Hexagons vs. Squares
Today's text is a result of discussions I had with my fellow game designers some years back, when our studio was embarking on a new prototyping spree. We specialized in city builder design and many of our ideas revolved around various builder game mechanics.
One of the pretty fundamental design decisions that needs to be made when starting to work on such a game is about the underlying grid your game is going to use. A rectangular grid seems like the obvious choice, the one you will most likely pick out of inertia and without thinking. However, there are loads of games that do not take this path. Unless you are doing something really experimental, your only remaining choice will be the hexagonal grid.
You may think that the choice between hexagons and rectangles is purely aesthetic, and that you are free to pick one or the other based simply on your personal preference. However, things are not quite as simple as they seem. The choice of underlying grid has much more profound implications on many aspects of game design. Choosing one over the other without being aware of these implications might come back to haunt you mercilessly down the line.
What follows is my attempt to lay out what you are actually choosing between.
Tessellation
Above all, let's first examine why we have just two choices. The answer to this question lies directly in the fabric of our universe, or at least in the way we are hardwired to perceive it. For reasons we do not quite fully understand yet, we feel most comfortable perceiving our universe in three spatial dimensions and describing it using so-called Euclidean geometry. Sorry, fans of Lobachevsky, Riemann and other eldritch horrors.
In this framework, a 2D plane is defined by three points. In other words, you can pick any three points and construct a plane over them. Three points obviously define the vertices of a triangle. A triangle is thus the first geometric shape that we can use to populate the plane. If we remove one of the points of the triangle, our plane collapses into a one dimensional line. All other shapes with straight edges can be constructed as a combination of two or more triangles.
The process of filling up a 2D plane with geometric shapes is known as tessellation. One of the defining prerequisites of tessellation is that the underlying plane is covered completely with shapes, without overlap and without gaps between them. If we are constructing a grid, we are trying to do exactly this: cover an area with a set of cells, i.e. shapes.
Furthermore, we are imposing one stricter requirement, namely that all shapes representing cells need to be identical and regular. Since we do not want any gaps in our grid, all angles meeting at one vertex, a point in the grid, need to add up to a full circle, i.e. 360º.
That single constraint does all the work. Run through the regular polygons and check their interior angles:
- Equilateral triangle — 60º. Six of them meet at a vertex. 6 × 60 = 360. Works.
- Square — 90º. Four meet at a vertex. 4 × 90 = 360. Works.
- Regular pentagon — 108º. Three give you 324º, four give you 432º. Neither is 360. Fails.
- Regular hexagon — 120º. Three meet at a vertex. 3 × 120 = 360. Works.
- Regular heptagon and everything above — the interior angle keeps growing past 120º, so three of them always overshoot 360º and two always fall short. Everything from seven sides up fails, forever.
Three shapes. That is the entire list, and it is not a matter of taste or convention. It is arithmetic. Anyone who has ever asked why strategy games do not use octagons has run into this wall — the answer is that octagons leave gaps, and gaps are not a grid.
Wasted Space
The reason equilateral triangles themselves are rarely used as grid elements in games has to do with the usable area of the polygon.
We are probably going to use our grid to visualize something in our game, presumably a map. This means we are going to place visual assets onto our grid tiles. If we do not want bits of our visual assets to stick out over the edges of the tile, the maximum size of the visual asset will be roughly determined by the area of the inscribed circle — the largest circle that fits inside the cell.
For the three polygons in question, the ratio between the inscribed circle and the polygon itself works out as follows:
- Equilateral triangle — 60.5%
- Square — 78.5%
- Regular hexagon — 90.7%
In other words, using triangles wastes roughly forty percent of every cell. Whatever you place in a triangular tile has to cower in the middle of it, and the three sharp corners are dead space you paid for and cannot use. Squares and hexagons thus remain the only two viable choices.
Notice something else in those numbers, though, because it is the first point in this whole discussion that runs in favour of hexagons: the hexagon is the most efficient of the three. It is the closest of the tessellating polygons to a circle, which means it wastes the least space around whatever you put inside it. It also has the lowest perimeter for a given area, which is precisely why bees build honeycombs out of hexagons rather than squares — less wax for the same storage.
So if the argument were purely about geometric efficiency, hexagons would win outright and we could all go home. The argument is not purely about geometric efficiency.
Scaling
An interesting quirk of geometry is that these three shapes do not behave in a similar way when you try to combine them.
We can combine four squares to create a bigger square. Likewise, you can combine four equilateral triangles to create another, larger equilateral triangle. However, you cannot combine any number of regular hexagons to create a bigger regular hexagon. It simply does not work. Seven hexagons arranged around a central one produce a lumpy, non-convex flower shape, not a hexagon.
This property has important implications on several fronts, from the way people perceive these grids to the way visual assets can be organized.
