2)
Any straight line can be extended infinitely.
3)
Given any straight-line segment, a circle can be drawn having the segment as
the radius and one endpoint as centre.
4)
All right angles are congruent (equal).
5)
Parallel postulate:
Through a point outside a given line one
and only one line
parallel to the first line can be drawn.
Riemann geometry differs from Euclid
ʼ
s geometry because of Euclid's 2nd and 5th
postulates.
Riemann geometry states that straight lines are bounded and there are
no parallel lines. To visualize this idea imagine doing geometry on a sphere instead
of a plane- this is after all, what mapmakers do every day. All lines of longitudes are
parallel on a plane sheet of paper. Yet they meet at poles when we consider the
surface to be spherical.
Another difference is that in Euclid
ʼ
s geometry the sum of the angles of a triangle is
equal to 180°. In Riemann Geometry the sum of the triangles is greater than 180°.
This allows lines to cross the same line at 90° and cross each other at a circle's pole.
We again return to the example of the sphere. All longitude lines cross the equator at
90° and meet at the north and south poles.

8
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a
Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small
triangle
on
the
face
of
the
earth,
the
sum
of
the
angles
is
very
nearly
180
°.(
)
Riemann geometry also played a significant role in Einstein's Theory of Relativity. If
outer space were a black hole or huge circle, space would be curved. This would
support Einstein's Theory of Relativity [
].
The Riemann Hypothesis
The most important question in number theory and possibly, in the whole of
mathematics is the
Riemann Hypothesis
. It is included in
David Hilbert
ʼ
s
list of the
most challenging problems for the twentieth century mathematicians. David Hilbert
said
If I were to awaken after having slept for 1000 years, my first question would be: Has
the Riemann Hypothesis been proven?
The Riemann Hypothesis states that the nontrivial roots of the Riemann zeta
function
5
defined on the complex plane C
6
all have real part
1
2
. The line Re (z)
5
Riemann zeta function: First discovered by Euler, the simplified equation for zeta
function is
ζ
(x) = 1+(1/2)
x
+(1/3)
x
+(1/4)
x
+…………

9
equaling
1
2
is called the critical line. In plain English, all of the complex zeroes of the
zeta function have real part
1
2
. The Riemann hypothesis has thus far resisted all
attempts to prove it. This still unproven hypothesis remains the most important and
beguiling, unresolved problem in mathematics. Proof of the Riemann hypothesis is
number 8 of Hilbert
ʼ
s Problems [
]
and number 1 of Smale
ʼ
s Problems [
.

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- Fall '09
- Math, Geometry, Riemann, Bernhard Riemann, Carl Friedrich Gauss, Georg Friedrich Bernhard Riemann