Global Hexagonal Grids are possible
I've looked at some more websites and emailed some people (only one of whom answered) and it seems that it is indeed possible to create hexagonal grids on a rotatable globe.
Uptopic, I discarded the possibility of hex grids because I mistakenly thought the resulting pentagons within the grid would render the grid useless in regard to wargaming.
Although the formation of pentagons apparently cannot be avoided, I've since discovered that the number of pentagons within a global hexagonal grid will always be limited to 12, regardless of how many hexagons make up the grid.
That's an important point, because it means that we can construct grids containing hundreds of thousands, even millions of hexagons, yet there will always be only 12 pentagons within the grid.
This is perfectly illustrated with the large globe graphic below:
I don't know how many hexagons comprise this grid, but it appears to be several thousand. Yet, in that sea of hexagons, there are only twelve pentagons, four of which can be seen as the red-colored polygons (not the circles, the dots), while the rest are similarly located in parts of the globe not visible.
Also shown in the above graphic, are certain other areas of (lesser) distortion.
From the point of view of a gamer, I think the 12 pentagons, as well as the other, lesser distortions, are something we could learn to live with, especially considering that these shortcomings seem minor, compared to the many benefits of being able to game on a rotatable hexagonal global grid.
It is also very important to remember that these hex grids can be infinitely subdivided into ever smaller hexagons, until a desired resolution (scale) is reached (see charts below).
Regarding the above globe, for example, it is my opinion that a further division or two would be necessary before a scale useable for wargaming is obtained. Regardless, the number of pentagons would remain at 12.
Which polyhedron to use?
Until now, I've limited the discussion to hex grids originally made from a polyhedron called an icosahedron, in particular, a regular icosahdron, which has 20 triangular faces of equal area, as well as 12 vertices, which are the points at which the areas meet.
(As an aside, it is at these 12 points that the resulting pentagons will always be located. I highly recommend interested readers take a look at a short set of pictures that illustrate how a global hex grid is easily made from an icosahedron, starting here and ending with slide 9, the picture of the 10,242-hex globe.)
(20 sides, 12 vertices)
I have since learned that global hex grids can apparently also be derived from dodecahedrons, which are kind of the opposite of icosahedrons, but have 12 equal-area pentagonal faces and 20 vertices. Presumably, such grids will also have a constant number of pentagons, but I haven't come across any detailed information on that.
(12 sides, 20 vertices)
Additionally, I had an email correspondence with someone who told me that grids can also be derived from other base solids, such as tetrahedrons, cubes and octahedrons, although I don't know whether those would indeed be hexagon grids or something else.
Regardless, most of the information dealing with hexagonal global grids available on the internet appears to indicate that these grids are mostly derived from icosahedrons, which I described above.
Here is a chart showing some of the characteristics of a global hex grid. I know this chart deals with a grid derived from an icosahedron, because it appears among the series of pictures I mentioned above, that show how a grid is made from an icosahedron.
The first column, "R", refers to the number of times the original icosahedron has been divided. The first division always appears to be labeled "0".
The second column gives the total number of hexagons ("cells") of the grid.
The sixth column gives the distance between hexes, center to center, which obviously also indicates the size of the hex. Thus, at resolution 5 there are a total of 40,962 hexes with each hex being about 120.5 kilometers (75 miles) across.
Here's another chart:
Note that this chart is of a hex grid that appears to be based on a dodecahedron, because the number of grid cells given at resolution 0 is 12, which corresponds to the original 12 faces of a dodecahedron.
Note also the different numbers of total cells (hexagons) given for each resolution as compared to the first chart above. This seems to indicate that various numbers of total hexagons can be derived, depending on what kind of polyhedron the grid is originally based on (icosahedrdon, dodecahedron, etc.).
Regardless of all of that, this is a most useful chart, because it gives a very good indication of the total numbers of hexagons (cells) that might be necessary on a global grid to attain a desired hex scale.
The chart is self-explanatory, considering the footnotes, and it's worthwhile to take a closer look.
The first column, "R", once agains indicates the resolution, or, how many times the original dodecahedron (in this case) has been divided.
