We first state that products are inherently two-dimensional, and
we then look at rectangles of various dimensions to represent certain
areas. We then cut the rectangles along the diagonals to create right
triangles. [Whether we actually do this or just pretend we did does not
matter, but it is the basis for the "construction" to follow.]
Now we travel from the origin in a coordinate plane a (positive)
distance x followed by a (positive) distance y, then return directly to
the origin along the hypotenuse of the triangle (diagonal of the
rectangle) that we have determined.
/|
/ |
/ | y The figure that you see may vary from that intended,
/ | and x and y would ordinarily be quite dissimilar.
/ |
0---------|
x
If only the horizontal movement is "backwards", that is (-x), the
triangle that is formed is the mirror image [L-R] of the above. It can be
matched with the first one _only_ by "flipping" it out of the plane. And
it will not fit with the first to form a rectangle (if x <> y) unless it
_is_ flipped.
Or if only the vertical movement is "backwards", that is (-y), the
triangle that is formed is the mirror image [U-D] of the first one. It
can be matched with the first one _only_ by flipping it out of the plane.
Finally if _both_ the horizontal and vertical movements are
opposites of the positive direction, that is (-x) and (-y), then the
triangle that is formed is a rotation [by pi radians :-) ] of the first.
It can be matched with the first one _without_ flipping it out of the
plane. And it will fit with the first to form a rectangle merely by a
gliding translation.
matthewg@sunyocc.edu
George E. Matthews, Onondaga Community College, Syracuse, NY13215
(315) 498-2381
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