In search of a symmetric homogeneous graph with a pivotal origin

Question

I'm trying to design a computer game and I need a symmetric homogeneous graph with a pivotal origin which will act as the map of the game (players will walk according to it).

Here's an example of what I'm looking for:

Let me describe this game:

1. Players can walk on the edges landing on vertices.
2. Blue dot marks the final destination.
3. Yellow dots show stages one prior to final one and they are all of the same value (difficulty level). This goes on as we get farther from the center. In other words the difficulty level drops by the number of edges remaining to the pivot.
4. The purpose of the game is to players fight their path through for the blue dot.

What are desired characteristics of this pattern:

1. It has a central point which can be considered as the final stage.
2. Number of vertices of the same value grows as we get farther from the origin.
3. All the vertices have the same number of neighbors (6, event though the number 6 itself is not important but to keep the game simple, the lower the better).

As perfect as this graph is, it still has got one problem. It's not completely fair. Take vertices 1 and 2 for example. Player 1 has only one path to next level while player two has 2 (one adjacent yellow dot by P1 and two adjacent yellow dots by P2).

Does anyone know of a similar pattern which is also fair?

Answer 1

Does the penrose tiling satisfy what you're looking for? Various other aperiodic tilings may also be of interest.

(Image courtesy of Wikipedia)