<p><pre><code> The second moment occurred soon after he turned twelve, when he
was given “a little book dealing with Euclidean plane
geometry.” The book’s “lucidity,” he wrote—the idea that a
mathematical assertion could “be proved with such certainty
that any doubt appeared to be out of the question”—provoked
“wonder of a totally different nature.”
</code></pre>
Mathematical proofs, first really introduced in Geometry classes for most US students around 9th or 10th grade, are what really hooked me on math. I understood everything up to that point (and after), but it was a structural understanding (the rules and syntax, essentially). Once we arrived at logic, I (as a very bored precocious student) spent most of my class time proving everything from the axioms we were given (for algebra or geometry) that I could.<p>Giving students the tools to play and explore freely (we had proof assignments, of course, but I took it far beyond that) is what really hooks them on a subject. Mandate that they <i>must</i> read these books, they'll hate it. Require them to read a book and a few excerpts of others, and give them access to a large library, and they'll read forever. Same with every other subject. That's where many of the real geniuses[0] of a field come from.<p>[0] Whether technically geniuses by IQ or just acknowledged as such for their understanding and creations.
BTW, the book that was briefly mentioned in the article, <i>Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise</i> by Schroeder (<a href="http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/0716721368" rel="nofollow">http://www.amazon.com/Fractals-Chaos-Power-Laws-Infinite/dp/...</a>) is truly excellent, it is written in the form of short pieces collected into chapters investigating a very wide-ranging set of phenomena, ranging from Brownian Motion to self-similarity, Cantor sets and cellular automata. For $10-$15 on Amazon, it's a great bargain!
here's a proof that the ratio of the triangle area to the square is scale invariant:<p>imagine covering a geometric shape with a bunch of little squares, each with side length s. you could imagine a limit argument, with the squares getting small enough to arbitrarily approximate the area of the shape. uniform scaling increases the distance between any two points by some factor k.<p>the area of each of the original squares is s^2, and the new area is (ks)^2 = k^2*s^2<p>thus the area of the triangle increases by a factor of k^2. the same argument will apply to the square drawn on the hypotenuse. thus this k^2 term will cancel out when you take the ratio.<p>(we could've used the area formula for a triangle here, but this argument applies more generally: for example we can deduce the area of a circle is proportional to radius^2 without deriving the formula)
"Arthur Eddington ... was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.”<p>The third person was <a href="https://en.wikipedia.org/wiki/Karl_Schwarzschild" rel="nofollow">https://en.wikipedia.org/wiki/Karl_Schwarzschild</a> .
Have any of Einstein's relatives proved remotely as clever as him?<p>aha, of course there's an article <a href="https://en.wikipedia.org/wiki/Einstein_family" rel="nofollow">https://en.wikipedia.org/wiki/Einstein_family</a>
My physics professor showed me that exact proof 20 years ago. I remember being amazed. He used alpha instead of F as the scale factor :)<p>Still, it's easier for me to believe that it was attributed to Einstein at some point in the past 60 years than it originated with him.