What follows is an attempt to answer a frequent question. The two applets below are supposed to serve as a backdrop for my answer. Besides the fact that both were written in Java and by the same author, the applets have something else in common. As, in fact, do other applets and a great many programming-free human activities and their results. With the customary reference to the lack of space, I'll try to make do with just those two applets. Hopefully, the two are sufficiently different to make the point. The first one is a puzzle. Draw an {N/D}
In the applet the symbol {N/D} appears in the lower right corner. Both
numbers can be modified by clicking a little off their center lines. When N
and D have a common factor, there are conflicting view
points as to whether the configuration of nodes and edges represents a
single polygon or a collection of polygons with fewer vertices. However,
for the puzzle we shall assume that N and D are mutually
prime. In the classical case, A clue to the puzzle lies in another The second applet is a demonstration of an optical illusion. In its original form as conceived by professor Misha Pavel of the Oregon Graduate Institute, a rotating square is occluded by four other squares that leave a cross-like opening to view the motion. Look at the applet. The impression you are supposed to get is that the square periodically expands and then shrinks and then expands again. That this is an illusion can be verified by changing the size of of the occluding objects. The explanation of the phenomenon can be found at its source. The narrower is the aperture, the more pronounced is the effect. The applet allows also for triangular occluders and the rotating shape. Surprisingly, the visual effect is quite different. With triangular occluders, the sides of the rotating shape seem to cave in. Also, when the occluders and the rotating object are of different shapes, the rotating object seems to wobble, not unlike what happens with rolling shapes of constant width. The wobbling seems to be more pronounced in the vertical than other directions. Now, back to the opening paragraph, what is that common thing shared by
the two applets? Well, both use the In the first applet, when the user presses a mouse button, the program
finds the node nearest to the cursor. If the As regard the second applet, its dependency on
whereas the functions cos() and sin() satisfy the fundamental identity ^{2}() + sin^{2}() = 1
which is but another form of the Pythagorean theorem. More directly the
theorem is used to compute the side Now, the question I have been trying to answer was:
A short answer is I know that the answer is not very satisfactory, because of a frequent
follow-up, Time had different impacts on Pythagoras' and Euclid's (geometric)
bequests to humankind. Euclid left us an all-time bestseller
that shaped mathematical thought over the space of two millennia. In the
19 Speaking metaphorically, an urgent request for a list of daily applications of the Pythagorean theorem is an indication of just how short the range of our vision may be. |