Hexagon Tiles: Like them, love them, or RAGGGEEEE!!!

I like the idea of hexes and getting rid of stacks of doom. Every hex-based turn-based game I've played has had superior combat compared to Civ (not saying much since I don't think Civ has ever done combat all that well). I just hope they nail it because it IS a big change and I wouldn't want to see the overall character of Civ change so drastically that the total game experience is messed up.

If you wanna check out hex-based combat and how it compares, try a free and extremely well done game called "battle of wesnoth." IMO wesnoth illustrates how well hex-based can be done - it has very good combat and AI. You need to be on top of your game even playing on easy levels or wesnoth's AI will flog you mercilessly. The tactical and strategic elements are so vastly superior (IMO) it's ridiculous.
 
Hi, I am the president of the PYXIS innovation and the inventor of the indexing used on the ISEA hexagonal tessellation that you are discussing on this thread. I thought I might be able to help you out.

We call the entire mechanism a digital Earth reference model (DERM) as it encompasses the principals of a digital as opposed to an analogue way of encoding information - think of the way a sound signal (music) is encoded digitally as opposed to analog and you will mostly get the difference between the DERM and conventional coordinate systems used to reference the Earth like lat and long. Of course gamers have been well aware of the difference between such things for many years and much of the original ideas for PYXIS goes back to my own gaming days in the 70's.

So our first job was to select an optimized discretization strategy - tiling - of the Earth surface. Hexagons are generally best under most criteria however they do not aggregate and decompose congruently - like squares and triangles can - into self similiar parts. How does one create a hierarchical multi-resolution hexagonal mesh? Further studies have suggested that the ideal criteria for a Earth tilling would have each cell the same area. This is also a challenge and you have discovered that its impossible to cover the Earth with the same tile and preserve both shape and area over 20 cells - this is why there are so many patents for golf balls.

We adopted a solution called the Snyder Equal Area Icosahedral Aperture 3 Hexagonal grid (ISEA3H) first proposed by some researchers at Oregon State University who were looking at sampling for biodiversity analysis. This is a base 3 system where the grid changes by a square root three distance between cells each resolution. The area is preserved in each cell - not a meaningful trait for gaming but essential in an discrete Earth reference where statistically valid sampling of information should be preserved at any point on the planet. As noticed the exception is the 12 pentagons that will form at the vertices of the icosahedron - these are strategically placed in large water bodies for the most part - at any rate they are really really tiny for most any application - though we do have to heed them heuristically.

The indexing (addressing), mathematics (from neighbour operations to advanced algebra) and quantization strategy (sampling) are all developed and available for the grid in our library. The core value being that real time data can be synchronized to the grid and therefor multi-source analysis of Earth observations is enabled without need for a manual preprocessing typical of digital cartography and GIS. I good application for this is Geospatial-Intelligence where spatial data is brought together on the fly.

I think that if one thinks of gaming as a simulation and one uses real data to feed into that simulation, there may be some value for a game. Its rather high fidelity for a game I would expect.

If you want to look at the DERM in more detail take the following steps:

1. Down load the prototype browser on our web site - worldview - https://www.pyxisinnovation.com/downloads.php

2. Click off the 2 default data sets

3. Activate the navigation pane by picking on it - then select the numbers 0, 2 and 4 on your key pad - this will show 4 resolutions of the ISEA3H grid - it is skipping intermediate resolutions. The entire grid is pretty dense.

4. At the bottom left there is an indicator of lat long, change this to PYXIS indexing. This will show you the index of each cell at the display resolution. Note that the display resolution is tiny - smaller than the pixels on your monitor - so you won't see them. There are 265 cells in the smallest grey cells you see on the screen.

5. Use your scroll button or +- buttons to zoom in. At resolution 40 the cells are 2 mm apart.

6. You can populate the cells with almost any conventional georeferenced imagery, vectors or terrain. Just drop the file from your windows exporer into the library on the left of the screen. Its a P2P system so you can also publish your data to others.

If there is anyone interested in further information I suggest contacting us at our office 1-877-389-6619 or see website and I can have one of our engineers provide you with more.

Hope that helps out and appreciate your interest.

Regards, Perry

The PYXIS hexagonal grid covers the entire Earth with hexagons and 12 pentagons - starting at Res 0 with 32 cells - down infinitesimally - for example at Res 29 there are 500,000,000 hexagonal cells + 12 pentagons a meter apart. The cells are all equal area except the pentagons are 5/6 the area of a hex. There are individual cell addresses and movement rules. Imagery, terrain and vectors can populate the cells quickly. There is a tile of cells that facilitates caching and transmitting data. The cell addresses are hierarchical so that you can aggregate and decompose information from one resolution to another in a database fashion.

Rendering each pixel as a hexagon cell is probably overkill for a game or at least several generation beyond what is out there now for a game. However, it would work.

