Comparing Geometries
In mathematics, there are many broad and partially undiscovered areas of
learning that are parts of many school curriculums. One of those topics is
geometry. In schools, mainly Euclidean is the only geometry taught, but there
are two other types as well. Geometry can be split into Euclidean, Spherical and
Hyperbolic. Euclidean geometry was the basis of all geometry as it is known
today. It was founded by Euclid and followed his five postulates, a
straight line
segment can
be drawn joining any two points, given any straight line
segment,
a circle can
be drawn having the segment as radius and
one endpoint as center, all right
angles are congruent, and
if two lines are drawn which intersect a
third in such a way that the sum of the inner angles on one side is less than
two right
angles,
then the two lines inevitably must intersect each
other on that side if extended far enough. The most famous of those five
postulates was the last postulate. This made many mathematicians think about
whether they could find a counterexample for it, paving the way for research
into spherical and hyperbolic geometries. These three geometries contrast the
similarities and differences of shapes, and are the basis for how any of them
work on any plane.
The first shape that has contrasting features in each of the geometries
is a triangle. Triangles are a type of polygon. Over all three geometries there
are a few points that are the same or similar. All triangles have three sides
and three angles, all three sides are the shortest distance between the two
points and they do not intersect except at the corners to make the angles, and
finally all three types can have two congruent triangles.
In Euclidean geometry, the angle sum is equal to 180°. This differs from
spherical, in which the angle sum is greater than 180° and different from
hyperbolic, which is triangles with an angle sum less than 180°. Also, in
spherical geometry there can be up to three right or obtuse angles, but in
Euclidean there is a maximum of one obtuse or right angle. Finally, in
hyperbolic geometry, as the angle measure gets smaller, the triangle gets
larger. This is different from spherical geometry, where the triangle gets
larger as the angles get larger. Similarly in Euclidean geometry the area is not
affected by the angle measures. Also, polygons have many similar and differing
characteristics across the three geometries. In both hyperbolic and Euclidean,
polygons have at least three sides, but in spherical, they only need at least
two sides. All three geometries believe polygons have any angle measure.
Polygons and triangles are two of the many shapes that reflect the differences
between Euclidean, spherical and hyperbolic geometry.
In addition to triangles and polygons, circles have many characteristics
that are shared and many that are not shared. Some of the shared characteristics
are all circles consist of one curved line that stays an equal distance from a
center point and they all can be made. In spherical geometry, the largest circle
is the great circle, then as the radius becomes larger than the great circle,
the area of the triangle decreases. This does not apply to Euclidean or
hyperbolic because they have an infinite amount of space for the circle’s radius
and area to be as large as it wants. Another difference is on a sphere, there is
no difference between the finite area and the infinite area because both are
circles. Finally, in all three geometries, it is possible to have more than one
congruent circle on a single plane. Circles are shapes that follow a specific
set of requirements and do not have any straight lines. They are also one of the
few shapes that can be formed in any geometry as long as curved lines are
permitted.
As stated prior, the fifth postulate of Euclid is the reason
mathematicians searched for and found the other two types of geometry. The fifth
postulate states that if there is a line and a point, there is only one line
going through the point that is parallel to the line. In spherical geometry,
there are no parallel lines because all lines are great circles which means they
go through the center, meaning they all cross. In hyperbolic geometry, there are
an infinite number of parallel lines crossing through a point that is parallel
to another line. This is because the raised planes allow for the lines to curve
away from the original line. This means that the lines are not tradition
parallel lines because they do not remain the same distance away from each
other. Although it seemed impossible, mathematicians found a counter example
that proved Euclid’s fifth postulate incorrect.
In Euclidean, spherical and hyperbolic geometries, there are many
characteristics of shapes and lines that show their differences and
similarities. Across the world, people try to disprove statements and postulates
as incorrect or correct. Most of the time this results in a new type of thinking
that spreads rapidly across the world. In this case, the new thinking was the
disproving of Euclid’s fifth postulate that spread the belief that other
geometries were in the world and they held the answers about 3D planes that
Euclidean could not answer.