Saturday, September 29, 2007

What Is Spcentral.org

Chapter IV: Building a New Geometry




Both spherical geometry and elliptic geometry are based on a fundamental basis: the parallel postulate of Euclid that "at a point outside a given line is only possible to draw a parallel to this line is modified to assume that at a point outside a given line can not draw a parallel to that line. However, there is a third possibility. This possibility is to assume that a point outside a given line can draw a parallel over to a line, which to many may seem surprising when first exposed to a geometry in which take this assertion as true. Interestingly, if we take this assumption as valid, using it to replace Euclid's fifth postulate, we can build another geometry known as hyperbolic geometry , perfectly consistent, within which we can deduce theorems as valid as those found in Euclidean geometry . This type of geometry, within which compliance with this alternative assumption, also known as Lobachevsky geometry, after the Russian mathematician Nicolai Ivanovich Lobachevsky (1792-1856) who discovered and developed quite a geometry of this type. Unfortunately, no Lobachevsky graphic conceived somehow somehow allowed to represent geometric objects in hyperbolic geometry, thereby displaying the theorems derived using only the pure algebraic symbolism became virtually impossible. This display only became possible with a very ingenious invention discussed below.

Let's start with the basic question: How can we build this new world, within which to comply with the simile of the other axioms of Euclidean geometry, and which can also carry out the construction of geometric figures as " triangles and "Quadrilateral" and where they can perform theorem proving just as rigorous and axiomatic as is done in Euclidean geometry?

The answer to this question is obtained by resorting to something in mathematics is called a transformation . Let's consider some type of transformation, investment plane in a circle , considering this investment as the generalization of a reflection about a straight line. In the following figures, we have first to the line L and a point P outside it:



The point P will be "reflected" below the line L putting a point P ' below this line have the same distance to line L as the distance that the line holds up to point P (usually taken to mean the projection of the image of a point, a segment, or even of a figure with the same letter but with the distinction of premiums, or adding the symbol ' to the letter.) Thus, each point, line, or even geometric figure above the line put L , will be "reflected" below the line L at another point, line, or even geometric figure which shall be a "reflection" of the original. In the diagram we also have a circle with respect to which it will conduct a similar discussion. Inside the circle will reflect all the points, lines, and even geometric figures, we can draw on outside it, for which project an external shape, point by point, into the circle. How do we know how far, within the circle, will the "image" of a point outside of it? We will do the following rule: OP

• OP '= r ²

where r is the radius of the circle. Thus, if we know the distance OP and know the radio r the circle, we can calculate the distance OP ' simply as:

OP' = r ² / OP

For those who have studied geometry analytical, this relationship might find it suspiciously familiar. And it is, because with a simple change of symbols, have the following:

y = 1 / x ²

And this, as students of analytic geometry as you may know, is the formula of a hyperbola , a curved line drawn on the map Two-dimensional Cartesian. Since we are building our new non-Euclidean geometry with a similar relationship, it is fair to call our new geometry and hyperbolic geometry .

This model of hyperbolic geometry has a name. This is the Poincaré hyperbolic disk , designed by the French mathematician Henri Poincaré (1854-1912). As you'll soon see, this is a model in which a straight line (in the Euclidean sense, drawn out of the disk) is being represented in the album as an arc whose ends are perpendicular to the disk's boundary. Henceforth, when talking about the circle of investment and talk about the Poincaré disk, it means you are talking about the same thing.

As we move into our new world, we must be very careful and be alert to the possibility that the axioms in Euclidean geometry, these "truths so obvious to be accepted as true without discussion," may no longer be valid in hyperbolic geometry, and most especially the "parallel postulate", which certainly have to be modified. And it will be modified in a spectacular way: "For a point outside a given straight line can be drawn from an infinitely large parallel lines to given line ". This immediately comes into conflict with what we were taught in school.

Formulémonos now an interesting question. How will the images of geometric objects to be thrown away daily in a circle of investment? start the simplest case of all, a straight line outside the Poincaré disk aligned with the center of inversion O disk.

Here the answer is obvious, since by the very definition of investment in a line anywhere outside aligned with the center disk O have another point within the image of the disk located on the imaginary line that together. So, the next line will AB image the line B'A ' within the disk (note that the points for each line are exchanged.)


