Geometry for Elementary School/Why are the constructions not correct?
In the previous chapters, we introduced constructions and proved their validity. Therefore, these constructions should work flawlessly. In this chapter, we will check whether the construction are indeed foolproof.
Testing a construction
- Draw a line of of length 10cm.
- Copy the line segment to a different point T.
- Measure the length of the segment you constructed.
I must admit that I never could copy the segment accurately. Some times the segment I constructed was of the length 10.5cm, I did even worse. A more talented person might get better results, but probably not exact.
How come the construction didn't work, at least in my case?
Our proof of the construction is correct. However, the construction is done in an ideal world. In this world, the lines and circles drawn are also ideal. They match the mathematical definition perfectly.
The circle I draw doesn't match the mathematical definition. Actually, many say that they don't match any definition of circle. When I try to use the construction, I'm using the wrong building blocks.
However, the construction are not useless in our far from ideal world. If we use approximation of a circle in the construction, we are getting and approximation of the segment copy. After all, even my copy is not too far from the original.
In the Euclidian geometry developed by the Greek the rule is used only to draw lines. One cannot measure the length of segments using the rulers as we did in this test. Therefore our test should be viewed as a criticism of the use of Euclidian geometry in the real world and not as part of that geometry.