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Now draw the same triangle on a curved surface. No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to
by fooker 5y ago
Now draw the same triangle on a curved surface.
No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to calculate area is that we are doing euclidean geometry.
We do not yet know if there is some similar assumption hiding behind our ignorance of physics.
- HPsquared 5y agoEven in a non-Euclidean geometry, wouldn't π still emerge (or be approached in the limit) in "flat" areas?
- FreeFull 5y agoWith regular spherical and hyperbolic geometry, yes, it is flat-ish if you look at a small enough area. On the other hand, if you have a different distance metric, the ratio of a circle's diameter to circumference can be a different constant. For example, in taxicab geometry, it is 4
- DonHopkins 5y agoIn Uber Geometry, it can vary from 5 to 14 during surge pricing.
- HPsquared 5y agoIn London geometry, the distance depends where you are in the one-way system.
- milesvp 5y agoI know this is sort of a low quality joke post, but I don’t think I’d ever thought about a directed taxi-cab distance before. I’m not yet convinced it’s a useful model outside GIS, but I may spend considerable time today thinking about it. Thank you.
- HPsquared 5y agoHappy to be of service, in filth it will be found!
- KineticLensman 5y agoTIL that Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes.
- HPsquared 5y agoIt depends if you define a circle as "constant distance from centre point" or "minimum (circumference/area)".
- aj3 5y agoEuclidian geometry would be the same in any universe though, as it does not depend on objective physical reality.
- biofox 5y agoIt could be argued that the Euclidean plane is its own universe.
- mypalmike 5y agoThe Euclidean plane is a conceptual model also not based on physics.
- coldtea 5y agoIt is based on observing (and needing to measure) plots of land. The abstraction came later. Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).
- gmadsen 5y agohow is general relativity related to any prior experience of the 1920s?
- belugacat 5y agoTrains and clocks, mostly!
- gmadsen 5y agothat is special relatively , not general
- RichardCA 5y agoThe maths that underpin GR were developed before Einstein. Without the maths you have no ontology, no way to evaluate the truth of a scientific claim. https://en.wikipedia.org/wiki/Tullio_Levi-Civita https://en.wikipedia.org/wiki/Tullio_Levi-Civita
- JoeAltmaier 5y agoYes yes. But you can ask that question, and we can answer it, regardless of whether our space is curved or flat. PI is the same in a flat space in any universe where you ask the question, because we're doing a mental exercise, not measuring it locally.
- TomSwirly 5y ago> PI is the same in a flat space in any universe pi is the same in _any_ space, flat or not. Indeed, the space we are in is probably not flat, but pi is unchanged.
- philipswood 5y agoTrue but Pi (mathematical constant Pi) isn't necessarily the ratio between the circumference and the diameter in those spaces.