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What are some creative ways of teaching/explaining this?
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When explaining planes, I ask the class if anyone has ever sat in a wobbly chair and challenged them to explain why it wobbled. They will usually say that it was because one of the legs was “messed up.” At this point I’ll pick up a “non-wobbly” chair in the room and show them that the reason it isn’t wobbly is because all the points touch the ground at the same time, hence they are coplanar. I’ll even place the chair to the ceiling showing that they all touch the ceiling at the same time. But the moment one of the caps comes off on one of the legs, it is no longer coplanar with the other three legs, creating the wobble.
I then go into why cameramen always use tripods to keep the camera steady. Three legs will always be planar, but not four.
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From Carl Boyer’s “A History of Mathematics”:
“The Pythagoreans had defined a point as “unity having position,” but Plato would rather think of it as the beginning of a line. The definition of a line as ‘breadthless length’ seems to have originated in the school of Plato…”
Later…
“The role of Plato in the history of mathematics is still bitterly disputed. Some regard him as an exceptionally profound and incisive thinker; others picture him as a mathematical pied piper who lured men away from problems concerning the world’s work and encouraged them in idle speculation.”
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