The possibility that there might be other alternative geometries different from Euclidean geometry where not met the validity of the fifth postulate was already considered in other times before Saccheri, Lambert and Gauss. One of them was spherical geometry, which can analyze the geometry of the surface of a balloon. In this geometry are called "flat" and "right" respectively, to the surface of the area and circumference of circles , appropriate terminology because in any of the geometries of "straight" line is the simplest that can joining two points and the "flat" surface is also simple. Not hard to see in this geometry two "straight" are always cut in two diametrically opposite points. This smells suspiciously like the hypothetical statement that at a point outside a given line is not possible to draw any line with which they will never cross. On the other hand, the sum of the angles of a triangle drawn on the surface of a sphere is always greater than two right angles (180 degrees). In a triangle bounded by a quarter of Ecuador terrestrial and arches of two meridians drawn to the North Pole, the sum of internal angles is 270 degrees. Since the surface has two dimensions is common to call two-dimensional geometry that studies the figures found on a particular area. The fact that there was a two-dimensional non-Euclidean geometry was not given great importance for the simple reason that the spherical geometry was studied in three-dimensional plane, which I assumed he was Euclidean, and this led to him not so much importance to non-Euclidean properties of the sphere. Ultimately, this would prove to be a glaring omission that would delay the discovery of non-Euclidean geometry as such.
spherical geometry is a special case of a more general geometry, a geometry that is based not on the surface of a sphere, but on the surface of an ellipsoid . An ellipsoid is a solid of revolution obtained by rotating an ellipse about one axis of symmetry. Understandably, this type of geometry called elliptic geometry .
The ellipsoid is based on the ellipse in two-dimensional geometry is defined as a curve such that from any point in the same amount of lines to two points within the same foci known as is a constant amount.
When the axle shafts and to b and c ellipsoid are all equal, then the ellipsoid becomes a sphere. Geometry developed
on the surface of an ellipsoid is known as elliptic geometry. Also known as Riemmaniana geometry, after the mathematician Bernhard Riemann who developed this type of geometry and I will talk more in depth.
Then there is the figure of an ellipsoid, which we imagined as a spherical ball of basketball to which we have flattened lying on the floor and applying pressure at the top to the foot (this should be added that our planet, Planet Earth is a sphere, is an ellipsoid, as it is flattened at the poles):
on the surface of the ellipsoid is drawn a triangle with vertices at the points A , B and C . In the bottom of the triangle, the two internal angles can be thought of as a 90-degree angles, as the ants would measure if we were walking on the surface of the ellipsoid, taking with us a good protractor to measure angles. The upper corner you can imagine how a small angle, say around 15 degrees. Thus, the sum of the interior angles of a triangle drawn on the surface of the ellipsoid is greater than 180 degrees . This contrasts sharply with one of the theorems of Euclidean geometry tells us that within a triangle the sum of the angles is always equal to 180 degrees. It can be shown rigorously that the corresponding theorem in elliptic geometry reads: in an ellipsoid, the sum of the angles of any triangle drawn on the surface of it will always be greater than 180 degrees . And this is not the only theorem that differs from its counterpart in Euclidean geometry. All other relevant theorems are also different . The fact that all the theorems are different is a direct consequence of that in elliptical geometry is not possible to draw a line parallel to a given line . Is a direct consequence of the curvature of space which is developing the non-Euclidean geometry.
How could build the equivalent of Euclidean geometry, axiomatic, on the surface of a sphere? We should start by defining the equivalent of a straight line on the surface of the sphere. In plane geometry, a line is defined as the smallest distance between two points. If you look good on the surface of the sphere, where we can only draw arcs between two points we can draw many bows, but all of them only one will be the shortest those points. It turns out that this arc is part of a circle of the sphere, defined as the one obtained by the field through its center with a plane that cuts into two equal parts. Earth meridians are great circles clearly. The following figure best illustrates the above:
In this case, the ball straight to and b are circles drawn on the surface of the sphere, while line c clearly not a straight and that is not a circle. The concept of the shortest distance between two points on the surface of a sphere can also be extended to an ellipsoid, which given two points on its surface can bind them with an arc that is part of a plane that cuts the ellipsoid passing through the center of symmetry. These arcs with minimum distances between two points, both the sphere and ellipsoid are called geodesic . An interesting feature of spherical geometry (elliptical) is that, as said earlier, not only is not possible to draw two lines that never cross, but even all straight lines in a spherical geometry (elliptical) will meet twice , as in the case of spherical straight to and b in the picture above. Defined
our "straight" line in elliptic geometry, we can start writing the axioms of our non-Euclidean geometry:
axioms of geometry ELIPTICA
Postulate 1: Between two different points may be drawn one and only one line.
