# What should a spherical grid on flat board look like?

In a game I have an idea for, players control satellites in orbit around Earth. In order to make orbits easier to abstract to the tabletop, I would like these satellites to move in a semi-random way around the board: Each satellite moves a set number of spaces, but in a random direction each turn. (Or perhaps if you spend some fuel, you can choose the direction)

I think this is good, however, I want a board that looks like a sphere, because a plain square, hex, or triangle board would spoil the visual appearance of being above a sphere.

Ideally, each place in this pattern would have 6 adjacent places (for a standard die), and there should not be any "edges" - you should be moved back to the centre if you keep moving in one direction. (This last feature is unlikely to be completely possible, I'll probably have to add a rule to avoid actually going off the edge.) My first thought was that this would be a spirograph of some sort, similar to this:

However, while this has the right appearance, it isn't suitable for a game board because its squares disappear to nothing near the edges and the centre. It also uses 4-sided spaces, going against my desire to use a d6 for determining directions.

Unfortunately, I haven't had any luck designing or finding a better solution. Could anyone help me to find such a board/pattern?

• If you want 6 neighbour places, then you'll need hexagonal cells. If you want the edges to wrap you can do that on a flat board, but it will represent the surface of a toroidal shape. Jan 5 at 19:14
• I'd appreciate it if the downvoters commented why they downvoted? I'd love to be able to improve. As far as I can tell, my question follows all the rules for this site - the only thing I can think of is maybe it's a better fit for puzzling SE? Jan 7 at 20:34
• I didn't downvote, but did you read What types of questions should I avoid asking? Jan 7 at 20:39
• This is the closest thing I know of that uses hexagons (d6 movement) but also looks like a sphere: web.archive.org/web/20220401131637/https://pub.ist.ac.at/~edels/… — note that it was an April's Fools Joke because their design would not really cover an entire sphere. But you don't need an entire sphere, only the front, and I think the design in that diagram might fit your needs. Jan 11 at 17:53
• Why would things in orbit move in a random direction in the game? Things in orbit follow a totally set trajectory (as long as you take into account minor gravitational influence from other celestial bodies) until there is a collision or they apply thrust.
– Zags
Jan 17 at 19:40

Your spirograph looks extremely similar to the boards used in spherical go. This is a spherical grid with two poles:

Properties:

• Every intersection has 4 neighbours.
• If you start on an intersection and follow a path from intersection to intersection, always leaving an intersection by the edge opposite to the one you entered, then you'll go through every edge exactly once and through every intersection exactly twice and end up where you started.
• If you ignore distance distortions, the inner pole and the outer pole are completely symmetric.

On https://www.govariants.com/ you can select "badukWithAbstractBoard", "Circular", and then adjust the number of nodes per ring and the number of rings.

Go is played on the intersections, not the squares.

You get 6 directions of movement if you allow moving along the 4 edges and along the 2 radial diagonals:

• For the solution of allowing movement in six directions, the final design should probably feature additional lines for the radial diagonals - this might make the spiralizing lines harder to see. Since the six-direction soft requirement comes from wanting to utilize a six-sided die, the original poster or others willing to utilize such a concept might consider to switch to a four-sided die instead in favor of a more easily readable board. Jan 18 at 12:16

https://en.wikipedia.org/wiki/List_of_map_projections

You could have two discs, one for each hemisphere, and colored marks to help find where you go onto the other one (if you go off one hemisphere on the red mark, you enter the other hemisphere on its red mark). Or you could use one disc, but then the outer edge represents the point antipodal to the center point (traditionally, the North Pole is at the center, and the South Pole is the singularity, and yes, one point gets smeared across an entire circle) and points close to the singularity will be distorted accordingly.

BTW, if you want to tile your board with hexagons, you pretty much have to have twelve of the spaces be pentagons.

You could still use your spirograph with a few modifications.

• Add an outer and inner circle. They represent the danger zones. If we think as the center of earth, the lines can't go anyway to the center fully. Think of the red rings as danger zones. Too close and you will crash into earth. Too far away and it will fly away.
• Add blue circles that to the top of each of your small diamond shapes. They could be the stable orbit and/ or launch distance.

