If you let one row of fish follow their nose in Escher’s woodcut “Circle Limit III”, all the rest are forced to swing around wildly.
The fish here are strung out along “hypercycles”: curves in the hyperbolic plane that are equidistant from a given hyperbolic line, but no two hypercycles here share the same line.
Escher’s woodcut shows the “Poincaré disk model” of the hyperbolic plane. In the “hyperboloid model”, the hyperbolic plane is the set of points (x,y,t) in 3-D spacetime such that:
x^2 + y^2 – t^2 = –1
In this model, if you pick a spacelike vector V and slice the hyperboloid with a plane that is orthogonal to V, then:
• if the plane passes through the origin, you get a hyperbolic line;
• if the plane does NOT pass through the origin, you get a hypercycle equidistant from that line.
The translations of the hyperbolic plane shown in the movie, which preserve one particular hypercycle, correspond to boosts in the 3D spacetime that preserve the vector V.
If any other hypercycle in the woodcut came from a plane with the same normal, then it too would be unchanged by the boosts.
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