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I like to know what is the math behind the outlines of these boards

https://www.boardgamegeek.com/image/4383021/spirit-island

https://www.boardgamegeek.com/image/4880004/spirit-island

https://www.boardgamegeek.com/image/2706606/spirit-island

https://www.boardgamegeek.com/image/3692997/spirit-island

https://www.boardgamegeek.com/image/4013584/spirit-island

I wonder how to come up with these exact shape and how to express this in a formula. The maker of the Boards used the term tessellation and mentioned that it was initial based on a hex map. I like how they fit nicely and still give the impression of uniqueness. Is there anything else interesting (from an mathematicians point of view) about these?

user2567875
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1 Answers1

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The underlying shape for each board piece is a rhombus with angles of $60°$ and $120°$.

$2$ of these form a parallelogram, $3$ can be put together to form a regular hexagon, and $4$ can form a larger rhombus.

$3$ of the edges of each rhombus are replaced with identical rotationally symmetric curves. The rotational symmetry ensures these edges fit together in all layouts.

The fourth edge of the rhombus is replaced with a different curve, but always forms the water boundary so will only appear on the edge of the layout and doesn't need to match the other edges.

nickgard
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