change hexagons to octagons

[PawelS]

I think it's better to use hexagons and the 12 pentagons than triangles.

 

[E66man]

You did say regular polygons, which equilateral triangles indisputably are.

I think we're running into a misunderstanding on what we are discussing here. The bottom line is that the icosahedron is the largest possible Platonic solid. You cannot make a larger polyhedron from regular polygons without mixing your polygon types. This is a mathematical fact.

What I think you are trying to state, is that you can take the face of any polyhedron and subdivide it into triangles to get the surface to have as many tiles as you want. In that case, you can probably make a polyhedron with an arbitrarily large number of faces, however, I don't know if it is provable that every regular polygon can be subdivided into *equilateral* triangles.

If you're willing to drop having your faces consisting of equilateral triangles or regular polygons, then you can have a surface of just about any arbitrary size. The downside is that you cannot guarantee consistency at the vertexes regarding how many triangles are meeting, so you would likely have to limit movement to the sides only which probably won't go over well.

 

[blind biker]

Why to the sides only?

 

[blind biker]

By the way, guys, just so that you understand where I'm coming from: I got my inspiration from the larger (larger than C60) fullerenes:

 

[PawelS]

By the way, guys, just so that you understand where I'm coming from: I got my inspiration from the larger (larger than C60) fullerenes:

This is what I meant too.

 

[E66man]

By the way, guys, just so that you understand where I'm coming from: I got my inspiration from the larger (larger than C60) fullerenes:

The C540 fullerene looks like it uses only hexagons, which is ideal for our purposes. Note however that this construct uses non-Euclidean geometry so the sum of the interior angles isn't necessarily 360 degrees. All of my earlier comments assume we're talking about Euclidean geometry.

I don't know enough about non-Euclidean geometry to make a lot of certain statements... for example, I don't know if it is provable or not that a construct larger than the C540 fullerene is possible that only uses hexagons. And all the Euclidean questions remain... do you allow for mixing polygon types? If so, then to prevent the issue of having some tiles with more neighbors than the others, you have to go with triangles as the base tile. And to answer the 'why only move through the sides' question, if it turns out to be impossible to subdivide the faces with equilateral triangles, then each vertex will have different numbers of triangles meeting. In this case, moving 'diagonally' gives you differing amounts of neighbors from tile to tile, which is the problem you wanted to avoid by doing the triangle subdivision in the first place.

I have several colleagues here with math doctorates so I've asked them if it is proved or unproved that you can subdivide regular polygons into equilateral triangles in non-Euclidean space, or if you can make polyhedra arbitrarily large with a single type of regular polygon in non-Euclidean space. Even if the answer to these questions is 'no', it seems like there are enough fullerenes using hexagons that you could get some decent map sizes.

http://antiprism.com has some algorithms and example pictures for generating these kinds of polyhedra, although you may notice that some of them don't provide an equal area per face.

 

[blind biker]

The C540 fullerene looks like it uses only hexagons, which is ideal for our purposes. Note however that this construct uses non-Euclidean geometry so the sum of the interior angles isn't necessarily 360 degrees. All of my earlier comments assume we're talking about Euclidean geometry.

I don't know enough about non-Euclidean geometry to make a lot of certain statements... for example, I don't know if it is provable or not that a construct larger than the C540 fullerene is possible that only uses hexagons. And all the Euclidean questions remain... do you allow for mixing polygon types? If so, then to prevent the issue of having some tiles with more neighbors than the others, you have to go with triangles as the base tile. And to answer the 'why only move through the sides' question, if it turns out to be impossible to subdivide the faces with equilateral triangles, then each vertex will have different numbers of triangles meeting. In this case, moving 'diagonally' gives you differing amounts of neighbors from tile to tile, which is the problem you wanted to avoid by doing the triangle subdivision in the first place.

I have several colleagues here with math doctorates so I've asked them if it is proved or unproved that you can subdivide regular polygons into equilateral triangles in non-Euclidean space, or if you can make polyhedra arbitrarily large with a single type of regular polygon in non-Euclidean space. Even if the answer to these questions is 'no', it seems like there are enough fullerenes using hexagons that you could get some decent map sizes.

http://antiprism.com has some algorithms and example pictures for generating these kinds of polyhedra, although you may notice that some of them don't provide an equal area per face.

