Science questions not worth a thread I: I'm a moron!

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You can derive it from Euclid's geometric axioms. (Archimedes calculated pi by inscribing circles with regular polygons).

EDIT: I think it was Archimedes not Aristotle ;)

Modern day it is defined from

smallest positive x such that

cos(x) = 0

then x = pi/2

(angles in radians, cos defined in terms of Taylor series).
 
I would think steppes and grasslands. A quick scan of Wiki doesn't obviously, say, except to suggest that the spread of heavy forests over grasslands is one theory on what drove them to extinction. But if you consider other large land plant eating mammals, they grow the largest in the grasslands. Grasslands provide more food than forests. So the African elephant is mostly a grasslands animal. Not sure about the Asian elephant.
 
Asian Elephants (and some African Elephants!) live in Jungle quite comfortably. I can't really picture them living on Steppes as those aren't quite grassland.
 
What is time dilation?
 
Time dilation is that if you see a friend holding a clock moving past you very quickly, and compare how fast it ticks to a clock which you're holding, the moving clock ticks slower. The crazy mind bending stuff comes in if you consider what your friend sees. For him, he isn't moving and you're the one moving past him with your clock. Therefore he sees your clock as the one running slow.
Relativity states that you are both right. I wont go into any more detail because it makes my head hurt.
 
Time dilation describes the fact that observers sometimes disagree on how fast events occur. This can happen in events described by special and general relativity.

In special relativity, as Lord Olleus describes, two observers moving at high, constant speeds will perceive events in their own reference frame (all things moving with the observer) to happen faster than events in the moving reference frame. This dilation is symetric: both observers will perceive the other as moving slower.

In general relativity, observers closer to a massive (heavy) body will experience time slower than observers far away from a massive body. The same is true of accelerating observers. The faster one accelerates, the slower time is experienced. This dilation is not symmetric, and it is possible to know if you are in a gravity well or being accelerated.

Time dilation always occurs at the same time as length contraction.
 
Time dilation describes the fact that observers sometimes disagree on how fast events occur. This can happen in events described by special and general relativity.

In special relativity, as Lord Olleus describes, two observers moving at high, constant speeds will perceive events in their own reference frame (all things moving with the observer) to happen faster than events in the moving reference frame. This dilation is symetric: both observers will perceive the other as moving slower.

In general relativity, observers closer to a massive (heavy) body will experience time slower than observers far away from a massive body. The same is true of accelerating observers. The faster one accelerates, the slower time is experienced. This dilation is not symmetric, and it is possible to know if you are in a gravity well or being accelerated.

Time dilation always occurs at the same time as length contraction.


So... Earth time would appear slower to a person repairing a satellite in orbit?
 
So... Earth time would appear slower to a person repairing a satellite in orbit?

Yes, and this effect has to be corrected for in GPS satellites.
 
So not just gps corrections as a result of the distance to the mass of the Earth.

The universe is an odd, odd place.
 
Or rather, human intuition is very limited as our everyday experience is based on a very narrow range of masses, lengths and time scales compared to what happens generally in the universe.
 
When people say that space is warped/folded, does this require a 4th spatial dimension?

In my physics class, a 2 dimensional object (notebook/paper) was used as an analogy, in a discussion on worm holes. One end of the paper folder over to create a shorter distance between opposite ends of the flat-world. But paper has to have that up/down third dimension to fold at all, right?

Is it problem because I'm taking the paper universe analogy too far?
 
No, it doesn't require an extra spatial dimension. "Space is folded/warped" means that it has non-Euclidean geometry. That means you cannot describe it by using three axes with 90 degree angles to each other, but have to describe it another way. For example the surface of a sphere is non-Euclidean (one effect is that the angle sum of a triangle is not 180 degrees). The surface just has two dimensions, and there is no need to introduce a third, but our minds need the third dimension to imagine it, because it tends to think in three Euclidean dimensions. But the extra dimension needed to imagine it is a limitation of our mind, not anything fundamental.
 
Can you elaborate? I know a bit about non-Euclidean geometry.

I know the space-time metric is non-Euclidean (or is it a pseudo metric hmmm, does the triangle inequality apply?)

d(p1, p2) = sqrt ( (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 - (t1-t2)^2 )

is that the reason?
 
Can you elaborate? I know a bit about non-Euclidean geometry.

Sadly, not that much. I couldn't stand the ugliness of General Relativity notation.

I know the space-time metric is non-Euclidean (or is it a pseudo metric hmmm, does the triangle inequality apply?)

d(p1, p2) = sqrt ( (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 - (t1-t2)^2 )

is that the reason?

That's the Special Relativity metric (depending on convention the spatial dimension can get the minus sign instead of the temporal dimension). It is sometimes called pseudo-Euclidean, as it only differs from the Euclidean metric in its signs. However this isn't really what people mean by warped space, as the Euclidean metric is still valid if you only consider the spatial dimension.

In General Relativity, however, the metric isn't just non-Euclidean, but it also varies from place to place as it depends on the mass and energy around it. If you solve the Einstein equation you get a tensor field for the metric and in general these metrics are non-Euclidean even if you just consider the spatial dimensions.
 
Other than chordates, are there any phyla with species that currently live in the ocean but whose ancestors were terrestrial? I would think there might be some arthropod examples of this, but are there?

EDIT: I should specify "fully aquatic", as in they don't spend any part of their life cycle on land.
 
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