Here is mid res sample with the lower res hex grid showing as an example. Note again that the actual pixels of the imagery are hexagonal providing the high fidelity environment of multi-sources of Earth observations to make up the scene.




Perry

I'm waiting for someone more knowledgeable than me about this subject to comment on these posts. Clearly the guy knows what he's talking about, but unfortunately I don't (I'll stick to discussing history and gameplay -- things I know about). People had no problem discussing the possibilities of a spherical hex globe earlier in this thread, but now that someone with experience on the issue has commented, those people seem to have disappeared. So would someone with some knowledge about the subject matter please acknowledge his intelligent posts and comment on them?
 
I'm waiting for someone more knowledgeable than me about this subject to comment on these posts. Clearly the guy knows what he's talking about, but unfortunately I don't (I'll stick to discussing history and gameplay -- things I know about). People had no problem discussing the possibilities of a spherical hex globe earlier in this thread, but now that someone with experience on the issue has commented, those people seem to have disappeared. So would someone with some knowledge about the subject matter please acknowledge his intelligent posts and comment on them?

Basically he's saying yes, if you have 12 pentagons in the globe, by adding more and more Hexagons it can be made to be more and more spherical.
 
Basically he's saying yes, if you have 12 pentagons in the globe, by adding more and more Hexagons it can be made to be more and more spherical.

Actually its spherical at any resolution. Its not like a Buckminster Fuller geodesic dome. The grid is projected to the sphere at any resolution, that is why you can see the grid of 3 resolutions in the picture - they all exist on the sphere.

Technically, the cells exist on the spherical plane cut by the geoid (http://en.wikipedia.org/wiki/Geoid) and terrain model - ie it IS the shape of the Earth.

As for the Pentagons - they are tiny and placed in the ocean - but in a game, as it is our own application, movement through a pentagon would require special rules if encountered. The Pentagons are unavoidable due to the realities of spherical geometry - you will always encounter some compromise of area/distance and shape.

Perry
 
Another problem for Civ purposes is that is impossible to represent a whole sphere on a flat plane without some kind of distortion. Whether you're using squares or hexes, if you're trying to represent the whole globe you either have to stretch some areas or make the map into weird shapes. Civ uses a Mercator projection, where tiles near the poles actually represent a smaller area of land than tiles near the equator. To simplify things, the game engine ignores that and just pretends all the tiles are the same. This Wikipedia article has some basic info on map projections: http://en.wikipedia.org/wiki/Map_projection
 
Correct strumhauke, it would not be rectilinear if you want to keep the hexes. It would something like this if you wanted to keep the hexes (sorry for poor resolution)...

 
Hi, I am the president of the PYXIS innovation and the inventor of the indexing used on the ISEA hexagonal tessellation that you are discussing on this thread. I thought I might be able to help you out.

We call the entire mechanism a digital Earth reference model (DERM) as it encompasses the principals of a digital as opposed to an analogue way of encoding information - think of the way a sound signal (music) is encoded digitally as opposed to analog and you will mostly get the difference between the DERM and conventional coordinate systems used to reference the Earth like lat and long. Of course gamers have been well aware of the difference between such things for many years and much of the original ideas for PYXIS goes back to my own gaming days in the 70's.

So our first job was to select an optimized discretization strategy - tiling - of the Earth surface. Hexagons are generally best under most criteria however they do not aggregate and decompose congruently - like squares and triangles can - into self similiar parts. How does one create a hierarchical multi-resolution hexagonal mesh? Further studies have suggested that the ideal criteria for a Earth tilling would have each cell the same area. This is also a challenge and you have discovered that its impossible to cover the Earth with the same tile and preserve both shape and area over 20 cells - this is why there are so many patents for golf balls.

We adopted a solution called the Snyder Equal Area Icosahedral Aperture 3 Hexagonal grid (ISEA3H) first proposed by some researchers at Oregon State University who were looking at sampling for biodiversity analysis. This is a base 3 system where the grid changes by a square root three distance between cells each resolution. The area is preserved in each cell - not a meaningful trait for gaming but essential in an discrete Earth reference where statistically valid sampling of information should be preserved at any point on the planet. As noticed the exception is the 12 pentagons that will form at the vertices of the icosahedron - these are strategically placed in large water bodies for the most part - at any rate they are really really tiny for most any application - though we do have to heed them heuristically.

The indexing (addressing), mathematics (from neighbour operations to advanced algebra) and quantization strategy (sampling) are all developed and available for the grid in our library. The core value being that real time data can be synchronized to the grid and therefor multi-source analysis of Earth observations is enabled without need for a manual preprocessing typical of digital cartography and GIS. I good application for this is Geospatial-Intelligence where spatial data is brought together on the fly.

I think that if one thinks of gaming as a simulation and one uses real data to feed into that simulation, there may be some value for a game. Its rather high fidelity for a game I would expect.