As far as I know, all of the fullerenes use pentagons (the famous 12 pentagons) in addition to the hexagons.

 

[E66man]

As far as I know, all of the fullerenes use pentagons (the famous 12 pentagons) in addition to the hexagons.

Really? Looking at C540 (the leftmost one in your picture) it doesn't look like there are any, but it can be hard to tell.

 

[TexasTiger]

This thread makes my head spin. There is a reason I do liberal arts and not math.

 

[PawelS]

If such kind of maps are used in Civ6, I think it should be required to place a city in one of these 12 tiles to build the Pentagon wonder

 

[E66man]

This thread makes my head spin. There is a reason I do liberal arts and not math.

There's a formula for evenly distributing points on a sphere, so I'm fairly convinced you can make a map of any large size with hexagons since it corresponds to a non-planar graph where every node has 6 connections, but I didn't want to post it to give readers a break

 

[No_such_reality]

Why not just decide what surface area we are playing and have the system use simulated movement and distances with factoring for terrain.

Movement could simply be labeled as 'units' of distance. Worker movement 2 units. Warriors - 1 unit. horsemen 4 units etc. Improvement, cities, etc, all cover various areas.

Much like you have the uneven borders for culture now, clicking any unit could display a boundary of the close curve representative of the selected Unit's (ie.Worker) movement units factored by distance and terrain type.

Tessellation becomes a non-issue as the surface type (2-D, 3-D) dictates the formulas necessary to construct the surface.

 

[E66man]

Tessellation becomes a non-issue as the surface type (2-D, 3-D) dictates the formulas necessary to construct the surface.

That would work with movement but I think you still need to have a tessellated surface or else you need to come up with a new system for assigning citizens to work land, and for deciding how much of the surface corresponds to a given terrain type or resource bonus.

 

[blind biker]

Really? Looking at C540 (the leftmost one in your picture) it doesn't look like there are any, but it can be hard to tell.

Oh yes. The fullerenes in the picture are actually all the same - C960. The figure on the right is just a curved version to make it spherical, which is what we're talking about. Check out the originating link (slightly) for more info.

 

[blind biker]

Look guys, the pentagons are necessary to "ball up" the hexagonal map, but it's not such a big deal, actually. Look at these capped carbon nanotubes:

The pentagons are only at the caps, while the whole waist is purely composed of hexagons. And even at the caps, there are only 6 pentagons at each side. If we imagine the caps to be the polar caps of our map, we can merrily declare the pentagons as impassable terrain, and all is honkydory.

 

[boehj]

What is really really sorely missing is jump/duck. I don't care if you give me triangles, squares, hexes, whatever... but I wanna be able to jump when someone shoots a cannonball at me. You could also have UU's that could curl up into a ball and just roll at cities and take 'em down. Have four highly promoted rollers, rolling all around Pangaea crushing every civ they meet and every cannonball just shooting right over them. Almost invincible I'd say.

The more I think about it, the more I think Civ 6 should be a side-scrolling platform game. Firaxis would get my $50 *tomorrow* if they gave me a commitment that jump/duck would be in the next episode... or even next patch?

 

[spoooq]

There's a formula for evenly distributing points on a sphere, so I'm fairly convinced you can make a map of any large size with hexagons since it corresponds to a non-planar graph where every node has 6 connections, but I didn't want to post it to give readers a break

For this, I have found a truly wonderful proof, but the post is too small to contain it.

 

[alpaca]

Why does everyone insist on regular polygons? Just deform them a bit and it should be possible to have them cover a sphere without needing any second sort. In the game, the tiles are usually deformed anyways because they follow rivers, hills and such, and gameplay-wise there is zero difference between a regular hexagon and a quite weird shape that's almost a triangle, for example.

 

[_Axel]

Well the 12 pentagons would'nt really be a big problem if they were on the poles, more of a challenge I think. People don't usually settle on there anyway, as they are impassable in CiV.

 

[blind biker]

For this, I have found a truly wonderful proof, but the post is too small to contain it.

This would only be cool if you actually died without leaving any other trace of your proof.