Most strategy games like chess, civilization 4 and 5, generals.io and others use a game grid design. It defines a map grid that you use to place units, buildings, etc. The grid has to consist of regular polygons to cover the entire map and leave no blank spaces. The question is, how many regular polygons are there that can cover entire map. First, I introduce several examples of games using grids in their design. Then, I will derive an answer for the question.
Triangular Game Grid Design for Strategy Games
There are games that are based on grids consisting of regular triangles. These are also called triangular grids or hex-triangle grids. They often leverage the unique geometry and connectivity of triangles. These grids are less common than rectangular or hexagonal grids, but they provide interesting gameplay mechanics.
Here’s a list of such games and types:
| Triominoes | A variant of dominoes where the tiles are triangular instead of rectangular. Players connect triangles by matching numbers on their edges. |
| Blokus Trigon | A version of the popular game Blokus, but on a triangular grid. Players place pieces made up of connected triangles while following specific rules. |
| Triangular Tile-Laying Games | Games where players lay tiles composed of triangles, similar to Carcassonne but with triangular grids. |
Unique Gameplay Aspects of Triangular Grids
- Connectivity: Each triangle connects to 3 or more neighboring triangles, offering more dynamic movement and adjacency rules than square grids.
- Geometry: Triangular grids allow unique tiling patterns and pathfinding mechanics.
- Flexibility: They can form larger geometric shapes, such as hexagons or larger triangles.
Rectangular Game Grid Design for Strategy Games
Games based on grids of regular rectangles are some of the most popular. Square grids are the most common. These games are widely recognized due to their simplicity and versatility. These grids allow for straightforward movement, adjacency rules, and gameplay mechanics. Here’s a categorized list of such games.
| Chess | A strategy game played on an 8×8 grid of squares. Movement is governed by specific rules for each piece, such as diagonal or straight-line moves. |
| Scrabble | A word game played on a 15×15 grid where players form words by placing letter tiles on the grid. |
| Civilization 4 | Maps consisting of square-like shapes where units and cities are positioned. |
Why Rectangular Grids Are Popular
- Simplicity: Easy to visualize and compute movement or adjacency.
- Versatility: Supports a variety of mechanics, from puzzles to strategy.
- Accessibility: Intuitive for players, as grids naturally align with human perception of space.
Hexagonal Game Grid Design for Strategy Games
Games based on hexagonal grids are popular for their unique geometry. They offer different movement patterns and adjacency rules. These grids provide distinct gameplay mechanics compared to square or triangular grids. Hexagonal grids are particularly useful for strategy games, as they allow for smoother, more natural movement in six directions. Here’s a categorized list of such games:
| Civilization 5 | Units arStarting with Civilization V, the maps use hexagonal grids. This allows for more natural terrain movement compared to square grids. |
| Catan | Players build settlements, roads, and cities on a hexagonal grid of resource tiles. The hex grid allows resource diversity and strategic placement of settlements. |
| Heroes of Might and Magic III | Though the overworld map uses squares, the combat grids in some versions are hexagonal. |
Unique Aspects of Hexagonal Grids
- Efficient Packing: Hex grids provide a compact and uniform tiling of space, often used for territory control or pathfinding.
- Balanced Distance: In a hex grid, all adjacent cells are equidistant, avoiding the diagonal distortion of square grids.
Comparison Between Different Polygonal Type Grids
One obvious advantage of hexagonal grids is that there is an equal distance from center of one cell to another. This advantage does not exist in rectangular and triangular grids. The distance between cells in the rectangular grids depends on the neighboring cells. Moving diagonally takes a larger distance than moving horizontally or vertically in a rectangular grid. You can move more efficiently by moving diagonally. Avoid moving horizontally or vertically if you want to travel faster on the map. Same goes for triangular grids where the distances from center of one triangle to centers of other triangles vary. This problem does not exist in the hexagonal grids.
There are different number of connections for each of the tree types of grids. Hexagonal grids take more steps to evaluate. There are 6 ways a unit can move from the current hexagonal tile. In rectangular and triangular grids there are fewer connections between adjacent tiles that share same side as a border.
Map Coverage with Regular Polygonal Cells
Now, the question is, how many regular polygons exist such that they cover the map entirely? This is useful when considering different options for game grid design.
General Formula for Polygonal Angles
First, we observe that for a regular polygon, the angle in between two sides is given by formula:
where k is the number of sides of a polygon. This follows from the break-down of the polygon into triangles:
Grid Map Coverage Formulation
Now, for the polygon to cover map we must find number n of polygons that cover 360 degrees when their edges touch:
In other words, we need number n to be a whole number. This number gives us the number of polygons required to cover the grid in the direction of 360 degrees nearby. The relation between number n and angle a is the following:
Substituting Expression 1 into a gives:
Simplifying:
Number Analysis
The expression 2 holds for k > 2 such that,
n = 6 when k = 3. This means it takes 6 regular triangles to cover 360 degrees of space (Figure 2).
n = 4 when k =4. This means it takes 4 regular rectangles to cover 360 degrees of space (Figure 2).
n = 3 when k = 6. This means it takes 3 hexagons to cover 360 degrees of space (Figure 2).
n is the number of k-sided regular polygons required to cover 360 degrees of space.
For k = 5, the number n is 10/3, which is not a whole number. So, no 5-sided regular polygons can cover the grid uniformly.
We now analyze cases when k > 6. There are two cases: k is even and k is odd. When k is odd, we have k – 2 odd. Odd denominator means n is not a whole number.
Second case when k is even, we can replace it with k = 6 + 2m. The expression 2 evaluates to:
One of 3 + m and 2 + m is odd and one is even. This means n is not a whole number.
In total, we have that only for k = 3, 4 and 6 we have n as a whole number. So only regular triangles, regular rectangles and regular hexagons can cover two-dimensional grid completely without blank spaces.
Conclusion
Grid-based structures play a crucial role in strategy games, defining movement rules, unit placement, and overall gameplay dynamics. This document explored three key types of grid systems. These systems are triangular, rectangular, and hexagonal. It analyzed their unique properties, advantages, and applications in popular games like Chess, Civilization, and Catan.
Only these three types of regular polygons can perfectly tile a 2D plane from a mathematical perspective. They ensure seamless map coverage without gaps. Hexagonal grids offer equal movement distances and efficient packing. Rectangular grids provide simplicity and ease of computation. This makes them the most widely used. Triangular grids, though less common, introduce unique pathfinding mechanics and strategic complexity.
Understanding these grids’ strengths and limitations is crucial. Game designers and developers aim to create balanced and engaging strategy games. The choice of grid structure significantly impacts the gameplay experience. It also affects strategic depth, whether designing a turn-based tactics game or creating an open-world exploration map.
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