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NON-EUCLIDEAN
GEOMETRIES
Austin Mahlum
SPACE MAP
 4 Common Axioms
 Euclidean Geometry
 Saccheri Quadrilaterals
 Birth of new geometries
 Logical Consistency
 Elliptic Geometry
 Hyperbolic Geometry
 Why should we study different geometries?
4 COMMON AXIOMS
 1. To draw a straight line from any point to any point.
 2. To produce [extend] a finite straight line continuously in a
straight line.
 3. To describe a circle with any center and distance [radius].
 4. That all right angles are equal to one another.
EUCLIDEAN GEOMETRY
 Also known as parabolic geometry
 What we are taught in high school
 5th postulate: Given any straight line and a point not on it, there “exists
one and only one straight line which passes” through that point and never
intersects the first line, no matter how far they are extended.
 Interior angle sum of a triangle is = 180 degrees. This is better known
as the triangle postulate and is equivalent to the 5th postulate.
SACCHERI
QUADRILATERALS
 Girolamo Saccheri(1667-1733)- Attempts to use proof by
contradiction to prove the triangle postulate, ends up with the
Saccheri quadrilaterals.
BIRTH OF NEW
GEOMETRIES
 Carl Gauss (1777-1855): Never published any formal work on the
matter, but had private letters that showed he was working on it.
 Nikolai Lobachevsky (1792-1856): In 1829 published “On the
Principles of Geometry”.
 Janos Bolyai (1802-1860): In 1832 published “The Science of Absolute
Space”.
 Georg Riemann (1826-1866): In 1867 his work “On the Hypotheses
which lie at the Bases of Geometry” was published.
LOGICAL CONSISTENCY
 Eugenio Beltrami (1835-1899): In 1868 published two memoirs
that created models that were developed using Euclidean space as
opposed to axiomatic fashion; however, these models worked with
Non-Euclidean geometries. As a result, if these new geometries were
not logically consistent, then Euclidean geometry was not consistent.
VISUAL AID
ELLIPTIC GEOMETRY
 Also known as Riemannian geometry
 Simplest model is a sphere
 5th postulate: Given any straight line and a point not on it, there
“exists no straight line which passes” through that point and never
intersects the first line, no matter how far they are extended.
ELLIPTIC GEOMETRY
 Theorem: A lune whose corner angle is Θ radians has an area of
2Θ𝑟 2 .
HYPERBOLIC GEOMETRY
 Also known as Lobachevsky-Bolyai-Gauss geoemetry
 Simplest model is a pseudosphere
 5th postulate: Given any straight line and a point not on it, there
“exists at least two straight lines which pass” through that point and
never intersects the first line, no matter how far they are extended.
HYPERBOLIC GEOMETRY
HYPERBOLIC GEOMETRY
 Proposition 23: To construct a rectilinear angle equal to a given rectilinear angle
on a given straight line and at a point on it.
 Theorem 1-3: The angle sum of a triangle does not exceed 180 degrees.
 Definition 1-1: A lambert quadrilateral is a quadrilateral with 3 right angles.
 Theorem 1-6: In hyperbolic geometry, the fourth angle of a Lambert
quadrilateral is acute, and each side adjacent to the acute angle is longer than the
opposite side.
HYPERBOLIC GEOMETRY
 Theorem 1-8: Given two lines, if there exists a transversal which cuts the lines
so as to form equal alternate interior angles or corresponding angles, then the lines
are parallel with a common perpendicular.
 Theorem 1-9: If two lines have a common perpendicular, there exists
transversals, other than the perpendicular, which cut the lines so as to form equal
alternate interior angles (or equal corresponding angles). Moreover, the only
transversals with this property are those which go through the point on that
perpendicular which is midway between the lines.
WHY STUDY DIFFERENT
GEOMETRIES?
 We do not know if our universe is shaped with Euclidean
geometry, in fact Albert Einstein’s General Theory of Relativity and
modern day String Theory depend on non-euclidean geometry to
function.
SOURCES
 http://www-history.mcs.st-and.ac.uk/HistTopics/NonEuclidean_geometry.html
 http://jwilson.coe.uga.edu/MATH7200/NonEuclideanCompanio
n/NonEuclideanCompanion.pdf
 http://lamington.wordpress.com/2010/04/10/hyperbolicgeometry-notes-2-triangles-and-gauss-bonnet/