Lastly, as much as negative is the opposite of positive, if the two ants live on a surface with negative curvature, the distance between them will grow faster than if they were on the flat plane. The situation is different if two ants are on the sphere, the distance between them does not grow as fast. the distance between them after 10 minutes is 10 times the distance between them after 1 minute. The plane has zero curvature means that if two ants starting at the same place in the plane decide to walk away from each other in different directions, assuming that they walk at constant speed, then the distance between them increases linearly, i.e. The plane has zero curvature, the sphere has positive curvature, and the traditional example given for a space with negative curvature is a saddle although a certain population in Brooklyn would prefer kale leaves as examples. There are three qualitatively different behaviors, which correspond to positive curvature, zero curvature and negative curvature. The curvature measures ``the rate of divergence of geodesics". So what is this thing called the curvature? The curvature is one of the most basic notions in geometry and is responsible for large-scale phenomena. This is, in a way, the simplest kind of geometry, and though it seems good enough for everyday life in a small village - the polar bear would not have seen anything strange had he stayed his whole life around the North Pole because he got tenure there - it is no longer an accurate description of the way things are on a larger scale - when the polar bear walked so far that he reached the equator, things got strange. What we learn in school is flat geometry, which only applies to flat objects such as the plane or the island of Manhattan. While it is true that many things we learn at school are false, this is not one of them. In the riddle, however, the bear walks along a triangle whose angles are all 90 degrees each, so the sum of the angles is 270 degrees instead of 180. Many of us can still remember the day we learned that if you add up the three angles in a triangle, you always get 180 degrees.
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