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 Euclidean-Geometry. 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/Non-Euclidean_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=c-B4TdjCAcidOuWj0YkH&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCcQ6AEwAzgK#v=onepage&q=non%20euclidean%20geometry%20zero%20infinite&f=false http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM28%2FS0013091500034829a.pdf&code=cc4d7bdc83db6f845d1724b76fbc7feb http://www.essortment.com/non-euclidean-geometry-60944.html C:\Documents and Settings\utente\Desktop\p4\comenius\geometrie-non-euclidee-lobacevskij-bolyai-e-la-diffusione-delle-nuove-teorie.html