Postulate 2:
Postulate 3: You can draw in a circle using elliptical surface any point as center, which can have any radius (this radius is an arc!). Postulate 4
: All right angles are equal.
Postulate 5: Through a point P , that does not belong to a line can not be another line that is parallel to the given line, as all lines drawn outside the given line will find it eventually .
Postulate 6: Two things are equal to one third equal.
Postulate 7: If we add same same, the resulting sums are also equal.
Postulate 8: If we subtract them equal equal, the remnant will be equal.
Postulate 9: Figures that can be matched are equal.
Postulate 10: The whole is greater than the sum of its parts.
Note that Postulate 2 we left temporarily empty while we decide what we will get there. The problem here is that while a line on the Euclidean plane can be extended indefinitely in both directions in the ecliptic plane can not do that without the finish line back to the point where he left after giving "full circle." This is the real reason why contradictions Saccheri quickly came when he investigated the "third" hypothesis on the assumption that within the ring, the Saccheri quadrilateral, the sum of internal angles is greater than 180 degrees. If I had changed here not only the fifth postulate, but the second, then the "third hypothesis" would also come with contradictions, and would have discovered the two non-Euclidean geometries alternative to plane geometry.
Once we have our axioms for Euclidean geometry, we can begin to apply conjecture and prove theorems. All you have to do is proceed in a similar way as we do in Euclidean geometry, and who received a good geometry course at an institution of secondary education, this does not be no problem.
For those who have been accustomed to thinking of their lives in two parallel lines as two lines equidistant , may find it difficult to accept the idea of \u200b\u200bthinking about the possibility of an alternative which will be impossible to keep two parallel lines are to be found sooner or later no matter how "parallel" to be. This can be seen best in the construction of a set of two lines perpendicular to them is as parallel lines on the basis that they will always be equidistant:
In this figure, we have a line AB , which by geometric construction has been drawn a parallel CD passing point p ', with an additional perpendicular from point p the line until the line AB CD. The fact that the line through the points and p p ' is perpendicular to both lines and AB CD is highlighted by the fact that it forms two right angles perpendicular exact (90 degrees ) with the two lines intersect with them. It is always possible to draw a perpendicular and in any geometry. But along the line AB We can draw another perpendicular through a different point as the point q , and measuring the distance between the points and p p ', we mark a point on this perpendicular to keep the same distance in first perpendicular between points and p p ', which serves to extend the parallel line CD right. After that, look elsewhere r and repeat the procedure, further prolonging CD straight to the right, so that will always remain equidistant from the line AB . We can repeat the procedure to infinity , And always will be two equidistant lines. It is also clear that we can only draw a line parallel to one another so that never cross and always remain equidistant, checking the "obvious truth" of the fifth postulate.
The problem with this reasoning, at least in spherical geometry, is that if we build on a globe two lines will always be equidistant, then at least one of these lines will not be a circle , and will not be a line on how you defined for a spherical geometry. This is precisely what happens in the figure showing circles traced on the surface of a sphere. In this figure, to is a circle and therefore a straight ball, but c not. All are drawn perpendicular to the ball straight to line perpendicular c be equidistant from the line, but as c is not a straight ball, it has no parallel in the sense spherical a word . At this point, a person with doubts could argue that if we Salim of the surface of the sphere and go to three dimensions, then it is always possible to draw two straight lines remain equidistant all the time, to infinity , without finding, as requested by the fifth postulate, which sounds logical until you meditate on the fact that this assumes that the three-dimensional space is Euclidean to infinity, an assumption which is extremely risky because nobody has been there. What assures us that, at enormous distances, three-dimensional space does not exhibit a spatial curvature as exhibiting the surface of the sphere? To compound matters further, we have to consider the disturbing fact that in three dimensions, we can draw two lines right now will always be equidistant and never cross, which, however not be parallel in our sense "intuitive" in the word. These are two curved lines that are between a relationship like that between the two strands of a braid , or that between the two strands of a helical pair DNA segment. You can not use a fancy "intuitive" what we mean by parallel lines, in the sense of Euclidean geometry, to "prove" so that space entire universe obeys this geometry.