It is easy to imagine a game asset representing a small building that can be replaced with a four times bigger asset representing a similar but much larger building. We can do this several times over to create larger and larger structures. This natural scaling property can be used in all manner of ways in games to achieve various gameplay features. Upgrade a house into a block. Merge four plots into an estate. Zoom the map out one level and have four tiles become one.
This is simply not possible with a hexagonal grid. Combining several hexagonal shapes will quickly break up the perception of regularity.
If your design involves multi-cell structures, upgrade paths that grow a building's footprint, or level-of-detail systems where the map aggregates as you zoom out, this single property may decide the whole argument for you before any other consideration gets a hearing.
Neighbours and Distance
Now we come to the part of the argument that hexagon partisans consider decisive, and they are not wrong to.
On a square grid, every cell has four neighbours across its edges. It also has four more that touch it only at a corner. That immediately forces a decision that has no clean answer: are diagonal cells adjacent or not?
Say no, and movement becomes blocky. Units shuffle in right angles like they are navigating a city grid, and a "range of three" looks like a diamond.
Say yes, and you have imported a problem. The diagonal is longer than the edge, by a factor of √2, roughly 1.414. If a diagonal step costs the same as an orthogonal one, then a unit moving diagonally covers about forty percent more ground for the same price. The optimal way to cross any distance becomes a zigzag, which is both unrealistic and slightly silly. A unit that marches fifty kilometres a day north suddenly manages seventy kilometres if it goes northwest instead.
Games have patched around this for decades. Charge 1.5 movement points for diagonals. Alternate the cost between 1 and 2. Forbid diagonal movement through two blocked corners. Every one of these is a workaround for a problem the geometry created, and each of them adds a rule the player has to learn.
There is a subtler version of the same problem, and it matters more than the distance does. On a square grid, two diagonal cells touch at a single point, not along an edge. Does fire spread through that point? Does a wall built on both diagonals block movement between them? Does a zone of control extend across it? Does a flood fill leak through? There is no correct answer, only a decision, and you will make that decision separately for every system in your game and get it slightly wrong in at least one of them.
A hexagonal grid has none of this. Every hexagon has exactly six neighbours. All six share a full edge. All six are the same distance away. There are no corner-contacts anywhere in the entire grid, because three hexagons meeting at a vertex all mutually share edges. The ambiguity does not get resolved — it never exists.
This is why hexes became the standard for tabletop wargames long before video games existed, and the reason the military adopted them for map exercises. Distances are honest. A range of three tiles describes a shape that is roughly circular rather than a square or a diamond. Artillery ranges, line of sight, movement allowances and zones of control all behave the same way in every direction, without a table of special cases.
There is a cost, of course. Six directions rather than eight. Some players experience this as a loss of freedom, and defensive lines that were leaky on a square grid become genuinely difficult to breach on hexes, which changes tactical play considerably.
Perception
As I have stated many times in my writing, video games are where mathematics and psychology meet to have a party. Pure mathematics provides just the basics. The psychology of human perception is where the true magic happens.
We perceive graphics constructed on a rectangular grid as more orderly, and graphics constructed on a hexagonal grid as more organic. This is rooted in the difference in scaling properties of squares versus hexagons. Combining several hexagons results in quasi-organic looking non-convex shapes.
This property of hexagonal grids can be exploited if we are aiming for exactly that, for example if we want to create a natural, irregular landscape. A hexagonal grid allows us to do this while keeping a relatively simple and elegant mathematical structure underneath.
On the other hand, if our goal is to create a more rigid, regular looking map, a rectangular grid might be the better choice.
This is evident in the genres where each grid type dominates. Compare the classic hex wargame — Panzer General and the whole Avalon Hill lineage it descends from — with a city builder like SimCity. In the wargame, the map is terrain: rivers, ridgelines, forests, none of which have any business being straight. In the city builder, the map is a city: blocks, roads meeting at right angles, rectangular buildings on rectangular plots. One subject matter is organic and the other is rectilinear, and the grids follow accordingly.
That is not a coincidence and it is worth stating as a rule of thumb. Match the grid to the geometry of the thing you are depicting. Nature is not rectilinear. Human construction very much is.
Furthermore, for some reason we humans seem to intuitively prefer rectangular shapes. Could it be because we are essentially quadrupedal creatures? Would insects prefer hexagonal spaces?
Rectangular houses are one of the most common features of architecture across a great many human cultures, followed closely by round floor plans of various huts. Hexagonal houses are extremely rare. Beehives, on the other hand, share a basic hexagonal plan.
Words for Directions
Try aligning four cardinal directions North, South, East and West to hexagons. Try giving movement instructions on a hexa map?
Simply put, our mental models seem to be better tuned to rectangular maps. Regardless of whether this is a result of our biology or simply a cultural artifact, it is reflected in the language we use.
