Hexes -> Hex-Tiled Maps

[PPQ_Purple]

It's not like a sphere.


It's analogous pseudo-toroid maps, except that it loops in three directions instead of 2.

Ok, at the point when you have to use a complex multi word phrase to describe something than it is definitively not something that most people want.

Lets look at the CIV4 options:

Flat
Cylinder
Doughnut

Notice the paten?


Now try saying: "analogous pseudo-toroid" a couple of dozen times fast.

 

[azzaman333]

Except it's named torodial, not doughnut, and as a result your pattern is invalid.

 

[Chalks]

Wow, and the award for the most confusing post of the month goes to...

 

[Trias]

It's not like a sphere.

It's analogous pseudo-toroid maps, except that it loops in three directions instead of 2. And except that it has to be a regular hexagon shape, where as a pseudo-toroid map can be rectangular (and has to be to resemble a torus). On such a map, an attack can come from any side.

Besides, there would not be a pole unless you made the entire top and bottom sixths, together one third of the map, into a large polar region.

Actually the wrapping you are proposing just produces a torus. A skewered one, but still a torus. The easiest way to see this is to look at the tiling you have drawn and notice that the map is basically the area between four of the read hexes, with the opposite sides of the resulting parallelogram identified.

 

[Souron]

Ok, at the point when you have to use a complex multi word phrase to describe something than it is definitively not something that most people want.

Lets look at the CIV4 options:

Flat
Cylinder
Doughnut

Notice the paten?


Now try saying: "analogous pseudo-toroid" a couple of dozen times fast.

I'm trying to emphasize that so called doughnut maps do not actually model a doughnut well. Real donuts have a smaller distance around the donut hole than the outside. The so called doughnut maps do not. Hex tiled maps do not have even a poor 3D analogue.

Anyway I seem to have swallowed the word "to" between those words.

 

[Souron]

Actually the wrapping you are proposing just produces a torus. A skewered one, but still a torus. The easiest way to see this is to look at the tiling you have drawn and notice that the map is basically the area between four of the read hexes, with the opposite sides of the resulting parallelogram identified.

Looks like you're right! Curious.

It's like tiling a map that has an odd number of rows, So that East and west don't quite line up.

I'll think on this.

 

[Yakk]

You cannot make a hexagonally locally connected map tile on a sphere.

It is a sad result from geometry. You need those 12 petnagons (for example) to make the map "sphere-like".

I think you can derive this from the Euler's characteristic of the map. Euler's characteristic is the number of faces+vertices, minus the number of edges, after you triangulate the cells.

When you triangulate a hexagon into 6 triangles (with a common center), you end up with 6 faces, 12 edges and 7 vertexes.

6 of the edges are shared with an adjacent hexagon (so count for "half"), and 6 of the vertexes are shared with 2 other adjacent hegagons (so count for 1/3).

6 faces -( 6 + 6/2 edges) + (1 + 6/3 vertexes) = 0

Ie, each sub-triangulated hexagon addes 0 to the Euler characteristic of the map, assuming the map is "locally hexagonally tiled". And a map with a Euler characteristic of 0 that is orientable is a donut (torus).

Toss in a pentagon:
5 internal faces - (5 - 5/2 edges) + (1+5/3 vertexes) = 1/6.

Add 12 pentagons, and your Euler Characteristic hits 2, which happens to be the Euler characteristic of the triangulated sphere.

...

The short story: look at a foot/soccer ball. See the 12 pentagons? Not there for no reason.

 

[Souron]

You cannot make a hexagonally locally connected map tile on a sphere.

It is a sad result from geometry. You need those 12 petnagons (for example) to make the map "sphere-like".

I think you can derive this from the Euler's characteristic of the map. Euler's characteristic is the number of faces+vertices, minus the number of edges, after you triangulate the cells.

When you triangulate a hexagon into 6 triangles (with a common center), you end up with 6 faces, 12 edges and 7 vertexes.

6 of the edges are shared with an adjacent hexagon (so count for "half"), and 6 of the vertexes are shared with 2 other adjacent hegagons (so count for 1/3).

6 faces -( 6 + 6/2 edges) + (1 + 6/3 vertexes) = 0

Ie, each sub-triangulated hexagon addes 0 to the Euler characteristic of the map, assuming the map is "locally hexagonally tiled". And a map with a Euler characteristic of 0 that is orientable is a donut (torus).

Toss in a pentagon:
5 internal faces - (5 - 5/2 edges) + (1+5/3 vertexes) = 1/6.

Add 12 pentagons, and your Euler Characteristic hits 2, which happens to be the Euler characteristic of the triangulated sphere.

...

The short story: look at a foot/soccer ball. See the 12 pentagons? Not there for no reason.

All true, but it has nothing to do with this thread.

 

[Yakk]

Then nothing described in your original post is very unique to hexes.

