On Mon, Jan 31, 2011 at 12:20 PM, Jonathan Crabtree < firstname.lastname@example.org> wrote:
> Hi Dave > > The question why people can't draw twice as big or twice as much is > interesting to me. > > Whether or not the question is unclear may depend on innate thinking > styles. That's my hypothesis. > > Mathematicians have the right to defend a square twice as big having four > times the area and claim the question is wrong. > > You might be interested in this thread on mathfuture, where Ed Cherlin tells the story of Apollo's altar:
> As I said before, we state: the ratio of circumference of a circle to its > diameter is pi. Like it is the most natural thing in the world. I think it > is EXTREMELY strange. and I will bet that the Greeks (or whoever discovered > pi) thought God must be having some fun.
Not so much. However, the cube root of 2 is known to be a prank of the Greek gods. Apollo demanded through his oracle at Delphi that the Greeks double the size of his cubical altar. First they doubled the side, which made it eight times larger, then they proved that 2^(1/3) cannot be constructed with ruler and compass, and sacrificed 100 bulls instead. Apollo is on record as being pleased with their solution, or so the Greeks wrote down in their histories.
Euclidean constructions allow addition, subtraction, multiplication, division, and square roots, just like a five-function calculator. It is not very difficult to prove that cube roots don't qualify, in somewhat the same way that one proves that square root of 2 is irrational.