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Question:
Dear Scientist, Teacher, or Person kind enough to respond, I am having trouble with the concept of skew lines in my current geometry class, and was wondering if anyone knew of some examples of skew lines in everyday life. I would prefer if these examples were skew lines in order to perform a function, and not just the for design. Thank You.



Replies:
How about in baseball? Imagine a batter attempting to bunt a pitch: holding the bat stationary, roughly at a right angle to the motion of the pitched ball. Let the bat represent one line, and let the path of the ball represent another line. (Let us pretend that the ball really does move in a straight line, and that the bat is very long.)

If the bunt attempt is successful, the two lines must intersect. But if the batter misses the ball, the ball must have gone above or below the bat, and in this case the two lines are skew.

Tim Mooney


For the record: "Skew Lines" are two or more non-parallel lines, or curves, that do not intersect. In 2-dimensional Euclidian geometry there are no skew lines. In 3 or higher dimensions, there are an infinite number of skew lines. For a visual example, consider a cube: The front horizontal edges and the rear vertical edges are "skew lines". There are general analytical tests for determining whether two (or more) lines (curves) are "skewed".

"Practical" examples arise in the analysis and design of networks. For example, if you drop a pile of sticks on the floor they will be in a tangled pile, but the pile of "n" sticks form a set of "n" skewed lines, since every stick is separated from every other stick by a distance of at least 2*R, where R is the radius of the strands.

A second example might be a 3-dimensional lattice of NxNxN dots. The problem is to determine how many of the point pairs (Ni, Nj) can be connected without intersecting a third point, that is, there is no other point Nk such that Nk falls on the line connecting Ni and Nj. This might be useful in designing how many "secure" links can be made between "i" and "j" to guarantee that a third party "k" cannot eavesdrop on the link between "i" and "j".

One added consideration is a pedagogical one. It isn't necessary to have a "practical example" for a mathematical concept or study. Mathematics, including geometry, considers the logical connections between the axioms and various conclusions that result from theorems involving the axioms. The "responsibility" of the mathematics is to ensure consistency, not necessarily to be concerned whether there is some "problem" or "example" out there to solve. Sometimes the math is driven by a problem, but also, often the math precedes the problem. A classical example of this type of "math-before-the-problem" is complex numbers, those numbers involving that curious number "i", the square root of (-1). That math was robustly developed before its application to physical problems was recognized. Now complex variable algebra is key to understanding most of modern physics and engineering.

Vince Calder



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