As anticipated at the beginning, spherical geometry (elliptical) can always construct a triangle using arcs of circles maximum, as shown in the following drawing in which has drawn a spherical triangle:
continuing
Note that the arcs of circles that define a spherical triangle so den full circle, always form another triangle on the other side of the field, something that certainly never happen in Euclidean geometry. In the drawing, the triangle A'B'C 'is called the antipode triangle ABC, and that can be obtained reflecting the original triangle through the center of the sphere. By symmetry, both spherical triangles must have the same area. Defined a spherical triangle in this way, then we have a spherical geometry theorem easy to prove compared to its Euclidean counterpart:
Euclidean geometry: "In any triangle, the sum of the interior angles equals two right angles (180 degrees). "
elliptic geometry: In any triangle, the sum of internal angles is greater two right angles. "Charting a
Saccheri quadrilateral whose base is aligned with the Earth's Ecuador, and raising her two lateral sides along the meridians land (so that the sides of the square are at right angles to the Earth Ecuador), we can see visually without difficulty as follows:
Euclidean geometry: "The sum of the angles of a quadrilateral equal to four right angles (360 degrees). "
elliptic geometry: "The sum of the angles of a Saccheri quadrilateral layout on the surface of a sphere is less four right angles (360 degrees)."
Euclidean geometry: "For every circle of radius r length the circumference is equal to π 2 r. "
elliptic geometry: "For every circle of radius r the length of the circumference is greater that 2 π r ."
If we built a Saccheri quadrilateral on the surface of the celestial sphere by matching the base of the ring with Ecuador (in which case the base will be the full equivalent of a straight line), the inner corners of the upper corners quadrilateral summit will be obtuse. This is equivalent to the hypothesis that was discarded by Sacchieri, and it was rejected not only by the fact that this hypothesis was in direct contradiction with Euclid's fifth postulate that one can trace a parallel to a given line through a point external to it, but because when they are extended in parallel both directions to infinity cross sooner or later, also contradicting the second postulate of Euclidean geometry. If Saccheri, rather than discard the possibility that your ring will also support obtuse angles at the summit, had proceeded to the modification of the second axiom of plane geometry of Euclid, had created before his astonished eyes another different geometry, spherical geometry, that is that we're seeing right here. As in the geometry developed by Saccheri, two lines "parallel" will have a line to be a common perpendicular both. In this region of space, the picture begins to emerge on Saccheri's geometry is two perpendicular parallel with a common and are showing it off like this:
Compare the situation we found for the geometry Saccheriana. With this, we are in a position to make an overall comparison of the three possible geometries: the Euclidean geometry, is only possible to draw a parallel line outside a given line passing through a point outside it (with a common perpendicular to both) without the two ever intersect, while in spherical geometry is not possible to draw any "parallel" a given line (also with a common perpendicular to both) that passes through a point outside it unable to prevent cross terms sooner or later, while in the hyperbolic geometry Saccheriana or possible as we saw in another post the map an infinite number of parallel to a given line from a point outside it (all of them coinciding with common perpendicular to the line given in section external reference). This covers all possible cases.
A dramatic difference between Euclidean geometry and hyperbolic geometry provides us Pythagorean theorem tells us that in any right triangle the square of the hypotenuse c equals the sum of the squares of the legs to and b :
c ² = a ² + c ²
The proof of this theorem makes use of Euclid's fifth postulate. But as the fifth postulate of Euclid is only valid for a flat geometry, it is clear that the Pythagorean theorem turns out to be a false statement in spherical geometry (elliptical). The equivalent spherical geometry of the Pythagorean theorem is the following relationship where R is the radius of the sphere
tells us that: In any right triangle drawn on the surface a sphere with radius R , the cosine of the ratio of the hypotenuse c and sphere radius R is the product of the cosines of the ratios between the legs and the radius of the sphere . Here the cosine has the same definition as that used in trigonometry. Using a mathematical series for evaluation cosine function as follows:
easy to demonstrate that to a large value of R (as the radius of the Earth), this relationship becomes very little margin for error in arithmetic the Pythagorean theorem, which confirms our everyday experience.
