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enter image description here As you stare at this, try to imagine yourself as one of the fish. You are exactly the same size and shape as every other fish, and you can swim in a straight line forever without ever seeing any change in your surroundings or in your fellow fish. But looking into the map from outside, the compression of distances makes you appear to be swerving along a circular path and to be shrinking as you go. Pg 63 of Tristan Needham's VDG

I don't get it, how would it be that I see all the fish are of same size if I was in the disc but not as an outside observer? Could we explain the above situation using principles from geometric optics?

tryst with freedom
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    I don't think geometric optics is particularly relevant here. Does it bother you that the straight lines in a world map are not actually straight in real life? –  Dec 01 '21 at 06:55
  • That's a great comment.. I think I just understood what a metric does now – tryst with freedom Dec 01 '21 at 09:33

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I think the I understood it a bit more now. In the real world, if we have a ruler, and we graduate it by defining a unit marking on it, then wherever we travel on the earth, the unit won't change.

Here, the issue is that, as we move from point to point on the beltrami poincare' disc, the rule shrinks and expands and so does one unit. It's sort of like how currencies work between countries, as we move from country to country, what is the purchasing power of one unit currency is different( eg: 1$ is certainly not the same value on as 1 euro).

Now, the metric formula $ds' = \frac{2ds}{1-r^2}$ tells us how to convert the unit distance at some point (measured by an outsider) with how that amount of distance is actually valued nearby in that point. Sort of like converting all foreign currency into the US dollar (example), and seeing how much it would be worth on that scale.

This causes what we think of as straight-line to be different because due to our unit not being fixed point to point, there are better ways to travel between two points than to take a straight line.

tryst with freedom
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  • Highly informal but it is the way I understood it, if anyone wishes to criticize my understanding, feel free. – tryst with freedom Dec 08 '21 at 11:49
  • If it matters, the conversion rate here is a little different than in the upper half-plane: $ds' = \dfrac{2, ds}{1 - (x^2+y^2)}$ at Cartesian location $(x, y)$. – Andrew D. Hwang Dec 08 '21 at 20:58
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    To complete your understanding, consider the following: maps of Earth are maps of a real place, and the map's metric formula describes how real Earth lengths are distorted when squished into a Euclidean plane. If you look at a map of Earth, you imagine being on/in a sphere, not a plane. Just like that, you can also think of the hyperbolic plane as a real place where every fish is the same shape and size, and geodesics are truly straight, not just "better ways to travel". The Poincaré disk is just the imperfect approximation you get if you try to squish all of that into a euclidean plane. – Magma Dec 09 '21 at 12:07