Almost all languages have words that unambiguously describe movement in rectangular space: up, down, left, right. No language has unique words for six directions in hexagonal space. Movement in rectangular space is easier to translate into words.
Get Stanislav Stankovic’s stories in your inbox
Join Medium for free to get updates from this writer.
Ok, fine, I am fully prepared for some wiseguy to point out a language of some obscure rainforest tribe that has words for six cardinal directions, but that is an exception that just reinforces the rule.
This is not a trivial matter, and I think it is the most underrated item on this entire list.
Your tutorial may need to give clear movement instructions. "Move two tiles to the right" is a sentence. The hexagonal equivalent is not. You end up saying "up and to the right," which is two words for one direction and is ambiguous about which of the two upward diagonals you meant.
More importantly, players talk to each other. They coordinate, they plan strategies, they write guides, they argue in forums about the correct opening. All of that runs on language, and hexagonal space does not have the vocabulary. Watch any hex-based community and you will find they have invented their own conventions — clock positions, compass headings, coordinate callouts — precisely because the natural language was not there.
If your game involves any meaningful amount of player-to-player communication about position, this is a real cost. It is invisible in a design document and extremely visible in a Discord server.
Finally, the quasi-organic appearance of hexagonal grids may have adverse effects on the overall user experience. As I already mentioned, rectangular grids tend to be perceived as more orderly. Piling and placing objects onto a hexagonal grid can result in a cluttered appearance. This becomes a significant problem if the player is expected to constantly locate particular items or locations on the grid.
Incidentally, this is a known tension in hex design. The Civilization V team deliberately obscured their tile borders and let terrain bleed between hexes to get a more organic look — and players immediately complained that they could no longer tell where the tiles were. Legibility and organic feel pull against each other, and you cannot have both at full strength.
The Cost of Building It
Everything above is design. This section is production, and it is the part that gets discovered three sprints in.
Coordinates. A square grid is a two dimensional array. Cell x, y. The neighbours are x±1 and y±1. Any programmer can hold the entire mental model at once, and every engine, tilemap tool and level editor supports it natively.
Hexagons need a coordinate system, and there are several, none of which is obviously correct. Offset coordinates look like a normal array but the neighbour offsets change depending on whether the row is odd or even, which is a permanent, inexhaustible source of off-by-one bugs. Axial coordinates are cleaner. Cube coordinates, where three axes sum to zero, make distance and rotation mathematics genuinely elegant, but require conversion at the boundaries and are unintuitive to anyone encountering them for the first time. This is all solved and well documented — Amit Patel's hexagonal grids reference is the canonical resource and I recommend it unreservedly — but "solved" is not the same as "free." It is a tax on every engineer who touches the map for the lifetime of the project.
Art. Square tiles pack into a texture atlas with zero waste. A hexagon inside its bounding box occupies exactly 75% of it, so a quarter of your tile texture memory is transparent corners. On top of that, seamless tiling art is harder on hexes, and any structure that spans multiple cells is a nightmare, because there is no rectangular multi-cell shape to build against.
Rotation. Square pieces rotate in clean 90º increments and a rotated square footprint is still a rectangle. Hex pieces rotate in 60º increments, and a multi-hex shape rotated by 60º occupies a genuinely different set of cells. If your game lets players rotate anything, budget for it.
Map edges. A rectangular grid has a rectangular boundary. A hex grid has a ragged one, and you will either accept the raggedness, mask it with terrain, or write special cases for the border rows.
Everything around the map. Screens are rectangular. Inventories are rectangular. Build menus, card hands, shop grids, and every other piece of UI in your game is on a rectangular grid whether you like it or not. A hex map is a hexagonal island in a rectangular ocean.
None of this makes hexes wrong. It makes them a decision with a price tag, and the price is paid by people who were not in the room when the decision was made.
The Civilization Case
The best available case study is Civilization, because it is the same design team, the same genre, and the same audience, and they switched.
Civilization I through IV ran on squares. Civilization V moved to hexes, and Civilization VI and VII have stayed there. The reasoning given at the time was exactly the argument in the adjacency section above: honest distances, no diagonal exploit, and — the marketing language of the era — more organic landscapes.
What makes it genuinely instructive is that the grid change did not arrive alone. Civ V also introduced one unit per tile, and those two decisions are load-bearing for each other. Building a defensive line on a square grid with diagonals is close to impossible, because there are always corner gaps to leak through. On hexes, six clean edges make a line an actual line. The grid change enabled a combat design that the old grid could not support.
That is the lesson worth taking. The grid was not swapped because hexes are prettier. It was swapped because a specific mechanic the team wanted became possible. And the trade was real: players lost two movement directions and spent a good while complaining about it.
