Euclidean Geometry
(the high school geometry we all know
and love) is the study of geometry based on definitions, undefined terms
(point, line and plane) and the assumptions of the mathematician Euclid
(330 B.C.)
Euclid's text Elements was the first systematic
discussion of geometry. While many of Euclid's
findings had been previously stated by earlier Greek
mathematicians, Euclid is credited with developing the
first comprehensive deductive system. Euclid's
approach to geometry consisted of proving all theorems
from a finite number of postulates (axioms).
Euclidean Geometry is the study of flat space.
We can easily illustrate these geometrical concepts by drawing on a flat
piece of paper or chalkboard. In flat space, we know such concepts
as:
 the shortest distance
between two points is one unique straight line.

 the sum of the angles in any triangle
equals 180 degrees.



 the concept of perpendicular to a line can be
illustrated as seen in the picture at the right.


In his text, Euclid stated his fifth postulate, the famous parallel postulate,
in the following manner:
If a straight line crossing
two straight lines makes the
interior angles on the same side
less than two right angles, the two
straight lines, if extended
indefinitely, meet on that side on
which are the angles less than the
two right angles.

Today, we know the parallel postulate
as simply stating:
Through a point not on a line, there is no more than
one line parallel to the line. 
The concepts in Euclid's geometry remained unchallenged
until the early 19th century. At that time, other
forms of geometry started to emerge, called
nonEuclidean geometries. It was no longer assumed
that Euclid's geometry could be used to describe all
physical space.
nonEuclidean geometries:
are any forms of geometry that contain a postulate
(axiom) which is equivalent to
the negation of the Euclidean parallel
postulate.
Examples:
1. Riemannian Geometry
(also called elliptic
geometry or spherical geometry): A nonEuclidean geometry using as its
parallel postulate any statement equivalent to the
following:
If l is any line and P is any point not on
l , then there
are no lines through P
that are parallel to l . 
Riemannian Geometry is named for the German mathematician, Bernhard
Riemann, who in 1889 rediscovered the work of Girolamo Saccheri (Italian)
showing certain flaws in Euclidean Geometry.

Riemannian Geometry is the study of
curved surfaces.
Consider what would happen if instead of
working on the Euclidean flat piece of paper, you work
on a curved surface, such as a sphere.
The study of Riemannian Geometry has a direct connection
to our daily existence since we live on a curved surface
called planet Earth.

What effect does working on a sphere, or a
curved space, have
on what we think of as geometrical truths?
 In
curved space, the sum of the angles of any triangle is
now always greater than 180°.
 On a sphere, there are no straight lines.
As soon as you start to draw a straight line, it curves
on the sphere.


In curved space, the shortest distance between any two
points (called a geodesic) is not
unique. For example, there are many geodesics
between the north and south poles of the Earth (lines of
longitude) that are not parallel since they
intersect at the poles.


 In curved space,
the concept of
perpendicular to a line can be
illustrated as seen in the picture
at the right.



2. Hyperbolic Geometry
(also called saddle
geometry or Lobachevskian geometry): A nonEuclidean geometry using as its parallel postulate any statement equivalent to the
following:
If
l is any line and P is any point not on
l , then there exists
at least two lines through P that are parallel to l
. 
Lobachevskian Geometry is named for the
Russian mathematician, Nicholas Lobachevsky, who, like
Riemann, furthered the studies of nonEuclidean
Geometry.

Hyperbolic Geometry is the study of a
saddle shaped space.
Consider what would happen if instead of
working on the Euclidean flat piece of paper, you work
on a curved surface shaped like the outer surface of a
saddle or a Pringle's potato chip.


Unlike Riemannian Geometry, it is more difficult to see
practical applications of Hyperbolic Geometry.
Hyperbolic geometry does, however, have
applications to certain areas of science
such as the orbit prediction of objects within
intense gradational fields, space travel
and astronomy. Einstein stated that space is
curved and his general theory of relativity
uses hyperbolic geometry.

What effect does working on a saddle
shaped surface have
on what we think of as geometrical truths?
 In hyperbolic geometry, the
sum of the angles of a triangle is less than 180°.
 In
hyperbolic geometry, triangles with the same angles have the same areas.

 There are no similar triangles in hyperbolic geometry.

 In
hyperbolic space, the concept of
perpendicular to a line can be
illustrated as seen in the picture
at the right.


 Lines
can be drawn in hyperbolic space that
are parallel (do not intersect).
Actually, many lines can be drawn
parallel to a given line through a given
point.

Graphically speaking, the hyperbolic saddle shape
is called a hyperbolic paraboloid,
as seen at the right.



It has been said that
some
of the works of artist M. C. Escher
illustrate hyperbolic geometry. In
his work Circle Limit III (follow
the link below), the effect of a hyperbolic
space's negative curve on the sum of the angles
in a triangle can be seen.
Escher's print illustrates a model
devised by French mathematician Henri
Poincare for visualizing the theorems of
hyperbolic geometry, the orthogonal
circle.
M. C. Escher web site:
http://www.mcescher.com
Choose Galleries: Recognition and
Success 19551972: Circle Limit
III
