. THE RECTANGLE PARADOX Two rectangles, shown in the top of the picture below, are cut and the pieces rearranged into another rectangle, shown in the bottom of the same picture. When comparing the AREAS we find them different by one square unit! How could this happen? This Paradox is being analyzed twice in LUCI MATH - UCLA Math Content Program for Teachers. It is discussed in Chapter 'Slopes' of "Perspectives on Algebra" (why?), and later in Chapter 'Pythagorean Theorem' of "Topics in Algebra and Geometry" (why?) If you tried hard to resolve this Paradox and still are cofused, draw the picture accurately on graph paper, and look carefully at the lines' intersections. You can see a mismatch with the gridlines. If it is still not very clear then CLICK at the picture below to see the animation. With the 'right click button' of your mouse you can control the animation: stop it, move back or forward frame by frame, print, and play again.

.. SMALLER PARADOXES To easier visualize what's going on here, consider smaller pieces. Instead of (5, 8, 13), use dimensions (3, 5, 8); or (2, 3, 5), or (1, 2, 3) to design similar rectangle paradoxes. Even the smallest triple (1, 1, 2) can be used. And larger sizes too, say, (8, 13, 21), or ... We can easier see the GAP in this case OVERLAP in this case And a GAP in this case O, Yes!.. You can go also to higher numbers for the sides and get even more delicate PARADOXES like this one with sizes (8,13,21):   The larger the sizes the harder to detect visually the mismatching effects. But the use of Slope or Pythagorean theorem will resolve the paradox. Do you see a pattern in these sizes/dimensions for a Rectangle Paradox? What are these numbers? They are 'famous'. In which cases do the pieces overlap, and when do they form a gap? In all cases the mismatched area is one square unit (why?) Can you calculate geometrically this 'mismatched' area ? Instead of the Fibonacci sequence mentioned above, one can use many other 'similar' integer sequences, and construct analogous paradoxes, but the mismatched area will be larger. What are these sequences?

... SIMPLER PARADOXES A different, simpler design can be used to create a similar paradox. Here is an example. One area unit is missing in the second shape. Instead of four pieces used in the pevious Paradox, this version uses only three pieces. It can be a square (as above), or a rectangle turning into another rectangle. You can easily create your own version of the Paradox with smaller (or larger) dimensions. Try it. A nice and simple algebraic formula is underlying this paradox in general. Which one? The same LUCIMATH approach with 'Slopes' or 'Pythagorean Theorem' can be used here too for a rigorous proof that the "paradox" is the result of overlapping or gaps. Can you think of another method of resolving these paradoxes? Can you think of an analogous 3-dimensional box-paradox (in space)?

Send questions and comments to mamikon@caltech.edu