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Final Post Regarding Orthoginal and Diagonal Movement on Exotic Grids

September 4, 2012

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

This is a Septagon Grid, or 7 GridThe 7 and 9 Grids provide unique challenges in that they have every deviation of the previous grids combined. 2 different shapes and the septagons come in two different orientations. Not terribly confusing for the grid, which can be tri-color coded for the three types of spaces. The septagons can be classified by the 7th angle pointing either left (white) or right (black)

A Rook on a 7 Grid

From a Septagon I observe 6 orthagonal dirrections. Movements that have been excluded are any movements that from the center of a space to the center of the next space would cross a corner. If the hour glass had been divided into two pentagons it is interesting to note that the movement could be radically different.

It’s not clear on this image, but in 5 of the dirrections the rook would cycle through the three types of squares in order. The sixth direction (dirrectly right in this case) only travels from septagon to septagon. Looking at the trajetory, it’s actually fairly arbitrary why I decided to not have that particular direction not travel onto the hourglass for its third movement (likewise other movements could not move onto the hour glasses). If, however, you draw out a trajectory along a single direction far enough, one trajectory falls closer to the original dirrection given by the initial move.

Another possible–still very debatable– rule is that given two equally valid paths the one that better fits the initial direction is then more valid. If they are equally split in this regard, then both paths are valid (as is the case of the fork in the road on the rightward movement in the diagram)

Furthermore a Rook on an Hour GlassMovement from the hourglass further supports that model as a valid path.

A Bishop on a 7 Grid

From an initial septagon,  4 corners provide Rank 1 Movement, 2 corners provide Rank 2 Movement, and the 7th corner provides Rank 0 Movement. The Rank 1 and 2 dirrections are fairly self explanatory (traveling along as few sides as possible from a corner in a single dirrection, along septagons of the same color) But the 7th corner provides a unique opportunity.

The 7th corner provides a space immediatly on the corner that is different from the initial shape. Which is the first diagonal move we’ve seen so far that allows the bishop to “change colors”

From there, in the diagram, the bishop continues Rank 0 movement onto another septagon, and runs out of diagonal moves in that strict dirrection. This would then call for a change of direction. To maintain a strict dirrection would require a Rank 3 move (skipping the white septagon). However this would ignore a Rank 1 move that is part of that Rank 3 path. From that Rank 1 move we find ourselves in the same position, that to alternate would require a major shift in the Rank of movment, as well as the bishop treading some ground that might be considered orthoginal. Once the Rank 1 movement is established, I find that dirrection most intuitive to the average bishop.

This brings up an interesting point for the 7 Grid. That for a Rook, dirrection has a major influene on corect path, where I belive preserving the movement rank of a Bishop is more important than maintaining dirrection. I might even be right!

A Bishop on an Hourglass

At first glance I seem to completely contradict my last statement with this diagram of a bishop on an hour glass, since I seem to be only adhering to 6 dirrections ( and rather orthoginal ones at that).

The deception of this image is that there are 10 cardinal dirrections for a bishop from the hourglass (impressive). Directly up and down are Rank 1 Movements. The other 4 directions are actually 8 closely related paths. From 4 corners you get standard Rank 1 Movement along the hourglass shapes. Then, from the Rank 0 movements on either side you get two paths each that follow the Rank 0 to Rank 1 Movement pattern we discuss for diagonal movement on a septagon. Important to note that because they are separate paths that they would seem to be able to slip past other pieces, but not be able to “go around”

Besides the ability to “change color” the primary characteristics of orthogonal and Diagonal are preserved and separate.

This is a Nonogon Tessellation, Starburst Grid, or 9 Grid

The 9 grid is probably my favorite. Just like the 7 grid it has 2 shapes and nonogons(9 sided shapes) of two different orientations. Tri-color coded and fantastic in its absurdity.

A Rook on a 9 Grid

The 9grid actually has an intuitive advantage over the 7grid, in that the star shapes present no side to side position onto the nonogons. Therefore, there is no orthogonal move onto or off of a star shape. The moves allowed to a Rook are then alternating moves between two dirrections. This creates 6 cardinal paths.

A Bishop on a 9Grid

The starbursts allow the bishop to only make Rank 0 movements. It must however alternate dirrection like the rook, but only between two movement options. This creates 6 cardinal paths, and not a single space of them overlap with a rook from the sae space. Ultimately this evens diagonal and orthoginal movement while maintaining two very different types of moves.

Bishop on a Starburst

The bishop does still have an advantage over a Rook on a 9 grid, however. Being able to move onto a star shape allows a couple options. Rank 1 movment to other star shapes is an intuitive movement. Rank 0 movement onto the surrounding nonogons is a little more tricky. It could conceivbly take an alternating trajectory, but you’ll notice that the combined dirrection of the alternating trajectory is almost indisinguishable from the Rank 1 movement from star to star. However, the 7 ad 9 gride have set precident for a single shift from Rank 0 to Ran 1 movement, so it feels reasonable that it would move onto adjacent nonogons but then would shift to a nonalernating path. Creating a fairly unique opportunity for the bishop to have advantage over orthoginal pieces.

A Decagon Tessellation, 10-8 Grid, or simply 10 Grid

The 10 Grid (or 10-6 Grid) Is composed of decagons and 6 sided hourglass shapes. It’s not quite as wacky as the odd-numbered 7 or 9 grids, However, it does have some interesting features beyond the Octo Grid, though you may notice some key similarities.

A Rook on a 10 Grid

Orthogonal movement is fairly straight forward. The only path that the rook cannot travel, is a perfectly lateral direction across an hourglass “crux” corner. All other 6 cardinal directions travel in perfectly straight lines without encountering a corner.

From the hourglass, the rook only has two legal moves, up and down.

A bishop on a 10 grid

Conversely,  the bishop can only move laterally from a decagon.

Bishop on the Hour Glass Space

However, from a six sided hourglass, a bishop’s movement mirrors the 6 cardinal directions of the rook.

The 10Grid very cleanly follows the foundations of Diagonal and Orthogonal Movement Rules

-From the same space, there is no overlap of diagonal/orthogonal movement.

-One space of Diagonal Movement would require two Orthogonal  moves.

but it has the additional feature that . . .

-One space of Orthogonal Movement would require two Diagonal moves.

The 10Grid has the feature that a bishop can reach any space on the board as easily as the rook. Both require one move to switch “colors” so to speak. Giving both pieces equal mobility with completely different movement paths. Out of all the grids I’ve covered so far, I feel that this is the one with the most potential for a game. Off the cuff, a chess like where one player has all bishops and the other player has all rooks would be instantly engaging.

In Review

Not all grids are intuitive maps for Orthogonal or Diagonal Movement. Our current definitions of Orthogonal and Diagonal need to be discussed and expanded to apply to further possibilities. However, further exploration yields many possibilities for games to surpass the same basic boards and create new systems of spatial interaction. By exploring new possibilities, we have the potential to circumvent some of the same old problem we encounter in game design.


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