Why would my dream life be
invaded by a geometry problem? The setup was a line segment, and the question
that invaded my dream life was, how many lines could intersect that line
segment? At first I imagined the intersecting lines to be all perpendicular and
parallel, and my dream thinking was that
the number of possible intersecting lines could not be infinite because (1) the
original line was only a line segment, and so not itself infinite; and (2)
while lines have no breadth or thickness, it seemed to me (but this could be
only commonsense thinking, which doesn’t always translate to mathematics) that
there would have to be space between the lines or they would simply fill in --
??? But they would not turn the group of lines into a solid, because there
would still be no thickness, or depth.... Does plane geometry care at all if a
surface is blank or filled in, or (and if I had to bet, I’d put my money on
this second disjunct) does it only care about lines and points?
Ah, but points have no length, breadth, or thickness! A point is not an object but a location.
So even the line segment could have, it seems, an infinite number of
intersecting, parallel, perpendicular lines. Do you buy it?
Next (still in my dream) I
started wondering about intersecting oblique lines. (What would be the smallest conceivable angle?
Would there be such a thing?) Would this generate a larger infinity of intersecting lines? Can infinity come in different sizes, bigger and smaller, or is infinity just always that -- infinity?
Finally, dragging myself out
of the dream and into wakeful consciousness, I searched around for a way to ask
the question that my geometry dream had posed, and here's what I came up with: What is the maximum number of
lines that can intersect any given line segment?
Here’s a question and answer
I found online that has bearing on my dream, but before following the link you might enjoy thinking about the question yourself. I mean, there's no exam involved here, not even a pop quiz.
