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...

 Fluid Intelligence has Made the Most Gains

So which kinds of intellectual performance have been pushed upward by the better environments of recent decades? Surprisingly, the steepest gains have not been found in the concrete skills that are directly taught in school, such as general knowledge, arithmetic, and vocabulary. They have been found in the abstract, fluid kinds of intelligence, the ones tapped by similarity questions (“What do an hour and a year have in common?”), analogies (“BIRD is to EGG as TREE is to what?”), and ...

 Euclid's Elements as a Game

"If video games had been around in 350 BC, Euclid would have made a video game," Devlin told me. The thirteen books of Euclid's Elements would have been the supplemental material, a PDF file that you could read if you wanted to. "People think I'm joking—I absolutely mean that. Euclid would not have written a textbook, he would have designed a video game." Peek at any of his proofs, Devlin said, and you'll quickly find that the great Greek mathematician, often called the father of geometry, ...

 Martin Rees: We'll Never Hit Barriers To Scientific Under...

We humans haven't changed much since our remote ancestors roamed the African savannah. Our brains evolved to cope with the human-scale environment. So it is surely remarkable that we can make sense of phenomena that confound everyday intuition: in particular, the minuscule atoms we're made of, and the vast cosmos that surrounds us. Nonetheless—and here I'm sticking my neck out—maybe some aspects of reality are intrinsically beyond us, in that their comprehension would require some post-h...

 Superstring Theory

It is time now to try to describe what a superstring really is. Here I run into the same difficulty which the geometer Euclid encountered 2,200 years ago. Euclid was trying to convey to his readers his idea of a geometrical point. For this purpose he gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." This definition would not be very helpful to somebody who was ignorant of geometry and wanted to understand what a point was. Euclid's notion ...

 The Kiss Precise

Four circles to the kissing come, The smaller are the benter. The bend is just the inverse of The distance from the centre. Though their intrigue left Euclid dumb There's now no need for rule of thumb. Since zero bend's a dead straight line And concave bends have minus sign, The sum of squares of all four bends Is half the square of their sum.