If you want to look at the DERM in more detail take the following steps:

1. Down load the prototype browser on our web site - worldview - https://www.pyxisinnovation.com/downloads.php

2. Click off the 2 default data sets

3. Activate the navigation pane by picking on it - then select the numbers 0, 2 and 4 on your key pad - this will show 4 resolutions of the ISEA3H grid - it is skipping intermediate resolutions. The entire grid is pretty dense.

4. At the bottom left there is an indicator of lat long, change this to PYXIS indexing. This will show you the index of each cell at the display resolution. Note that the display resolution is tiny - smaller than the pixels on your monitor - so you won't see them. There are 265 cells in the smallest grey cells you see on the screen.

5. Use your scroll button or +- buttons to zoom in. At resolution 40 the cells are 2 mm apart.

6. You can populate the cells with almost any conventional georeferenced imagery, vectors or terrain. Just drop the file from your windows exporer into the library on the left of the screen. Its a P2P system so you can also publish your data to others.

If there is anyone interested in further information I suggest contacting us at our office 1-877-389-6619 or see website and I can have one of our engineers provide you with more.

Hope that helps out and appreciate your interest.

Regards, Perry

Two questions. As you increase the number of hexes while reducing the size of the pentagons, does that not also:
- make the world flatter away from the pentagons yet still angular close to them, or if you want to prevent that, force you to use clearly less regular hexagons?
- thereby cause more deformation if you map Earth onto your surface?

The method you describe is feasible and similar things have been done, but one way or another you will encounter the same mathematical limitations.
 
Basically he's saying yes, if you have 12 pentagons in the globe, by adding more and more Hexagons it can be made to be more and more spherical.

That is, I fear, not the case. (The subgrids you see in the images are, I guess, exactly that - subgrids, not the next iteration of the generator. They are not more spherical than the main grid, and cannot be continued on the pentagons.)
 
Two questions. As you increase the number of hexes while reducing the size of the pentagons, does that not also:
- make the world flatter away from the pentagons yet still angular close to them, or if you want to prevent that, force you to use clearly less regular hexagons?
- thereby cause more deformation if you map Earth onto your surface?

hmmmmm...I hope I understand this question. The Grid you are seeing is just one very coarse resolution of the grid. The grid that is populated by the data, so take an image of the world, a tiff, it is sampled into the grid, each pixel. A group of such hexagonal pixel form a bitmap. Similarly, a terrain model is used to populate the wireframe. The more detailed the terrain model the more the model more accurately represent the real Earth surface. The large grid you see is a very coarse level of the grid.

There is no correlation about flatter near the Pentagons. The grid is projected to the actual surface of the Earth represented by the terrain. The Pentagons have no effect.

IE its the resolution of the data that provides the accuracy of the surface, not the the grid.


The method you describe is feasible and similar things have been done, but one way or another you will encounter the same mathematical limitations.

There is one significant limitation to spherical partitioning and this may be what you are referring to. I mentioned it above. That is that, beyond 20 cells (the triangles of an icosahedron) one can not partition the Earth with cells of the same area and exact shape. One must compromise. In the method we are deploying, we are preserving area of each hexagonal cell, allowing 12 pentagonal cells, and allowing some minimum distortion in the hexagons. You can see in images below that the hexagons are not regular/perfect hexagons. In a game you might want to preserve distance instead of area.

This grid is called the ISEA3H. You can google it. Trying to find the language that translates....this is a grid like geographic coordinates, you would not expect the lat, long grid to change behaviour as you look at more resolutions. But if you have accurate data, say a engineering survey, it will be more accurately portrayed on the grid. Its the data not the grid that changes.

We have a reproducible multi-resolution hexagonal grid, generated and addressed with a formula. Always in the exact same place on the Earth no matter what the resolution the data is.

thereby cause more deformation if you map Earth onto your surface?

I may have answered this above. Yes the grid cells are distorted uniformly, not the Earth.

Originally Posted by Krikkitone View Post
Basically he's saying yes, if you have 12 pentagons in the globe, by adding more and more Hexagons it can be made to be more and more spherical.

That is, I fear, not the case. (The subgrids you see in the images are, I guess, exactly that - subgrids, not the next iteration of the generator. They are not more spherical than the main grid, and cannot be continued on the pentagons.)

I think I agree and above may explain that the grid is generated on a surface, however the surface is generated from a hexagonal gird, so as one zooms in you will sample more detailed data and the surfae will therefor be more accurate. The grid remains in the same place.

On the Pentagon comment, Sorry I am not sure I understand this point. The hexes or pents are all the same except the shape of course. You might think of any cell as a hexagonal (or pentagon) prism or ray that extends from the centre if the Earth and the surface cuts the prism giving its shape. At a perfect sphere, the cells are all equal area + 12 pentagons of 5/6 the area.