Now comes a question a little more difficult. How will the image of an exterior line that is not aligned with the center of the disk? The answer may seem surprising. Will as the arc of a circle . To prove this statement, draw a line perpendicular to a line L external so that that line from a point perpendicular A passing through the center of inversion O the disk, which A point will have a picture A ' within the disc:




Now distinguish any point P on the line outside L , which will also have an image inside the disc at the point opposite P ':




Since, by the definition of investment:

OA • OA' = r ²

OP • OP '= r ²

both expressions is also clear that:

OA '/ OP' = OP / OA

This proportionality tells us that the two triangles and OP'A OAP ' are similar triangles. This being the case, then the angle OP'A ' is also a right angle by the relation of similarity, which makes the inner triangle is a triangle. But this means that, by a theorem of geometry, A'P ' is the string of a circle through the points A' , P 'O and being OA' diameter the circumference. The theorem tells us that we're calling the following: For any right triangle whose hypotenuse is made to coincide with the diameter of a circle, the three vertices of the triangle are points on the circle's circumference . Unfortunately, most geometry books used in schools and higher education, this theorem is not only not proven, but appears not even mentioned, why will proceed to the proof of this theorem before proceeding forward. To do this, suppose that a triangle lie down on its base, matching the base with the diameter of a circle. What we have to do is show that the triangle will necessarily a right triangle, which is to demonstrate that the angle opposite the base of the triangle is an angle of 90 degrees. Be a triangle

OA'P 'inscribed in a circle, with its base coinciding with the diameter of the circle with center at point C. Lie a line from point A 'to the midpoint of the line OP', ie the point C which is the center of the circle:



This divides the triangle into two triangles inscribed isosceles triangle OA 'C and P'CA triangle. " Which are isosceles triangles is the same definition of isosceles triangle, "a triangle with two sides equal and one unequal. "OA'C In the triangle sides OC and A'C are equal, because both sides equal to the radius of the circle, while OA 'is the uneven side, can say the same for another isosceles triangle. In Euclidean geometry we can invoke the theorem that says "in every triangle the sum of internal angles is 180 degrees." This means we can relate the internal angles within each isosceles triangle as follows:

r + m + u = 180 °

n + q + s = 180 °

Adding both sides of the equalities, we have:

r + m + u + n + q + s = 360 °

r + u + q + s + (m + n) = 360 °
myn
But the angles are supplementary angles, which together add up to 180 °, thus:

r + u + q + s = 180 °

Here we invoke another theorem of plane geometry that, as usual, it appears shown in most textbooks dealing with the subject, says: "all isosceles triangle the sides opposite the equal sides are equal. " In other words, that:


u = r q = s

Making these substitutions, we have:

u + u + q + q = 180

2u + 2q = 180 °

u + q = 90 °

This tells us as the internal angle for the apex A is a right angle, or whether the triangle is a triangle, which concludes the proof.

Thus, returning to the investment record, we conclude that a straight line segment exterior to the disc between points and A B will be projected into the record as a small circular arc of the way follows:



Note that so that it can form a complete circle within the disk, it is necessary that our exterior line extends infinitely in both directions, which ends at infinity tend to be "played" right in the center O Poincaré disk. A segment of an external line will image a circular arc within a disk of investment, and if the line is infinitely large then its image will form a complete circle within the disc that passes through the center of the disc . Since

figures outside the record are in turn reverse their corresponding images within the disc, we conclude that every circle drawn within the Poincaré disk passing through the center of will drive out of the disk image a straight line extending to infinity in both directions .

Interestingly, a circle in the Poincaré disk not passing through the center O will drive out of the disk image another circle, and is what is shown below.

To avoid confusion, a distinction is here to record investment as the circle k , the outer circumference of the disc of investment will be designated as the circumference c , and the image of the circumference c investment within the disc will be denoted as c '.

Next, draw two intersecting to cut to the circumference c in points and A B one and points M N and the other, so that just happen to be extended through the center of disk k investment. For the demonstration is not necessary to work with the two drying, and therefore we will use only one of them, the secant OMN. The images of the points and M N within the disc will M ' and N' , respectively. This we have shown in the figure below:


By the same definition of investment, the following two relations must be true:

OM • OM '= r ² = a

ON • ON' = r ² = a

Note that, since the radius r investment disk is constant, we made it equal to a constant amount to . Now

memberwise match these expressions:

OM • OM '= ON • ON'

OM / ON '= ON / OM' ___ (1)

After that, multiply memberwise equalities:

OM • OM ' • ON • ON' = a * a = a ²

(OM • ON) • (OM ' • ON') = a ² ___ (2)

now resort to a theorem in plane geometry which is shown in many books of text (a geometry text that does not contain the proof of this theorem does not even deserve to be classified as moderately good): If from a point O outside a circle are drawn to the circumference of drying, the product of the distances from O to the intersections of each circle is drying with a constant amount . In other words, the product

