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| Topology is the study of geometrical properties
that remain unaffected by manipulating the shape or size of a figure. Deforming
the figure can be accomplished by bending, squeezing, stretching, and compressing,
but not by breaking or tearing.
Thus a circle is topologically equivalent
to an ellipse because any circle can be invariably deformed to become a
ellipse. Both are simple closed curves. It is possible to start
at any point on a simple closed curve and travel over every other point
of the figure exactly once before returning to the starting point. Simple
closed curves do not cross themselves.
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| One of the fundamental concepts of topology
is so simple that it sounds trivial.
A simple closed
curve in a plane
One region is said to be inside the simple closed curve, and the other region outside. It seems fairly obvious that to go from a point on one side of the curve to a point on the other side, one must necessarily cross the curve. |
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| The result is called the Jordan Curve
Theorem in honor of the French mathematician Camille Jordan (1838-1922).
The term Jordan Curve is another name for a simple closed curve.
Although this theorem is obvious, it has proven to be extremely important to topologists and provides recreational math enthusiasts with answers to many classic problems. |
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| The task in the Three Utilities Problem
is to draw lines connecting three utilities (water, gas and electricity)
to each of three houses without any of the connections crossing each
other.
The task appears to be simple because in the real world utility companies perform these connections daily. However our world is a three-dimensional one where utility lines are able to go under and over each other. We must make the required connections on a flat two-dimensional piece of paper. |
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| To begin, let us draw utility lines from some of the utilities to some of the houses. The figure at the right shows all three utilties connected to houses 1 and 2. |
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| If we shade the simple closed
curve that passes through house 1, water, house 2, and electricity, we
find that house 3 is inside this curve and that it has not yet been
joined to gas, which is outside the curve. According to the Jordan
Curve Theorem, it is impossible to connect them without crossing the curve.
Regardless of how such utility connections are made, a situation of this sort ALWAYS arises. We have shown that the problem has no solution. |
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| A couple of observations will
lead to another interesting result:
If two points on the same side of a simple closed curve are joined, the curve will be crossed an even number of times or it will not be crossed at all (zero crossings, the minimum possible). Each time we cross the curve and then return to the original side, we add two crossings to the minimum count. |
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| If two points on opposite sides of a simple closed curve are joined, the curve will be crossed an odd number of times. Here the mimimum number of possible crossings is one. As before, each time we cross the curve and then return to the original side, we add two crossings to the minimum count. |  |
| These observations
lead to a simple rule for finding out whether a point is inside or outside
a simple closed curve maze. Join the point to an arbitary point outside
the curve (off the drawing) and count the number of times the connection
crosses the curve. If the number of crossings
is even, the two points lie on the same side of the curve,
and the given point must lie outside the curve.
If
the number of crossings is odd, the two points lie on opposite
sides of the curve, and the given point must
lie inside the curve.
The count is not affected if the connection is tangent to or just touches the curve at any time. Some authors require the user to draw a straight line when joining the points ... but being "straight" is irrelevant in topology. Use this rule to determine whether the points A and B are inside or outside the amazing Fishy Maze (requires Adobe Acrobat Reader). Shade the inside of the simple closed curve to verify your result. |
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