You end up with a pseudo-toroid, just like square tiles.

I guess the distance back to the starting location is roughly equivalent regardless of what direction you go in? But that was true on square toroid maps.

 

[Souron]

Actually the wrapping you are proposing just produces a torus. A skewered one, but still a torus. The easiest way to see this is to look at the tiling you have drawn and notice that the map is basically the area between four of the read hexes, with the opposite sides of the resulting parallelogram identified.

Thinking about this made me realize a few things:
1) You're right, this kind of tiling is still a torus.
2) Civ 2 and 3 would be better without east-west movement. This would effectively make the map hexagonal. A shame the developers did not realize this.
3) This kind of tiling is achievable with a square map, though the advantage of traveling the same distance along the diagonals as horizontally to return to the same point is lost.

 

[Ahriman]

A flat hex map with opposite sides mapping?

Sounds pretty confusing. I don't see any particular advantage.

 

[azzaman333]

A flat hex map with opposite sides mapping?

Sounds pretty confusing. I don't see any particular advantage.

Every direction is a border between you and someone else, making it (hypothetically) particularly useful for multiplayer games.

 

[Decius]

I'd like to hear more about the radioactive monkeys.

 

[Yakk]

Me, I'm hoping for Klein bottle maps.

North? We know not of this "North" you speak of.

 

[Souron]

A Klien bottle would be indistinguishable from a torus unless you had a way to dig through the surface to the other side. Which would be kinda cool. I'd need an interesting back story.

 

[Trias]

A Klien bottle would be indistinguishable from a torus unless you had a way to dig through the surface to the other side. Which would be kinda cool. I'd need an interesting back story.

I think he's suggesting a klein bottle with both side identified. That is scrolling around the world one from east to west would switch North and South, etc. This works for a Meobius strip, but for a Klein bottle it has some extreem weirdness around the two vertices.

 

[Ribannah]

It's not like a sphere.

It's analogous to pseudo-toroid maps, except that it loops in three directions instead of 2. And except that it has to be a regular hexagon shape, where as a pseudo-toroid map can be rectangular (and has to be to resemble a torus). On such a map, an attack can come from any side.

Besides, there would not be a pole unless you made the entire top and bottom sixths, together one third of the map, into a large polar region.

This is something that I proposed way back when Civ3 was in the making.

It is actually possible to view the hexagon-tiles as a sphere that looks like a three-dimensional globe from every direction, except that it isn't (which would be clear from the backside if you were only able to see it), and you can't map Earth onto it.

Poles can be placed anywhere you want, IIRC one would go in the middle of one superhexagon and the other in a corner.

 

[Yakk]

So, you can treat hexes as a simple grid, where every 2nd entry is "offset south half a square". So long as your width is a multiple of 2, things remain pretty sane.

In order to make a mobius strip, you just flip the map as you cross the meridian. Which way is north depends on where you are looking from, and how many times you scrolled from west to east (or east to west) over the meridian.

In order to make the mobius strip into a Klein bottle, all you do is add in "when you go out the top, you end at the bottom at the same location", and vice-versa. Nothing really strange.

The projective plane is wierder. In it, you start with the basic mobius strip, but when you cross the north-south meridian, you also flip the map east-west.

All of them would look locally utterly normal. You could even display a "mini map" in the corner, centred on wherever you are, that looks normal. It is just that as you scroll around the map, you could "get back home" and have east-west or north-south reversed (or both).

You could probably do the above with a grand hexagon instead of a square. How you identify sides isn't that important.

 

[Souron]

Edit: I don't get it. It seems to me that if you make a dot above the center of the mobious strip, and go around the loop, the dot will still be on the left. If we call the direction we go around the loop east, then the dot must be in the direction of north. But Wikipedia seems to agree with you.

Edit2:Oh, you're suggesting that both sides of the strip be represented by the same tile. That would be odd because not only would north and south flip, but so would up and down. If you ignore that aspect ... It might be interesting.

 

[Yakk]

Yep, up and down flip.

So pick a 3x3 window of this:

A D X C F Z A D X
B E Y B E Y B E Y
C F Z A D X C F Z

what tile is north of what tile becomes a local property, not a global one. That would be a mobius civ map.

A Klien would be a mobius, but you can go from the top to the bottom.

A projective plane would be ...

A D X C F Z A D X
B E Y B E Y B E Y
C F Z A D X C F Z
X D A Z F C X D A
Y E B Y E B Y E B
Z F C X D A Z F C
A D X C F Z A D X
B E Y B E Y B E Y
C F Z A D X C F Z

pick any 3x3 region of the above. You'll note that the corners of a projective plane look wrong, because you can go from a tile to itself in 1 square. In short, projective planes don't play happy with an infinite flat uniform tiling of hexagons or squares (and cannot be made to play happy, because their Euler characteristic isn't 0).