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A color diagram illustrating how the length of physical ("proper") rulers varies on the Lobachevsky upper half plane. Perpendicular x and y axes are drawn in black on a light gray-blue grid. The x-axis is labeled "y=0." Two dashed red lines parallel to the y-axis and extending up from the x-axis indicate fixed "coordinate distance" - they are everywhere separated by 20 grid boxes. Near the top of the grip they are separated by one long ruler. Just below, they are separated by two rulers. Closer to the x-axis they are separated by four rulers.

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A color diagram illustrating how the length of physical ("proper") rulers varies on the Lobachevsky upper half plane. Perpendicular x and y axes are drawn in black on a light gray-blue grid. The x-axis is labeled "y=0." Two dashed red lines parallel to the y-axis and extending up from the x-axis indicate fixed "coordinate distance" - they are everywhere separated by 20 grid boxes. Near the top of the grip they are separated by one long ruler. Just below, they are separated by two rulers. Closer to the x-axis they are separated by four rulers.
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Robert McNees (mcnees@mastodon.social)'s status on Thursday, 01-Dec-2022 15:53:28 UTC Robert McNees
in reply to

First you take the top half of the x-y plane: all the points -∞ < x < ∞ and 0 < y < ∞. Then you introduce a special rule for the distance between points.
Here's where it gets weird.
As you get closer to the x-axis (y=0) the rulers get shorter and shorter, so that the distances measured with them get longer and longer.
The "proper distance" between two points is measured with these rulers. It is not the same as "coordinate distance" – the number of squares separating those points on a grid.

In conversation Thursday, 01-Dec-2022 15:53:28 UTC from mastodon.social permalink