Math CentralQuandaries & Queries


Question from Graham, a student:

I am working on a math fair project.
Can a line segment curve over two planes?
Such as, if I had a three dimensional L bracket and I drew a line segment
on it with a marker starting on the bottom of the L and had it curve
around the corner and up the top, would it still be considered one line
segment? Or is that two line segments?
Is there a rule that a line segment can only occupy one plane?
Thank you.

In Euclidean geometry, line segments do not "curve." The basic axiom for three-dimensional space states that

If a plane contains the points A and B, then it contains all the points on the line AB (which includes the line segment of points between A and B as well as the points on the extension of that segment in both directions).

Any line is contained in the infinitely many planes that contain it entirely, and only those: think of drawing a line on a piece of paper and folding the paper along the line -- then with the line kept fixed, all positions of the flaps represent planes containing that line.

On the other hand, if you do not want to use the axioms of Euclid you are free to make up your own axioms. Every surface defines a natural geometry (called the intrinsic geometry), where a "line" joining two points traces the shortest distance between those two points while staying on the surface. (Such a "line" is called a geodesic.) If point A is on one face of a polyhedron and B is on the adjacent face, then the shortest distance between the two points could be found by unfolding the faces so that they lie in the same plane, drawing the Euclidean straight line from A to B, then folding the faces back to their starting position. This is like the well-known problem of the spider in one corner of a cubical room and a fly in the opposite corner -- you are to determine the shortest path the spider can take to reach the fly. (The solution is to unfold the cube so that there is a straight-line path between the spider and fly.) Alternatively, if you stretch a rubber band between two points on a surface, it will minimize its energy and contract to the shortest distance between those points.


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