And seen, as it turns out, really is the operative word. Puzzled, out of answers, checked the pockets, nothing of help to be found there. Tried out a graphic calculator, realized I didn’t know how to operate one. Stared at the ceiling for a while, then gave up. Which is where my father, the genius, the mathematician, the guy who probably thought he was the helpful co-pilot on my high-school calculus homework, but must not have realized he really was actually flying the plane, has the answer. And here it is:
The 8x8 square is divided into two congruent large right triangles with legs of length 3 and 8 and two congruent trapezoids. The trapezoids are each composed of a rectangle with sides 3 and 5 and a small right triangle with legs 2 and 5.I’m still confused, but I do love that man.
The 5x13 figure can only be constructed from these components if the larger 3-8 triangles are geometrically similar to the smaller 2-5 triangles, that is, the angles of the large triangles must be the same as those of the smaller triangles. That is only true for right triangles if the leg ratios 3:8 and 2:5 are equal, which they are not. This we learned in Euclidean Geometry.
Conversely, if the 5x13 figure was cut into components, where all the triangles were indeed similar, then you couldn’t build the 8x8 square from them. But you could probably fake it like they did.