People have asked me, why is there a hexagon at the start of the game, and why are there 13 tiles, 8 wide and 5 narrow in a player's hand? And what about the boma lines?
The two types of tiles (called shields in the game) are a special type of tiling called Penrose P3 aperiodic tiling. Sir Roger Penrose, who lives here in Oxford, is a Nobel-prize winning physicist for his work on black holes, but also a top-drawer mathematician, He found the tiles back in the early 1970s.
By aperiodic tiling, I mean tiling that does not repeat the more tiles you put down. If you think about the common floor tiles which are squares, then the tiles can fill the playing surface without any gaps, in a repeating pattern, as is the case with games like Carcassonne and Marram. This type of tiling is periodic.
The Elongo tiling can also fill the playing surface with no gaps, but only if certain rules are kept.
One of those rules is the fact that you need a "seed" at the start of the tiling to enforce aperiodicity, otherwise the tiling could become periodic, which would not make for interesting shapes. There are an infinite number of these seeds, but there are 8 small ones where the seed is made up of 7 or less tiles. The simplest ones have three tiles, both create a hexagon shape, one with two wide tiles and one narrow one, and the other with 2 narrow tiles and 1 wide one. Others have nicknames, like the 4-tile crown, and two different 5-tile shapes called stars. These shapes are all created naturally as the game goes on. In the design, I opted for the simplest shape, the hexagon.
You may have heard of the golden ratio, called phi, which appears in art, and in nature. Think of a Nautilus shell or the Fibonacci sequence. Mathematically it is 1+the square root of 5, all divided by two, which works out at approximately 1.618033.. At the heart of the mathematics for Penrose tiling is the 5-fold symmetry that you will see in the game when you lay 5 wide tiles in a circle, The golden ratio is the ratio of tiles - the ratio of wide to narrow - that is needed to enforce the filling of the playing surface. In the design of the game, I mimic the golden ratio by choosing a ratio of 8:5 which is 1.6, quite close to 1.618033.. Given that the game is short, this approximation is acceptable. Hence 8 wide tiles and 5 narrow ones.
Penrose tiling is also special for another reason. 5 is a magic number so pentagon shapes appear, but so can four-sided shapes and 3-sided shapes, which is why you see bomas which enclose 3-, 4- and 5 land patches.
The final rule that enforces aperiodicity relates to the way you can lay tiles to join each other. This is why you see two sets of dotted lines, one black and one brown, on the tiles. This restricts the ways you can join the tiles. I designed the lines so that they could join up to create the bomas creating a different set of patterns other than the land and its colours.
I hope this provides some explanations to those who were interested.
The two types of tiles (called shields in the game) are a special type of tiling called Penrose P3 aperiodic tiling. Sir Roger Penrose, who lives here in Oxford, is a Nobel-prize winning physicist for his work on black holes, but also a top-drawer mathematician, He found the tiles back in the early 1970s.
By aperiodic tiling, I mean tiling that does not repeat the more tiles you put down. If you think about the common floor tiles which are squares, then the tiles can fill the playing surface without any gaps, in a repeating pattern, as is the case with games like Carcassonne and Marram. This type of tiling is periodic.
The Elongo tiling can also fill the playing surface with no gaps, but only if certain rules are kept.
One of those rules is the fact that you need a "seed" at the start of the tiling to enforce aperiodicity, otherwise the tiling could become periodic, which would not make for interesting shapes. There are an infinite number of these seeds, but there are 8 small ones where the seed is made up of 7 or less tiles. The simplest ones have three tiles, both create a hexagon shape, one with two wide tiles and one narrow one, and the other with 2 narrow tiles and 1 wide one. Others have nicknames, like the 4-tile crown, and two different 5-tile shapes called stars. These shapes are all created naturally as the game goes on. In the design, I opted for the simplest shape, the hexagon.
You may have heard of the golden ratio, called phi, which appears in art, and in nature. Think of a Nautilus shell or the Fibonacci sequence. Mathematically it is 1+the square root of 5, all divided by two, which works out at approximately 1.618033.. At the heart of the mathematics for Penrose tiling is the 5-fold symmetry that you will see in the game when you lay 5 wide tiles in a circle, The golden ratio is the ratio of tiles - the ratio of wide to narrow - that is needed to enforce the filling of the playing surface. In the design of the game, I mimic the golden ratio by choosing a ratio of 8:5 which is 1.6, quite close to 1.618033.. Given that the game is short, this approximation is acceptable. Hence 8 wide tiles and 5 narrow ones.
Penrose tiling is also special for another reason. 5 is a magic number so pentagon shapes appear, but so can four-sided shapes and 3-sided shapes, which is why you see bomas which enclose 3-, 4- and 5 land patches.
The final rule that enforces aperiodicity relates to the way you can lay tiles to join each other. This is why you see two sets of dotted lines, one black and one brown, on the tiles. This restricts the ways you can join the tiles. I designed the lines so that they could join up to create the bomas creating a different set of patterns other than the land and its colours.
I hope this provides some explanations to those who were interested.