The second column shows the area of each hexagon at a given resolution, in square kilometers.
The third column gives the total number of hexagons (given as "cells", presumably because it includes those pesky 12 pentagons).
The fourth column, according to footnote #2, gives the "diameter of a spherical cap of the same area as a hexagon of the specified resolution". In other words, it closely approximates the cross-dimension of a hex at a given resolution, which is what we need to know to determine scale.
So, looking at the chart, it seems there is a "sweet spot" between resolutions 8-12, inclusive, that might be useful for the purpose of wargaming.
At resolution 8, the fourth column shows a value of 99 kilometers (62 miles), which roughly corresponds to the cross-dimensional size of the hexagon. I think this resolution might be the smallest one on this chart that would be useful for making a global wargame, perhaps at the corps/army level.
Level 9 might be even better, with hex sizes at about 57 kilometers (35 miles), probably still good for corps/army-level games, which is what a global game should probably be. The number of hexes at this resolution is given at over 196,000. This is a large number, to be sure, but not much larger than the 150,000+ hexes in the Panzer Campaign Moscow '41 game - so certainly still doable, especially considering that Moscow '41 was designed for a 10-year-old game engine.
At level 10, the hex scale is about 33 kilometers (20 miles), with over 590,000 hexes.
Level 11 has 19 km (12 mile) hexes, numbering over 1.7 million hexes.
Level 12 has 11 km (6.8 mile) hexes, numbering over 5.3 million hexes.
I think anything larger than that would be unplayable at a global scale.
Those then, are the possibilities. It should be kept in mind, once again, that the second chart appears to reflect a grid derived from a dodecahedron, as opposed to a grid made from an icosahedron, which would have different numbers of hexagons and hex sizes for a given resolution, as shown in the first chart.
More graphical examples
An example of a global climate model geodesic grid with a color-coded plot of the observed sea-surface temperature distribution. The continents are depicted in white. This grid has 10,242 cells, each of which is roughly 240 km across. Twelve of the cells are pentagons; the rest are hexagons. (source)
Four resolutions with approximate hexagon areas of:
(a) 210,000 km2
(b) 70,000 km2
(c) 23,000 km2
(d) 7,800 km2
(Note that, regarding globe (d) in the above picture, the 7,800 square-kilometer hexes would be about 88 kilometers (55 miles) across and thus at a scale that could be used for a corps/army level game. So this is what it approximately could look like, although there would obviously be a zoom function.)
What's possible and summary
Other than a standard globe, I can imagine a game that starts by letting the player select a number of parameters, much like in many existing flat-map games.
Firstly, perhaps several scales might be possible, in order to create shorter games (smaller scales) or games of longer duration (larger scales). Edit: On second thought, that might be too much too ask for.
Percentage of land/water, types of terrain, various weather effects which would create cold polar regions and hotter equatorial regions and everything inbetween and all with their applicable terrain (frozen oceans at the poles, etc.).
In short, everthing that has been done for various games with flat maps in the past, but applied to a rotatable global hex grid.
Basically, players would have the ability to create planets. The effect of seasons might also be fairly accurately modeled. Daylight/nightime might be too large a timescale to be playable at the global level, imo.
I can imagine games in which nuclear submarines and bombers would traverse arctic oceans, underneath pack ice in the case of subs, to launch their attacks.
It certainly seems possbile, from everything I've been able to learn, that hexagonal global grids can be made for use in wargaming, with a minimal amount of distortions.
Hexagonal global grids are only starting to be used in scientific applications, according to what I've read online, although the concept has been around since the 1970's.
No one, to the best of my knowledge, has done this in regard to computer gaming, wargames or otherwise. There are games that use global area maps, and according to another poster uptopic, a naval game that seems to use a hex grid in a limited way, but nothing close to what I described above.
So, any designers care to chime in here? What are your thoughts on this?
Spherical Geodesic Grids:A New Approach to Modeling the Climate
(especially the pages on how hex grids are made, here)
Discrete Global Grid Research at Terra Cognita
English/Metric Conversion Calculator
< Message edited by BoredStiff -- 10/5/2008 2:38:15 AM >
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