Moreover, to obtain the area of \u200b\u200ba triangle drawn on the surface of a sphere, an important theorem in spherical geometry is the theorem Girard, which is not difficult to prove, and which tells us "If R is the radius of a sphere, and to , and b c are the angles of a triangle (measured in radians) whose sides are segments of great circles of the sphere, S intone area of \u200b\u200bthe triangle is given by the relationship:
S = (a + b + c - π ) R ² "
To give my readers an idea on how to carry out the proof of a theorem in geometry non-Euclidean if it is a spherical geometry , below present the proof of the theorem of Girard. To do this, first we will set the spherical angle as the angle formed by two great circles intersect at one point. In the following figure we have two spherical angles that form the vertices A and A ' (which generate two areas on opposite sides of the field):
Similarly, we can form spherical angles at the corners and B B ':
and vertices and C C' :
The total area the surface of the sphere is given by the sum of the areas generated by the spherical angles (two "lunes" on each side), to which you have to subtract the areas of spherical triangles at the corners and ABC A'B'C ' (four times the area of \u200b\u200bthe spherical triangle ABC ) to eliminate duplication happens to have added the areas generated by the three spherical angles:
S = area A + area A' + area B + area B '+ area C + area C'
- 4 (area of \u200b\u200bthe spherical triangle ABC)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
S = 2 (area A) + 2 (area B) - 2 (area B)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
It is obvious that the area of \u200b\u200ba "lune" determined by an angle sphere is proportional to the spherical angle. If we know beforehand that the total area of \u200b\u200ba spherical surface is given by
S = 4 π R ²
then the area determined by a spherical angle p (where we assume that this angle is measured in radians) must be 2pR ² (thus, when the spherical angle has a magnitude of 2 π radians or 360 degrees, completely covering the surface of the sphere, the area formed by the spherical angle is 4 π R ² the total area of \u200b\u200bthe sphere).
Thus, the above equation can be rewritten as follows:
4 π R ² = 2 (area A) + 2 (area B) - 2 (area B)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
4 π R ² = 2 (2AR ²) + 2 (2BR ²) + 2 (2cr ²)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
4 π R ² = 4AR ² + 4br ² + 4CR ²
- 4 (area of \u200b\u200bthe spherical triangle ABC)
- 4 (area of \u200b\u200bthe spherical triangle ABC)
area of \u200b\u200bthe spherical triangle ABC = S
S = (a + b + c - π ) R ²
S = (a + b + c - π ) R ²
With this theorem we can deduce important results. One is as follows: "If two triangles (spherical) have internal angles equal, then their areas will also be equal." This leads to an important conclusion: Unlike as in Euclidean geometry, on the surface of a sphere there are no similar triangles by having equal internal angles, triangles necessarily equal.
And regarding the ring drawn on the surface of a sphere, using a procedure similar to that used to demonstrate Girard's theorem can be shown that for a spherical quadrilateral with internal angles to , b, c and d (measured in radians) S your area will be given by the relationship:
S = (a + b + c + d - 2 π ) R ²
Also with this relationship we obtain the important result that on the surface of a sphere there are no quadrilaterals similar, if they have all four corners for the same then they should have the same area, and to be the same size must be equal. The latter relationship can not be used even as an approximation to the area of \u200b\u200ba square drawn on a plane for the case an area as a radio R very large as the Earth, because in that case as the radius R is huge sum of internal angles of the quadrilateral (a + b + c + d) is approaching the value of π 2 and the difference will become very small, and as we all know the result of the huge quantity by a tiny amount can give us something in between, for example: 4500000000000000000
by
0.000000000000000015 produces 67.5 ("square meters land for a garden boy?).
The simple formula ( base by height) for the area of \u200b\u200ba Euclidean ring (plane) can be derived as a special case of the formula for the area of \u200b\u200ba spherical quadrilateral (Girard theorem) even when the radius of the sphere radius approaches infinity, the same way that the formula for the area of \u200b\u200ba spherical quadrilateral can be derived as a special case of the formula for a Euclidean ring. It is precisely this kind of difficulties which Saccheri was tempted to refuse to elliptic geometry as a geometry logically consistent.
Since the only difference between Euclidean geometry and spherical geometry (Elliptical) is the parallel postulate, the fifth postulate, this means that any demonstration Euclidean geometry that does not make use of the fifth postulate is a proof as valid, with the same result in spherical geometry (elliptical) . This allows us to incorporate into almost free of spherical geometry (elliptical) the proofs of many theorems of Euclidean geometry, some of them laborious. Similarly, this means that all theorems in Euclidean geometry, making use of the fifth postulate will be false in spherical geometry (elliptical) .
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