OM • ON = b ___ (3)

not change by moving the point M along the circumference c , remains a constant quantity, which is why We have made equal to a constant b . Substituting (3) in (2):

b • (OM ' • ON ') = a ²

OM' • ON '= a ² / b ___ (4)

Dividing (3) between (4) limb from limb and rearranging, we have:

(OM • ON) / (OM ' • ON') = b / (a \u200b\u200b² / b)

(OM / ON ') • (ON / OM' ) = b ² / a ² ___ (5)

Finally, we can replace (1) to (5) to obtain

(OM / ON ') * (OM / ON') = b ² / a ²

(OM / ON ') ² = b ² / a ²

OM / ON' = b / a

OM / ON '= constant

latter tells us that the figures described by the points M and N ' are similar, which is proved the theorem. As the moving point M consists either of the points that form the circle c concluded that the point N ' also describe a circle c' .

Here it should be noted an important fact: in the circle of investment, the shortest distance between points A ' and B ' is not the straight line drawn between these points that we see from above, but the arc circumference joining those points, that is, the hyperbolic straight . What is an inhabitant of "hyperbolic universe" is not the same as what we see from above and from outside. In fact, the investment center of the disc corresponds to a point at infinity, an inhabitant of hyperbolic universe would never reach to reach the center of the disc, nor reach to get to the edge of it, but to us such things seem possible .

So far I have raised here about the images inside the circle investment will post daily geometric objects outside the circle. Forget what happens outside the circle of investment and then see what would occur within the circle. It can be shown that, within the circle of investment, the "shortest distance between two points", which would be a hyperbolic line , the route followed by a light beam launched from a flashlight by a resident of hyperbolic universe, will the path of an arc of a circle that go "to infinity" (towards the edge of the disc) will do it closer to a perpendicular drawn out to the edge tangent disc. The proof of the latter requires going into detail on something known as the calculus of variations , where the task is to get the features you need to fill the imaginary line followed by a light beam to minimize travel time to reach one point to another, thereby providing the distance "short" within two points inside the disk. (In relation to the use of a ray of light to find the route that minimizes travel time between two points, which occurs under a principle of physics known as the Principle of Fermat , please refer to Supplement # 1 post at the end of this blog.) If the light beam is launched in the opposite direction will be the same. For us, that ray of light in the universe hyperbolic travels along a straight line (as the inhabitant of the universe) we will see from the outside following what appears to be a curved line. Therefore, the jump from the plane Euclidean hyperbolic universe, our thinking has to be relevant to a new reality (just as a resident of hyperbolic universe that we see from the inside have to adjust their way of seeing us who live outside its peculiar universe.) There

interesting interactive program (implemented in a software routine known as applet) of Professor Paul Garrett that allows us to immediately draw lines at will in a disk hyperbolic investment. The link provides this educational tool is as follows:

garrett/a02/H2.html

http://www.math.umn.edu/ ~ All you need to do is click the mouse cursor to placed somewhere within the record, and holding it down you drag the cursor drawing the line. If the cursor is held, the hyperbolic line may be enlarged, reduced, or relocated as we wish.

Those familiar with the definitions given in analytic geometry of several of the curves there studied, perhaps at the moment to see the image within the circle of investment from an outer circumference c is also another circle c 'may be wondering how this is possible. After all, in Euclidean geometry, the definition of a circle is:

"Circumference is the locus of points such as distance to a fixed point called the center is constant ."

This definition is understood that all the radii drawn from the center of the circle to the same circle have the same length. But if the way in which investment is defined all distances are altered in carrying out investment, and even all the lines except those that are aligned with the center of the disc end up being converted into arcs, then how come we keep having a circumference having been investment out of an outer circumference? How is it possible that the radios remain unchanged for a circle also remain within a disk of investment? The answer to this is that equidistant radii corresponding to the outer circumference of fact are indeed distorted, at first glance but here the distortion is not obvious. To clarify the matter, then shows the interior circumference with several of his "spoke" routes within it:




As shown, the radius of this circle c are not "straight lines" in the sense which are used in Euclidean geometry. Moreover, the "center" of the circumference c He was not even matches the central point to which we are accustomed to call "center." However, within the disk of investment, the definition of circumference remains strictly valid, provided we are willing to change our thinking by adapting to the new geometry, geometry hyperbolic. Our definition of what is still a circle within a disk of investment has to be amended as follows:

" hyperbolic circumference is the locus of points such as hyperbolic distance to a fixed point called the center is constant . "

can show that, within a unit investment disk (with radius R = 1), a "circle" with intrinsic radio r a circumference length C:

C = 2 • π • sinh (r )

where sinh (r ), the hyperbolic sine of r is a function defined as follows:


(hyperbolic trigonometric functions are something that grows naturally in the study of non-Euclidean hyperbolic geometry.)

