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An Intro to Diagonal and Orthogonal Movement on Exotic Grids Part 1

May 14, 2012

Part 1: Squares Hexagons and Triangles
Part 2: Pentagons and Octogons
Part 3: Septagins, Nonogons, and Decagons

This is a Grid. A Checkerboard, etc.

A grid is one of three regular tessellations, meaning that it is a infinitely repeating pattern of the exact same shape. The squares are alternately colored black and white.

A Rook on a Grid

On a normal grid, orthogonal movement occurs in a single direction across the edges of squares. Notice that an orthogonal move always alternates color.

A Bishop on a Grid

Diagonal movement occurs across corners of squares in a single direction.

Notice it does not cross any edges to accomplish this.

Also a bishop that only moves diagonally will only ever remain on a single colored square.

This is a Hexgrid

This is the modern grid of choice for many of the most popular board games. Notably, Settlers of Catan and many war games. A regular tessellation of repeating hexagons of three distinct colors.

A Rook on a Hex Grid

A rook travels across one the the six sides of the hexagon and travels in a single direction along that trajectory. Like on the grid, orthogonal movement cycles between each color in sequence.

A Bishop on a Hex Grid

On a hex grid, no two hexagons share a corner in the same way as the original grid. Without crossing an edge, diagonal movement can still occur across a corner and traveling along an edge.

Though this is certainly diagonal movement, it is subtly different than diagonal movement on a square grid. To differentiate these differences, diagonal movements can be ranked, according to the number of edges that a single move would touch.

In these examples, the square grid would use a Rank 0 Diagonal Movement, while a Hex grid would use a Rank 1 diagonal movement. Ranking becomes important when a grid offers more than one rank of diagonal movement.

Like the regular square grid, a bishop that only moves diagonally will only ever remain on one color of hexagon.

This is a Triangle Grid

A triangle grid is infrequently used in some obscure board games. This is the third and final regular tessellation, it is composed of triangles of four colors and two orientations. Having two orientations for the triangles makes this grid significantly different from it’s counterparts, but is still a very viable game board.

A Rook on a Triangle Grid

At first glance, the rooks movement seems obvious, but the triangles shift in orientation creates a major problem. If a rook were to travel in a single direction along a triangle grid, it would encounter a corner. A strict interpretation of orthogonal would suggest an orthogonal move could only occur one square at a time. However, by alternating between just two directions  to create an orthogonal path with a combined single direction.

Both orthogonal and diagonal movements can cycle through directions when encountering an obstacle, and it gains an alternating rank based on the number of directions necessary to combine into a single repeating direction.  A triangle grid require at least an Orthogonal Alternating 2 movement. Two triangles of movement would give a single repeating direction.

Following Alt2 Orthogonal movement follows the same rules as the previous grids in that it cycles through each color of the grid successively.

A Bishop on a Triangle Grid


Likewise, a diagonal movement traveling in one direction would encounter an edge, therefore it is appropriate to use two directions. The movement shown above is a Rank 0 Diagonal Alternating 2. For short hand sake, I will call it a 0/2 Diagonal move. Again, this type of movement follows normal diagonal rules of remaining on a single color.

Some other common things to note about these grids.

-From the same square, a rook and a bishop share 0 movement options.

-A diagonal movement would require two orthogonal movements.

-Diagonal moves are always between spaces of the same color.

-Bishops and rooks have the same number of paths, but rooks have the capacity to move onto more spaces.

Tune in next time for Pent Grids, Octo Grids, and more!

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