Transcript zero and infinity in the non euclidean geometry

ZERO AND INFINITY IN THE NON
EUCLIDEAN GEOMETRY
V postulate
If a line segment intersects two straight lines forming
two interior angles on the same side that sum to less
than two right angles, then the two lines, if extended
indefinitely, meet on that side on which the angles sum
to less than two right angles
A geometry where the parallel postulate
cannot hold is known as a non EuclideanGeometry. Geometry that is independent of
Euclid's fifth postulate
Consequences:
• The negation of the V postulate must
mean one of two possibilities: either
there exists no parallel to a given
line, or there exists more than one
such parallel.
Difficulties of V postulate
Towards the end of the seventeenth century, several
mathematicians began to suspect that the parallel postulate might
be redundant and complicated. But It was not anyone really
doubted the validity of the postulate;
In the following centuries, numerous attempts would be made to
prove the parallel postulate in one or another of its many
equivalent versions. But All would fail.
For example, Girolamo Saccheri (1667-1733) decided to attack the
problem by the so called indirect method of proof. To prove a given
proposition, we temporarily assume that the proposition is false,
and therefore its negation is true. We then proceed to show that
this assumption leads to a contradiction;
Saccheri’s work represents a turning point because :
a. He created the possibility to theoryze de invalidity of
the fifth postulate.
b. He gave birth, involuntarily, to the effective synthesis of
the non Euclidean Geometry.
c. he introduced the idea to found a geometry based on the
non logic-contradiction and it doesn’t on the intuitive
evidence.
Gauss
• “Can there be a system of plane geometry in which, trough a point not on
a line, there is more than one line parallel to the given line?” This is the
question.
• Gauss was the first to examine the V postulate’s problem when he was 15
years old.
• He never published any of his non-Euclidean works because he knew the
mathematical precedent was against him.
• “in the theory of parallels we are even now
further than Euclid. This is a shameful part of
matematics” Gauss
Farkas and Janos Bolyai
Gauss discussed the theory of parallels with his friend, the mathematician
Farkas Bolyai. He taught his son, János Bolyai, mathematics but, despite
advising his son not to waste one hour's time on that problem of the problem
of the fifth postulate, János Bolyai did work on the problem.
• “For God’s sake, I beg you to stop your search.
We should avoid and fear the matter of the
parallels as well as we should avoid and fear
the matter of the passions. It is also able to
take all your time and your happiness,
depriving you of your good health.”
Farkas Bolyai to his son Janos
Riemannian Geometry
• Bernhard Riemann – ninenteen century
• He Looked at negation of first part of Parallel
Postulate
• “can there be a system of plane geometry in which,
through a point not on a line, there are no parallels
to the given line?
• Saccheri already found contradiction, but based on
fact that straight lines were infinite
• Riemann deduced that “extended continuously” did
not mean “infinetely long”
The main difference between spherical
geometry and Euclidean Geometry is that
instead of describing a plane as a flat
surface a plane is a sphere.
On a sphere, the sum of the angles of
a triangle is not equal to 180°. 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. The surface of a sphere
can be represented by a collection of
two dimensional maps. Therefore it is
a two dimensional manifold
Spherical Geometry
The main difference
between spherical
geometry and
Euclidean Geometry
is that instead of
describing a plane as
a flat surface a plane
is a sphere.
°
• A line is a great circle on
the sphere. A great
circle is any circle on a
sphere that has the
same center as the
sphere
• Points are exactly the
same, just on a sphere.
Hyperbolic geometry
• Hyperbolic geometry was created in the
first half of the nineteenth century. In the
midst of attempts to understand Euclid's
axiomatic basis for geometry. It is one
type of non-Euclidean geometry, that is,
a geometry that discards one of Euclid's
axioms.
In hyperbolic geometry there are at least
two distinct lines through P which do not
intersect l, so the parallel postulate is false
.
• A characteristic
property of hyperbolic
geometry is that the
angles of a triangle
add to less than a
straight angle. In the
limit as the vertices go
to infinity, there are
even ideal hyperbolic
triangles in which all
three angles are 0°.
Black holes
Einstein found in
non-Euclidean
geometry a
geometric basis for
the understanding of
physical time and
space.
BLACK HOLE
A black hole is a region of space
from which nothing, not even
light, can escape. The theory of
general relativity predicts that a
sufficiently compact mass will
deform space time to form a
black hole. Around a black hole
there is an undetectable surface
called an event horizon that
marks the point of no return. It is
called "black" because it absorbs
all the light that hits the horizon,
reflecting nothing, just like a
perfect black body in
thermodynamics.
Now we see another type of non
Euclidean Geometry, it doesn’t
use the V postulate’s negation
but it is very irregular and
different to the normal
geometry:
FRACTALS
Fractals
A fractal is "a rough or
fragmented geometric shape
that can be split into parts, each
of which is (at least
approximately) a reduced-size
copy of the whole," a property
called self-similarity.
A fractal often has the following
features:
It is too irregular to be easily described in
traditional Euclidean geometric language.
It is self-similar (at least
approximately or stochastically).
It has a Hausdorff dimension which is
greater than its topological dimension
the perimeter is infinite
It has a simple and recursive
definition.
It has a fine structure at arbitrarily
small scales.
°
Koch curve
• "It is this similarity between the whole and
its parts, even infinitesimal ones, that makes
us consider this curve of von Koch as a line
truly marvelous among all. If it were gifted
with life, it would not be possible to destroy it
without annihilating it whole, for it would be
continually reborn from the depths of its
triangles, just as life in the universe is.“
“Von koch”
Definition:
• The Koch snowflake (also known as the Koch star
and Koch island) is a mathematical curve and one
of the earliest fractal curves to have been
described.
• The Koch curve has an infinite length because
each time the steps above are performed on each
line segment of the figure there are four times as
many line segments, the length of each being
one-third the length of the segments in the
previous stage.
°
• Hence the total length increases by one
third and thus the length at step n will be
(4/3)n of the original triangle perimeter:
the fractal dimension is log 4/log 3 ≈
1.26, greater than the dimension of a line
but less than Peano's space-filling curve.
• Therefore the infinite perimeter of the
Koch triangle encloses a finite area.
Koch curve’s CONSTRUCTION
• The Koch curve can be constructed by starting
with an equilateral triangle, then recursively
altering each line segment as follows:
• divide the line segment into three segments of
equal length.
• draw an equilateral triangle that has the middle
segment from step 1 as its base and points
outward.
• remove the line segment that is the base of the
triangle from step 2.
• After one iteration of this process, the result is a
shape similar to the Star of David.
• The Koch curve is the limit approached as the
above steps are followed over and over again.
FROM:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/NonEuclidean_geometry.html
http://www.cut-the-knot.org/triangle/pythpar/Model.shtml
http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html
http://books.google.it/books?id=lXjF7JnHQoIC&pg=PA127&lpg=PA127&dq=
non+euclidean+geometry+zero+infinite&source=bl&ots=-0tIBTHzk&sig=O17QoAEqIa_bvrjxHH4aRsOlNEY&hl=it&ei=cB4TdjCAcidOuWj0YkH&sa=X&oi=book_result&ct=result&resnum=4&ved=0
CCcQ6AEwAzgK#v=onepage&q=non%20euclidean%20geometry%20zero
%20infinite&f=false
http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM28%2F
S0013091500034829a.pdf&code=cc4d7bdc83db6f845d1724b76fbc7feb
http://www.essortment.com/non-euclidean-geometry-60944.html
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