We learned Euclid’s
five postulates in high school geometry. While I enjoyed the topic, it wasn’t
until recently how much I have grown to appreciate how powerful these five
simple statements are. Their power comes from the fact that Euclidian Geometry,
a system of hundreds of proven geometric theorems, can be built up these five
postulates.
The five postulates
are:
1.
A
straight segment (line) can be drawn between any two points.
2.
A segment
can be extended indefinitely in either direction.
3.
A circle
can be constructed from a center point and its radius (a line segment).
4.
All
right angles are equal.
5.
Given
a line and a point not on the line, there exists exactly one line through the
point that is parallel to the initial line.
Construction of
objects in Euclidian Geometry and associated proofs are completed using only a
compass and an unmarked straightedge.
The fifth
postulate is also expressed as “if
a straight line falling on two straight lines make the interior angles on the
same side less than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which the angles are less than two right
angles.”
The fifth postulate is key
to making Euclidian geometry plane geometry. Consider if one changes the fifth
postulate to “Given a line and
a point not on the line, there exists no line through the point that is
parallel to the initial line.” In this case, we would have elliptical geometry.
If one changes the
fifth postulate to” Given a line and a point not on the line, there exists at
least two lines through the point that is parallel to the initial line,” we
have hyperbolic
geometry.
An interesting
difference in these alternative geometries is the sum of angles of triangle. In
Euclidian geometry, the sum is 180 degrees. In elliptical geometry, the sum is
greater than 180 and in hyperbolic, the sum is less than 180.
Other posts mentioning Euclid:
Math Vacation: Abraham Lincoln - A President Trained by Euclid (jamesmacmath.blogspot.com)
Math Vacation: Perfect Number Generator (jamesmacmath.blogspot.com)
Math Vacation: Book Review: A Mathematician's Apology (jamesmacmath.blogspot.com)
Math Vacation: Dilcue's Pizza (The Lazy Caterer Sequence) (jamesmacmath.blogspot.com)
Math Vacation: Interior Angles of a Triangle (jamesmacmath.blogspot.com)
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