Just a quick paint demonstration

• I like this solution, because it looks spherical AND allows for a simple deorbiting mechanic, which I hadn't worked out how to implement yet. But I probably won't use it because of the 4-sided nature of the spirograph's pattern. Jan 15 at 1:42
• @Infinite_Maelstrom the spirograph is providing 4 directions, the blue circle are adding another 2 for a total of 6 as requested by the question. Jan 15 at 7:02
• Ahh I see. that would work too, I might try it. I almost have a solution thanks to all the help I've recieved here, I'll probably post it as an answer after I test it in a few games. Jan 16 at 5:29

What you have are two problems, first tiling a hexagonal grid onto a sphere, and second, projecting that sphere onto a flat plane in a way that preserves some appearance of a sphere.

The first is not fully possible, as the closest approximation of a hexagonal grid on a sphere needs 12 pentagons at the vertices of an icosohedron (this is detailed in Sahr, K., White, D., & Kimerling, A. J. (2003). Geodesic discrete global grid systems.). If you are set on a hexagonal grid system as the basis you will need to work around this limitation.

Alternatively adjusting the plan for a hex grid to something like a triangular grid would allow for a D6 to be divided into two sets of 1:2:3, and the tiling becomes simpler (although you do need to adjust the tiling slightly to account for your final projection)

Once you've worked out the issue of which tiling you want to use you're left with a much easier task of selecting the projection that best matches your vision.

The above Mollweide projection works relatively well with a triangular grid, but it may also be useful to consider options that produce a projection with squared edges like a Wulff net, or Peirce Quincuncial projection.

You can get something that roughly approximates spherical geometry with a diamond of hexagons (given that you want 6 adjacencies), where movement "wraps" east to west and "reflects" north to south. Specifically, orient the hexagons with one side vertical. Going east or west off of the side of the board puts you on the other side of the same row. Going e.g. south-east off of the side of the board puts you on the other side of the next row down. Going south-east off of the bottom hex puts you on that same hex, but moving north-east.

With this movement setup, not all paths will pass through the center, but they should all loop back on themselves.

There are other variations that could achieve similar results. For example, you could remove a few rows from the top and bottom of the diamond, and then when you go off the grid to the north or south, you go half way around the top row (including east-west wraparound) and switch the north-south movement direction. You could even do that with a rectangle of hexes. The primary point here is the east-west "wrapping" movement and the north-south "reflecting" movement.

• A path across the polar point should not go back over itself facing the opposite direction, if the intent is to model an orbit around a planet.
– Nij
Jan 8 at 5:14
• It's an approximation. There is no perfect mapping here, and this is one that I thought would be easy to deal with in gameplay. There are variations that work here too: you could truncate the diamond at the top and bottom, and work out where you end up when you go off the top. The more hexes you have in those rows, the harder it is to keep track of what ends up where. Or maybe if you would go off the top, you instead switch the direction and move in that direction. I think gameplay testing would be needed to determine the best setup here. Jan 8 at 5:26

Its a complex issue with many attempts to solve it. There is no solution that gets all the wanted features together so its a matter of trade-offs. Red Blob Games is talking about several options that could do the job, in my opinion the one that fit best is the Flat rendering:

a hexagonal grid everywhere, but the underlying behavior is approximately a sphere.

Basically, a hexagonal grid with "teleports" which connected as shown in the triangles mini-map and the sphere.

• You've been around here long enough to know what Stack Exchange thinks of answers which don't make sense without reading an external link :) Jan 8 at 11:38
• @PhilipKendall, you right, Thanks for letting me know I need to edit my answer, is this addition make sense: Basically, its a hexagonal grid with "teleports". Teleports are connected according to the triangle mini-map. Jan 8 at 11:49
• This is just a description of the implementation details of the suggestion in your other answer. Why is it a separate answer? Also, to clarify, are you suggesting that the hex grid on the physical game board have the overall shape of an unfolded icosahedron? Jan 8 at 17:15
• @murgatroid99, no, those are two different solutions. You just comment on the other solution that it is putting a tile grid on an actual sphere instead of putting a grid onto a physical flat surface. This solution is putting a grid onto a physical flat surface. So I dont get why you think those are the same solution. Jan 8 at 21:03
• @murgatroid99, I suggest that the hex grid on a physical game board with teleporters in the right locations have the overall shape of an unfolded icosahedron. Jan 8 at 21:04