This is a short note for the record.
I've been preparing a page on the indivisibles (not ready yet), when my eighth grader boy came up to share his solutions to a couple of olympiad problems. We talked a few minutes about the olympiad and then, sensing his mood, I decided to catch the moment and expand a little of my own. So I told of Torricelli's paradox:
Imagine a 2×1 rectangle ABCD with diagonal AC drawn:
The diagonal splits the rectangle into two equal triangles, right? Pick a point on the diagonal, say F. Now, draw the perpendiculars FE and FG, one to side AB, the other to AD. Because of the similarities of triangles, FE is twice as long as FG. This is true regardless of the choice of F. When F runs over the diagonal, lines FE fill up triangle ABC while lines FG fill up triangle ACD. There are as many lines FE as the are lines FG. It follows that the area of triangle ABC is twice as big as the area of triangle ACD. But we know that the triangles are congruent. What gives?
What do you think was the boy's reaction? He smiled and said:
Pinocchio says: "My nose is going to grow."