Perhaps this series of screen shots helps...

This one is showing the Icosahedron with the 3 resolutions of the grid generated on the sphere. Notice the pents at the vertices of the Icosahedron.



I will tilt and zoom in a bit...



This next one is populated with NASA's blue marble (not an overlay but sampled into hexagonal pixels) and a 2 minute (points every 2 minute of longitude) terrain model populating the mesh. So essentially the actual shape of the Earth - no longer a sphere. If you look close on the right side you will still see the one of the pointed tip of the icosahedron sticking out of the water...near the bottom of the zoom control. The vetrex on the land you can not see because the surface is higher than the geode/mean sea level.



If I zoom in further the data is sampled at a finer resolution. I just displayed one resolution of the hex grid here as it is pretty dense. This is the only place that the pentagon is located on land as opposed water. Can you see it?



Finally I am adding a high res of the Calgary tower, still same grid but finer resolution...

 
You can't really limit the pentagons to the poles. If you fix the north pole to be one of the vertices the other pentagons are completely determined. They could possibly appear on landmasses as well.
 
Thanks ppeterson for the additional images, they are quite impressive. :)

Yes, you can reply to the issue I raised as "we take the actual Earth surface, it's the grid that is deformed". I a game, however, one would have the hexagons all of equal shape and size, and hence see a deformed Earth map, like if it were mapped onto a standard C540 fullerine rather than the more spherical C60. To make it rounder, one would need extra pentagons, paired to heptagons (which yields stronger fullerines, btw).
 
You can't really limit the pentagons to the poles. If you fix the north pole to be one of the vertices the other pentagons are completely determined. They could possibly appear on landmasses as well.
Well, you can, but that gives a cilindrical world. To have equal axes, the pentagons need to be equidistant, be it by themselves or in pairs. This means that in addition to the poles, you can pick one longitude and thereby attempt to place most of the pentagons in oceans, deserts or mountain ranges.
 
In a game, however, one would have the hexagons all of equal shape and size, and hence see a deformed Earth map, like if it were mapped onto a standard C540 fullerine rather than the more spherical C60. To make it rounder, one would need extra pentagons, paired to heptagons (which yields stronger fullerines, btw).

Yes agree to your point, with a polite however; even in a fullerine area and/or distance is compromised.

Thank you for the pleasure to discuss this topic in your forum. There is a lot to learn from the gaming world that can be brought into real decision support tools. In the future I hope that we all have access to enough information that we can see, analyze and simulate the plans and actions we make and consider how these affect the Earth, its life, and its resources.

May all your wars be on a game board, peace and good fun to you, Perry
 
Thanks, Perry. :)

If I zoom in further the data is sampled at a finer resolution. I just displayed one resolution of the hex grid here as it is pretty dense. This is the only place that the pentagon is located on land as opposed water. Can you see it?
Yes, it is some distance from the lower left corner, breaking the hexagon lines.
 
Yes agree to your point, with a polite however; even in a fullerine area and/or distance is compromised.

Thank you for the pleasure to discuss this topic in your forum. There is a lot to learn from the gaming world that can be brought into real decision support tools. In the future I hope that we all have access to enough information that we can see, analyze and simulate the plans and actions we make and consider how these affect the Earth, its life, and its resources.

May all your wars be on a game board, peace and good fun to you, Perry

:goodjob: your organization is brilliant, this is incredible!
 
Well, you can, but that gives a cilindrical world. To have equal axes, the pentagons need to be equidistant, be it by themselves or in pairs. This means that in addition to the poles, you can pick one longitude and thereby attempt to place most of the pentagons in oceans, deserts or mountain ranges.

I thought that the vertices have to be the vertices of a regular polyhedron, otherwise some of the triangular faces that you are tiling will have a larger area than the others and more hexes than the others. If you place all of them on the same longitude you won't even get a polyhedron.
 
But say I wanna make just New Zealand. How do I do that? If it's all hexes, I can't use normal image files can I? I thought that was kinda the reason they made squares in IV instead of Rhombi?

Of course you can. Why wouldn't you be able to? A grid is a grid is a grid.



See? I just made that. It's easy. Of course, in the game, the tiles will be individual cells and it won't be just a grid applied overtop of an image, but rather, a bunch of cells that are used to approximate the image. But that's true of squares, too.

They used squares in civ4 because squares had always been used in civ. Alot of people wanted hexes in civ4, but they were making big changes (religion, civics, suicide catapults etc) and you have to be careful about making too many changes at once - civ players are a conservative bunch who don't like too much change too fast.

Squares were probably used in civ1 to save a few kb (yes, kb - it was released in 1991, so development was probably in 1989-90).
 
Sadly the long, skinny shape of NZ generally doesn't make for the best gameplay in my experience.
Still, hexes will be able to do it more justice (shapewise) than squares ever could.
 
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