Here is the idea: Given a point as the origin and a length of distance 1, it is relatively straightforward to use the straightedge and compass to construct all points on a number line whose coordinates are rational numbers (ignoring, as mathematicians tend to do, the impossibility of actually plotting infinitely many points in only a finite amount of time). At the ripe old age of 20 myself, I had achieved considerably less impressive mathematical accomplishments, but I at least understood Wantzel’s proof. His proof used cutting-edge mathematics of the time, the foundations of which were laid by his French contemporary Évariste Galois, who died at 20 in a duel that may have involved an unhappy love affair. Finally, in 1837, Pierre Laurent Wantzel explained why no one had succeeded by proving that it was impossible. For more than 2,000 years no one managed to solve it. To construct the larger cube, you have to figure out a way to draw one of its sides with the new required length, which is ∛2 (the cube root of two), using just the straightedge and compass as tools. Here that value might as well be 1 because it is the only unit of measurement given.
To double a cube’s volume, you start with its side length. The catch for this particular puzzle is that any points or lengths appearing in the final drawing must have been either present at the start or constructable from previously provided information. A straightedge can be used to extend a line segment in any direction, and a compass can be used to draw a circle with any radius from the chosen center. In fact, the challenge is more than two millennia old, attributed to Plato by way of Plutarch. A man was standing near a few brainteasers he had scribbled on the wall, one of which asked for the construction, with an imaginary straightedge and compass, of a cube with a volume twice that of a different, given cube. On a crisp fall New England day during my junior year of college, I was walking past a subway entrance when a math problem caught my eye.
0 kommentar(er)
