Last week, in Fundamental Understanding of Mathematics XLVI we took a look at the area of parallelograms, and found a formula for figuring out their area, based on what we knew about triangles.
In the comments, Joffan pointed out that
And we're quite close, here, to having the idea that a general shear transform, where one direction of parallel lines preserve their spacing and direction, also preserves all areas.
Let's take a look at what Joffan is talking about, below the fold.
We began this development of the idea of area starting with a triangle. We noticed that, if the top point of a triangle remained a set height above the base of the triangle, the area of the triangle remained the same.
As long as the triangle's height and base remains the same, the top point can be anywhere along the line that is height H above the base. Anywhere. And the triangle's area remains the same.
Next we took a copy of the triangle, flipped it over and attached it to the first triangle, to make a parallelogram.
Just as we can move the point of the triangle along the upper "height" line, and keep the same area, so can we move a Base of the parallelogram anywhere along one of the "height" lines, to create new shaped parallelograms that have the same area as the original.
The green parallelogram has the same area as the blue parallelogram. Changing the shape of a parallelogram in this way, by sliding one base along a parallel line, is called a shear transformation.
I suspect the term comes from engineering, where there are three kinds of forces that tend to break things: compressive force, which squeezes the two bases closer together, tensile force, which tries to pull the bases further apart, and sheer force, which makes one of the bases slide sideways. In the case of the parallelogram, one of the bases is sliding sideways.
So, here we have shown that parallelograms with different shapes have the same area, using the fact that triangles with different shapes have the same area. Now, of course we have to warn people that these are not just any old parallelograms, but parallelograms with the same bases and height, just as the triangles were not any old triangles, but triangles with the same base and height.
This is how mathematics works. Earlier, simpler ideas are used to prove later, slightly more complicated ideas. We lay a foundation, then build the first floor.
Have fun in the comments