** 2.7.3 Elliptic Parallel Postulate
***Printout*

*The postulate on parallels...was in antiquity
the final solution of a problem that must have preoccupied Greek mathematics for
a long period before Euclid.
—*

** Elliptic Parallel Postulate.
**Any two lines intersect in at least one point.

An
important note is how elliptic geometry differs in an important way from either
Euclidean geometry or hyperbolic geometry. Whereas, Euclidean geometry and hyperbolic
geometry are neutral geometries with the addition of a parallel postulate,
elliptic geometry cannot be a neutral geometry due to
Theorem 2.14, which stated
that parallel lines exist in a neutral geometry. Hence, the Elliptic Parallel
Postulate is
inconsistent with the axioms of a neutral geometry. Elliptic
geometry requires a different set of axioms for the axiomatic system to be
consistent and contain an elliptic parallel postulate.

*
*Georg Friedrich Bernhard Riemann (1826–1866) was
the first to recognize that the geometry on the surface of a sphere, spherical
geometry, is a type of non-Euclidean geometry. This is the reason we name the
spherical model for elliptic geometry after him, the
Riemann Sphere. (To help with the visualization of the concepts in this
section, use a ball or a globe with rubber bands or string.) Click here
to download
Spherical Easel
a java exploration of the Riemann Sphere model. In a spherical
model:

- Two lines, which are great circles, intersect
in two points called
*poles*or*antipodal points*. - A line is unbounded, by this it is meant that a line has no endpoints. A great circle has no beginning and no end.
- A line has finite length.
- The concept of betweenness of points does not make sense. What does it mean for a point to be between two other points on a line (great circle)?

From these properties of a sphere, we see that
in order to formulate a consistent axiomatic system, several of the axioms from a
neutral geometry need to be dropped or modified, whether using either Hilbert's
or Birkhoff's axioms. The incidence axiom that *"any two points determine a
unique line,"* needs to be modified to read *"any two points determine at
least one line."* Hilbert's Axioms of Order (betweenness of points) may be
replaced with axioms of separation that give the properties of how points of a
line separate each other. (For a listing of separation axioms see *Euclidean
and Non-Euclidean Geometries Development and History* by
Greenberg.) With these modifications made to the
axiom system, the Elliptic Parallel Postulate may be added to form a consistent
system. Often spherical geometry is called *double
elliptic* *geometry*, since two
distinct lines intersect in two points.

One problem with the spherical geometry model is
that two lines intersect in more than one point. *
*Felix Klein (1849–1925)
modified the model by identifying each pair of antipodal points as a single
point, see the Modified Riemann Sphere. With this
model, the axiom that any two points determine a unique line is satisfied.
Often
an elliptic geometry that satisfies this axiom is called a
*single elliptic geometry*. Note that with this model, a line no
longer separates the plane into distinct half-planes, due to the association of
antipodal points as a single point.

Klein formulated another model for elliptic geometry through the use of a
circle. The model is similar to the Poincaré Disk. Given a Euclidean circle, a
point in the model is of two types: a point in the interior of the Euclidean
circle or a point formed by the identification of two antipodal points which are
the endpoints of a diameter of the Euclidean circle. The lines are of two types:
diameters of the Euclidean circle or arcs of Euclidean circles that intersect
the given Euclidean circle at the endpoints of diameters of the given circle.
The model on the left illustrates four lines, two of each type. The model can be
viewed as taking the Modified Riemann Sphere and flattening onto a Euclidean
plane. Click here for a *
*javasketchpad
construction that uses the Klein model.

** Exercise 2.75.** In the
Riemann Sphere, what properties are true about all lines perpendicular to a
given line?

** Exercise 2.76. **How
does a Möbius strip relate to the Modified Riemann Sphere?

** Exercise 2.77. **
Describe

** Exercise 2.78. **
Find an upper bound for the sum of the measures of the angles of a triangle in
the Riemann Sphere.

** Exercise 2.79.** Use a
ball to represent the Riemann Sphere, construct a Saccheri quadrilateral on the
ball. Are the summit angles acute, right, or obtuse? Is the length of the summit
more or less than the length of the base?

*Non-Euclidean space is the false invention of
demons, who gladly furnish the dark understanding of the non-Euclideans with
false knowledge... The non-Euclideans, like the ancient sophists, seem unaware
that their understandings have become obscured by the promptings of the evil
spirits.
—*

2.7.2 Hyperbolic Parallel Postulate2.8 Euclidean, Hyperbolic, Elliptic Geometries |

© Copyright 2005, 2006 - Timothy Peil |