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Erzengeldeslichtes

Planetary Division, Geodesic?

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I'm trying to divide a planet (sphere) into sections. These sections then represent where a unit is, and where it can move, kind of like the maps in Civilization. The problem is that the planet is 3 dimensional, and can be viewed from outer space as well as planet side. I intended to use an Icosa geodesic sphere, but spotted a problem. I shall illustrate it below (or try to):
Now, on the right you can see two light purple triangles, and as is said in the legend, only one should be usable, but... which one? The problem, of course, is that it was originally an Icosahedron, so there are only 5 triangles at the icosahedron's original vertices, wheras when it's split up each vertex has 6 triangles around it. If we use the, in a full planet, much more proliferent correct triangles, it works fine, as on the left. But if we use one of the 5 around an original vertex (of which there are 12, so a total of 60 "bad" faces), then we have an extra face possible. I can't find anything better than an Icosahedron on the 'net, so if I didn't have a forum to ask on I'd probably just have the unit able to move to both. But since I can, I'll ask: Does anyone know of any other options? Please?

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Squirm    481
I think the move your worried about looks a bit odd anyway... you've got red triangles which are closer to the unit than some of the purple ones... I would have only those moves which cross an edge (3 moves per triangle), or all moves to triangles touching the corner (making 12 moves for most triangles, and 11 moves for the others).

Look at the first image (the one your happy with).
There is a red triangle near the blue one which can be reached in 2 moves, either by going left twice, or by going right and then jumping all the way across, or by first jumping straight past it and then coming back, but you can't get there in 1 move...

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Lab-Rat    270
Have you looked into the structure for buckminsterfullerene? These spheres are made up from pentagons and hexagons. You could have the connections made only through the edges. Although most of the images you'll find will be of C60, probably having too few faces for your needs, I seem to recall that the structure can be scaled up to have many more faces.

Edit: Pictures of the scaled up structre seem few and far between, so here is a clicky to the best I could find.

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szinkopa    198
There are only 5 regular bodies. The other "almost regular" ones (like yours, or the soccer ball) are derived from them. So any option apart from the regular bodies you choose, you will have similar problems.

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Quote:
Original post by Squirm
I think the move your worried about looks a bit odd anyway... you've got red triangles which are closer to the unit than some of the purple ones... I would have only those moves which cross an edge (3 moves per triangle), or all moves to triangles touching the corner (making 12 moves for most triangles, and 11 moves for the others).

Look at the first image (the one your happy with).
There is a red triangle near the blue one which can be reached in 2 moves, either by going left twice, or by going right and then jumping all the way across, or by first jumping straight past it and then coming back, but you can't get there in 1 move...


Yeah, but I can't really see a move from the center of the triangle to the center of one of the red ones, the line would go straight through one of those that's highlighted purple. The three purple at the points instead of the edges can have a straight line between the center of the triangles.

I'm considering other options from my Making a planet thread. I'm considering simply not being able to move to one of the light purple triangles and saying that every planetoid has precisely 12 "difficult movement areas" that cause problems. And of course, I'm considering the move to only 3.

szinkopa:
What are the 5 regular bodies? Tetrahedron, Octahedron, Icosahedron, what else? (Cube, sphere?)

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alvaro    21263
Let's use a different polyhedron. Take your original polhedron and consider the middle point of each edge. Join them to form a new polyhedron made of hexagons and pentagons. Consider that you can move between two cells iff they share a side.

I would find moving in that planet a lot more intuitive.

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szinkopa    198
The 5 regular convex polyhedrons:

Tetrahedron: 4 faces, 3 edges per face, 3 edges per vertex

Cube == Hexaherdon 6 faces, 4 edges per face, 3 edges per vertex
Octahedron 8 faces, 3 edges per face, 4 edges per vertex

Dodecahedron 12 faces, 5 edges per face, 3 edges per vertex
Icosahedon 20 faces, 3 edges per face, 5 edges per vertex

Here they are

There are 4 concave regular polyhedra.

Sphere is also regular, yes.

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Quote:
Original post by alvaro
Let's use a different polyhedron. Take your original polhedron and consider the middle point of each edge. Join them to form a new polyhedron made of hexagons and pentagons. Consider that you can move between two cells iff they share a side.

I would find moving in that planet a lot more intuitive.


I tried making a polyhedron using the midpoints, but it's not pentagons and hexagons, it's pentagons and triangles:



Now, I like the dodecahedron, it uses pentagons, with the pentagons meeting on the edges, but is it possible to use pentagons to make a larger sphere?

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alvaro    21263
Quote:
Original post by Erzengeldeslichtes
I tried making a polyhedron using the midpoints, but it's not pentagons and hexagons, it's pentagons and triangles:

Ooops! You are right. You have to use two points in each edge, at a certain distance from the vertices that I haven't determined yet. The result should look like a soccer ball.


[Edited by - alvaro on July 6, 2004 6:19:35 PM]

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Aph3x    288
What about (using your original diagram) placing the units on the vertices rather than the faces, with valid steps following an edge to a neighbouring vertex?

Or have I missed something..?

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Quote:
Original post by Erzengeldeslichtes
Now, I like the dodecahedron, it uses pentagons, with the pentagons meeting on the edges, but is it possible to use pentagons to make a larger sphere?

You can always chop up the faces of a polyhedron to make "sides" with whatever number of faces you want, but if you want it to be a regular polyehedron, with just pentagons, the answer is no.

If you put regular pentagons edge-to edge, you get this:

http://mathworld.wolfram.com/p1img1955.gif

When you fold that together, and keep adding more pentagons (12 total), you eventually close the polyhedron and get an dodecahedron:

http://en.wikipedia.org/wiki/Dodecahedron

There's no way to add any more regular pentagons. You can chop each pentagon into 5 triangles (cut corner to centre), but this has the same problems as the icosahedron.

My suggestion is to let units move to directly adjacent triangles only, on a map built by subdividing a regular polyhedron built from equilateral triangles into four smaller equilateral triangles. Start with a tetrahedron, octahedron or icosahedron. This way every triangle will always have 3 adjacent triangles, no matter how much you subdivide.

You do still pay a price for this though, in that if you keep subdividing, the triangles near the original vertices of the polyhedron become distorted when you "blow out" the new vertices you make to the surface of the sphere you're trying to approximate. It'll look weird, but it'll be a uniform set of movement options... though distance calculations might be a little funky. The icosahedron is the least affected by this problem, so you should probly start with that. Octahedrons are nice, though, in that they have a built in equator and perpendicular meridians.

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