If you cannot name the mechanic your grid choice is enabling, you are probably choosing on aesthetics, and that is fine — but be honest with yourself that this is what is happening.
So Which One?
There is no universal answer, which is why this text is longer than "use hexes" or "use squares." But the decision does decompose into a handful of questions with fairly clear answers.
Choose squares when:
- You are depicting human construction. Buildings, roads, plots, city blocks, farms, factories. Rectilinear subject matter wants a rectilinear grid.
- Your design needs multi-cell structures, footprint upgrades, or aggregation when zooming out.
- Players will need to talk to each other about position.
- Readability under clutter matters — the player must repeatedly find a specific thing on a busy board.
- Your team is small and the engineering budget is tight.
Choose hexes when:
- You are depicting terrain. Landscape, wilderness, territory, anything nature made.
- Fair distance in all directions is mechanically important. Ranged combat, movement allowances, line of sight, area of effect.
- Adjacency itself is a design resource — six neighbours give you a richer set of relationships than four, which is exactly how Civilization VI's district adjacency bonuses work.
- You want to eliminate the corner-contact ambiguity entirely, which matters enormously for propagation systems like fire, disease, influence or supply.
- Zones of control and defensive lines need to actually hold.
- You feel superior to other humans.
- You are addicted to board games.
- You hate yourself.
And in the interest of completeness: there is a third path, though the draft version of this text claimed there were only two. You can abandon the regular grid. Frostpunk arranges its city radially around a heat source, which is a grid derived from the fiction rather than from geometry. Townscaper builds on an irregular quadrilateral grid that produces charmingly wonky results. Cities: Skylines lets you draw roads freely and derives the plots afterwards. All of these cost more to build and none of them are safe defaults, but "unless you are doing something really experimental" is doing a lot of work in my opening paragraph, and it is worth knowing that the experimental door exists.
Conclusion
The grid is the most quietly consequential decision in this class of game, and it tends to get made in the first week by whoever set up the prototype, on the basis of what was easiest to code that afternoon.
That is what makes it worth writing about. Almost every other early decision can be revisited. You can retheme a game, rebalance an economy, redesign a progression system, and replace an art style, all painfully but all possible. The grid is different, because everything else is built on top of it. Movement rules assume it. Art assets are cut to it. Multi-cell structures are shaped by it. The UI is laid out around it. By the time anyone realises the grid was wrong, changing it means rebuilding the game.
So it deserves an hour of explicit discussion before anyone opens an editor. Not a long hour. Just long enough to answer: what are we depicting, does fair distance matter, will things grow across multiple cells, and will players need to talk about where things are.
Answer those four and the grid chooses itself.
Key Takeaways
- There are exactly three regular tessellations, and it is arithmetic, not convention. Triangles, squares and hexagons. Pentagons and everything above seven sides cannot close 360º at a vertex.
- Triangles waste about forty percent of every cell, which is why they never survive contact with an art pipeline.
- Hexagons are geometrically the most efficient at 90.7% usable area against the square's 78.5%. If efficiency were the only criterion, hexes would win outright.
- Squares scale and hexagons do not. Four squares make a bigger square; no number of hexagons makes a bigger hexagon. If your design has footprint upgrades, multi-cell buildings or zoom aggregation, this may decide everything on its own.
- The diagonal problem is the strongest argument for hexes. On squares, a diagonal step is 1.414 times longer than an orthogonal one, so equal-cost diagonals reward zigzagging.
- Corner contact is the subtler and more damaging version of that problem. Diagonal squares touch at a point, so every propagation system in your game needs its own arbitrary ruling. Hexes share only edges and the ambiguity never arises.
- Match the grid to the geometry of what you are depicting. Nature is not rectilinear; human construction is. That single heuristic resolves most cases.
- No language has words for six directions. If players must communicate about position, hexes impose a permanent, invisible tax on your tutorial and your community.
- Legibility and organic feel pull against each other. Obscure hex borders for a natural look and players lose track of the grid.
- Hexes carry a real production cost in coordinate systems, texture waste of twenty-five percent, rotation, ragged map edges, and the mismatch with an otherwise rectangular UI.
- Civilization switched grids to enable a mechanic, not for looks. One unit per tile needs clean edges to build a line. If you cannot name the mechanic your grid enables, you are choosing aesthetically.
- The grid is nearly irreversible. Everything else in the game gets built on top of it, so spend the hour arguing about it before anyone opens an editor.
Links
- Hexagonal Grids — Red Blob Games — Amit Patel's reference on hex coordinate systems, the best resource on the subject by a wide margin
- Hexagon boards vs square boards explained — a clear treatment of why adjacency degree, not distance, is the underlying issue
- Civilization V's move to hexes — the series has stayed hexagonal from Civ V through Civ VII
