each hexagon has six neighbors and the path to each of them is the same. Compare this to a classic game board consisting of squares. Here, each square has eight neighbors, four on each side and four diagonally. So the first question is how a piece can be moved here. If you only move across the sides, each step covers the same distance. However, if you want to move diagonally, the distance is √2 times (about 1.42 times) that of the other neighbors.

Is that a bad thing? In many cases it is not. In games like chess, it's even a feature. The different movements of the pieces are designed for exactly this kind of grid and harmonize wonderfully with it. Real distances don't play a role here. However, if game pieces need to be able to move as freely as possible over a fixed distance, then a grid of squares quickly reaches its limits. If you only allow movement across the horizontal and vertical neighbors, the distance to squares in between increases enormously. If you also allow movement across the diagonal, you can no longer divide movements into equally sized steps.

Hexagonal grids reduce the first problem and eliminate the second. Even with hexagonal grids, there is not always a linear path from one grid to another. However, the six possible basic directions allow a much better approximation. The largest possible deviation here is 15.5 % compared to a maximum of 41.4 % with squares. And the individual steps are always the same distance. Hexagonal grids also have disadvantages. For example, it is more difficult to map rectangular structures such as buildings and corridors. Here you often have to work with split fields. The coordinate systems for hexagons are also somewhat more complex if they are required.

== Cube coordinates ==

[[File:Hexagons-cube-coordinates.jpg|right]]