Hyperbolic Compass

[Fulgur14]

Today I wondered on how a compass in hyperbolic world would look and work.

On Earth, you have a compass that points North. But hyperbolic world doesn't have any objectively important directions like North.

So the basic compass design is what we see in Caribbean: a thing showing certain direction towards the horizon.

That's of limited usefulness by itself, but what if we combine multiple such compasses together?

Two compasses, aimed in different directions, would specify a line. You could guess your distance to the line from the angle between the two pointers, with the angle being 180 degrees directly on the line. This double compass could be useful for finding your way to a specific Great Wall, for example.

A triple or quadruple compass would then have the ability to pinpoint a specific hyperbolic location. With any three directions, you have an absolute triangle and in the center of this triangle, the three pointers would have 120-degree angles between each pair. And with any four directions, you can construct a pair of intersecting lines and specify the point of intersection -- in this point, the pointers will form two pairs with 180-degree angle within each.

[Rantorgh]

This raises the question of what is directing them? Any conventional field in hyperbolic space would increase in strength exponentially as you went towards the "origin". (Related: Yendor keys are similarly odd; you can detect them 100 tiles away but they don't harm you when you're carrying them.)

The mice must have a biological equivalent to find the Prince. There are enough around for you to reliably find one (trillions at least), but clearly they can't all have been close enough to see him then remembered the route back to him.

[Fulgur14]

Originally posted by simon_clarkstone:

This raises the question of what is directing them? Any conventional field in hyperbolic space would increase in strength exponentially as you went towards the "origin". (Related: Yendor keys are similarly odd; you can detect them 100 tiles away but they don't harm you when you're carrying them.)

The mice must have a biological equivalent to find the Prince. There are enough around for you to reliably find one (trillions at least), but clearly they can't all have been close enough to see him then remembered the route back to him.

The requisite number of mice can be reduced if you assume they can detect both the Prince and you -- in that case, there won't be trillions of mice escaping the perimeter of prison.

[tricosahedron]

I don't think the target marker for OoY keys has to be explained in a scientific way; Orbs grant you magic powers, and I assume touching an OoY casts some kind of spell that tells you in which direction the key is, or something like that. Not sure how the treasure compasses in the Caribbean can be explained, though.

Animals that have adapted to a hyperbolic world may have a very good ability to backtrack. Also, maybe there's some relation between the Orb of Love, and the mice's ability to find the prince / princess.

[Fulgur14]

Originally posted by tricosahedron:

I don't think the target marker for OoY keys has to be explained in a scientific way; Orbs grant you magic powers, and I assume touching an OoY casts some kind of spell that tells you in which direction the key is, or something like that. Not sure how the treasure compasses in the Caribbean can be explained, though.

Animals that have adapted to a hyperbolic world may have a very good ability to backtrack. Also, maybe there's some relation between the Orb of Love, and the mice's ability to find the prince / princess.

Basicaly, there are three strategies for hyperbolic animals: never get too far from your lair, have no lair and be on the move for your whole life, or use specific tracks to not get lost.

[Toph]

Note that you lose track of the key if you move too far from the path between it and the corresponding Orb, even if you're technically still within 100 tiles of it. This suggests that whatever force is guiding you, it pervades only a narrow region around that line, not the entire 100-tile circle around the key.