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Hyperbolic spaces are a bit freaky because of how huge they are. Living on a flat plane (like we do for scales smaller than a few thousand kilometers), area gro
by kmm 7y ago
Hyperbolic spaces are a bit freaky because of how huge they are. Living on a flat plane (like we do for scales smaller than a few thousand kilometers), area grows quadratically with the radius, i.e. if you venture out twice as far, the area you cover gets four times bigger. Go 10 times further, and there is 100 times more area to get lost in.
On a hyperbolic plane however, area grows exponentially when you stray further than a few times the characteristic length of the space. Increasing your range from one such length to two, the area covered grows by a factor of about 5. Going from ten to twenty however, increases the area by about ~20000!
If the characteristic length were a kilometer, you would never be able to get more than a few dozen kilometers away from your home before being irretrievably lost. The area around you 10 kilometers away would have the area of Manhattan. A circle with radius 20 kilometers would be slightly bigger than Egypt. And everything within 50 kilometers would be spread out over an area about the size of the Solar System. Unless you left a trail of breadcrumbs, you'd never be able to retrace your steps.
- bilekas 7y agoMadness... Now lets calculate the quantity of breadcrumbs required!
- danharaj 7y agoIt would perhaps be a relief from the viciousness of close quarters living if we lived on a hyperbolic surface rather than a puny sphere. Make a day's hike away from brutal conflict, take a detour, and never be accosted again.
- gabcoh 7y agoThis sounds like an interesting premise for a sci-fi story. Does anyone know of any books with this premise of living on a hyperbolic plane?
- harmoat 7y agoDoes the area grow less than quadratically with the radius on the surface of a shpere?
- bobbylarrybobby 7y agoOn a sphere of radius R, the rate of growth of area approaches 0 as the distance from you approaches 𝜋R (half the circumference), and generally decreases past 𝜋R/2 (quarter circumference)
- kmm 7y agoTo be precise, it's explicitly 2pi R^2 ( 1 - cos(r / R) ) , where r is the radius of the circle on the sphere, and R is the radius of the sphere itself. If r is small compared to R, cos(r / R) = 1 - (r/R)^2 / 2 + O((r/R)^4) so we recover the usual flat circle formula for the area pi r^2 , which is quadratic in r. Only if the r gets comparable to R does the curve flatten. So a circular area with radius 1000 km would only be 0.2% smaller on Earth than its flat equivalent. Not a useful way to prove the Earth is round. https://www.wolframalpha.com/input/?i=plot+pi*x%5E2%2C+2*pi%281+-+cos%28x%29%29+from+0+to+pi https://www.wolframalpha.com/input/?i=plot+pi*x%5E2%2C+2*pi%...
- SamReidHughes 7y agoYou can feel this when panning because the relative direction of a point on the outside edge of the display swivels around like crazy.