Triangles on Earth Exceed 180 Degrees

The idea that space and time can be curved or warped is fairly recent. For more than 2,000 years the axioms of Euclidean geometry were considered to be selfevident. As those of you who were forced to learn geometry at school may remember, one of the consequences of these axioms is that the angles ot a triangle add up to 180 degrees. However, in the last century people began to realise that other forms of geometry were possible in which the angles of a triangle need not add up to i8o degrees...

 Earth's Curve, Flat Maps, and the Fastest Route Between T...

Imagine, say, that you wanted to travel from New York to Madrid, two cities that are at almost the same latitude. If the earth were flat, the shortest route would be to head straight east. If you did that, you would arrive in Madrid after traveling 3,707 miles. But due to the earth's curvature, there is a path that on a flat map looks curved and hence longer, but which is actually shorter. You can get there in 3,605 miles if you follow the great-circle route. which is to first head northeast,...

 How People are Paid for Their Attention

The result of the new economics is that people are often paid for their attention, implicitly or explicitly. They get to see television free in return for watching commercials. Their magazines and newspapers are subsidized or supplied free by advertisers. Nowadays bus shelters, baseball stadiums, and even those little refresher towelettes on airlines such as Lufthansa are supported by advertisers eager for your attention. You are also rewarded with content according to the "quality" of attent...