Many tabletop games use repeated shape systems. Tile placement games usually use tiles of one shape that fit together perfectly, perhaps in multiple ways. While some board games have a regular printed grid for players to move around.
This blog is something of an exploration and investigation into tessellating shapes and how they might inspire us to create new games.
Regular self-tessellating polygons
Let’s quickly look at some definitions:
- A regular shape is a shape where all the angles (vertices) are the same size and all the edges are the same length.
- A polygon is a 2-dimensional shape, with 3 or more edges, where all the edges are straight lines.
- Self-tessellation means that that specific shape can be fitted together over and over to cover an area leaving no gaps.
There are only three regular polygons that will self-tessellate: squares, triangles and hexagons.
In tabletop games, we’re particularly used to seeing tessellating squares and hexagons. Many traditional games, like Chess and Go are played on a square grid. We are familiar with the square tiles of Carcassonne and Qwirkle. Catan, Tantrix and Hive all use hexagons, as do many war games.
The paths that grids create
If a game uses a grid for movement, or network creation, we have to think about what pathways the grid creates.
Square grids are popular, because you could travel along the edges of the squares in a straight line, but you can also travel from centre to centre of each connecting square in a straight line.
Hexagonal grids give us two different options. We could travel along the edges of the hexagons in a zig zag path – this is how we build roads in Catan (black hex edges in the diagram). Or we could travel from the centre of one hex to the centre of an adjacent hex – allowing us to travel along straight lines at three different angles (red lines in the diagram).
Planar Duality
In the hex diagram above, you may notice that the red paths that run through the centre of each hex themselves form a triangular grid. Moreover, if we take a triangular grid and draw paths from the centre of each triangle to those adjacent, we get a hexagonal grid.
This is always true for triangles and hexagons. These two shapes have ‘planar duality’. They are essentially the inverse of each other within the 2d plane. Planar duality is part of graph theory and is, in my opinion, endlessly fascinating. I highly recommend that any game designer who is even vaguely interested in mathematics lose themselves in graph theory.
We don’t see so many games based on triangular grids, because the pathways from the centre of one triangle to the next move in zigzag patterns as they follow the edges of hexagons. This makes travelling around the grid a little less intuitive.
My own game, Digsaw, uses a triangle grid, but players use the edges of the triangles to cut along, thereby using straight paths. When creating Digsaw, I experimented with different kinds of grids, you can find out more by reading my designer diary.
Semi-regular tessellation
While we only have three regular self-tessellating polygons, many regular polygons can tessellate with each other to create a variety of patterns. Squares and triangles with the same length edge will tessellate with each other in all sorts of different ways.
Octagons and squares tessellate to create a regular pattern. Note that this pattern has a planar inverse of right angled triangles – therefore creating straight pathways connecting the centres of adjacent polygons.
You can also combine 3 regular polygons to create a tessellating pattern.
Tessellating irregular polygons
Once we dispense with the restriction of only using regular polygons, we are met with an infinite number of tessellating shapes, providing a vast array of beautiful patterns.
Of course, if you’re going to cut tiles of a specific shape for players to put together in a tableau, remember that there may be multiple ways for these shapes to tessellate. Does your gameplay allow this? Or do you need to define legal tile placement in your ruleset?
I have a tile placement game (prototype) in which players create their own aviary. Tiles are hexagonal, but each hexagon has three rhombuses of different (or sometimes the same) colours/types. The placement of the hexagons is regular – they only tessellate in one way. But the hexagons can be rotated to create different areas of the same colour rhombus. As you can see, the areas that these rhombus shapes create are varied and interesting. Having certain sections of the hexagon coloured ensures the placement rules are intuitive, but with six different orientations for each tile, there’s still plenty of choice for the player to create multiple different coloured areas.
Finding planar inverses
The process of finding the inverse of a repeated tessellating pattern is simple. 1) Draw out the repeating pattern. 2) Place a dot in the centre of each shape. 3) Join the dots on adjacent shapes with straight lines. Shapes are adjacent if they share an edge.
When you repeat the process on the planar inverse, you get the original pattern again.
I find this kind of activity a useful design exercise. I learn about the relationships between different shapes and draw inspiration that might make its way into a game.
Shapes with curved edges
Of course, it’s also possible to find shapes with curved edges that tessellate. However, if you go through the process above of trying to find a planar inverse, then repeat this, you will not get the same shape as you started with. Our process here involves only straight lines in polygons with no internal angle that is greater than 180 degrees. We’re going directly from the centre of one shape to the centre of the next. Where we have curved lines, we can create a pathway of curved lines, but it’s not a strict inverse because of the infinite number of ways that you can curve a line.
As a tessellation lover, I really want to see all the tessellating shapes, patterns and planar inverses that you create. Please do share them in the comments below.