To see how we change this all that to which we had been used in flat Euclidean geometry, then we have some values numerical hyperbolic sine: sinh

(0.1) = 0.100167

sinh (0.5) = 0.521

sinh (0.8) = 0.888

Since for any value of x its hyperbolic sine is always an amount greater than x , hyperbolic geometry in a circle always have a length greater than a Euclidean circle with the same radio . This contrasts opposite to what happens in spherical geometry, where a circle drawn on the surface of a sphere will always have a length less than a Euclidean circle with the same radius.

investment If the disc is not unitary, and has a radius R different from unity, then the relationship to the length of a circumference of a circle "with radius r have intrinsic worth:

C = 2 • π • sinh (r / R )


not take long, carrying out "projections" of the circle inward investment through this process, to realize that there are some "truths" that simply does not can be obtained from other "principles" rather elementary. These truths will be our starting point to formalize our new geometry and to carry out the demonstration of theorems in a manner similar to how you performed in Euclidean geometry. Let's see if my readers can live with the following axiom system:


AXIOMS \u200b\u200bof hyperbolic geometry


Postulate 1: For two different points can be drawn one and only one, Hyperbolic straight.

Postulate 2: A hyperbolic line can be extended indefinitely in both directions without ever touching reach its endpoints.

Postulate 3: You can draw a circle using any point as center, which can have any radius.

Postulate 4: All right angles are equal.

Postulate 5: Through the point P , which belongs to a hyperbolic line, two lines can be drawn parallel hyperbolic P .

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 which can be matched are equal.

Postulate 10: The whole is greater than the sum of its parts.


The first postulate (axiom) is obvious. For this, we project a straight line just outside the circle (straight in the Euclidean sense) into the circle, point by point through the process of investment. You can only draw a line in the hyperbolic circle corresponds to the outer straight Euclidean. And if the line outside the circle that we are projecting inward is not situated within a line drawn from the center of the circle, discover an interesting thing: all lines hyperbolic circle drawn inside the circular arcs are being .

The second assumption is also obvious from an exceptional situation: in our investment are moving everything to enormous distances, at infinity, the circle inward . In fact, the infinite fail to see the universe in the distance for the edge of the circle. This means that as we move along a hyperbolic line to the edge of the circle, our projection out of the circle widens gaining great distances. Increasingly microscopic movements within the circle to get closer to the edge corresponding to distances increasingly immeasurable beyond. And as the edge of the circle is infinite, any hyperbolic line can be left "outside" the back to meet itself. Thus, while a straight Euclidean always found itself at infinity, the endpoints of a line hyperbolic never come to play even if they are extended.

Regarding the third assumption, a circle on a hyperbolic plane is defined as the locus of points such that they are all the same distance from a point called the center, as well as in the Euclidean plane. Here my readers will have a doubt. What form it takes, within the circle of investment, a conventional Euclidean circle route out of it? I'll leave you with questions to encourage them to play a bit with a circle of investment and make the strokes required to see what you get. Would you be amazed to discover that a circle in a circle projected investment will remain a circle, except that the center will appear to be "off center"?

The fourth postulate of our hyperbolic geometry require tracing geometric figures outside the circle of investment, then projected inward to determine whether it stands or whether to discard it.

The fifth postulate of our hyperbolic geometry, "by P point, which belongs to a hyperbolic line, two lines can be drawn parallel hyperbolic P" is not so obvious. But Euclid's fifth postulate was neither. However, you can check by drawing some lines inside and outside the circle of investment to discover this truth that we must accept as true, the demonstration is not possible.

The remaining postulates, sixth, seventh, eighth, ninth and tenth, should remain valid in our hyperbolic geometry, because otherwise the whole pyramid on which are mathematics and even built the same logic we can come down.

With these postulates of hyperbolic geometry we have developed, all we need to continue to prove theorems in hyperbolic geometry as Saccheri did, some of which may actually correspond without modification to the theorems that show in Euclidean geometry, while others will have to be modified. First we will review the basics. In hyperbolic geometry defined as "parallel lines" those endless lines drawn within a hyperbolic plane that never intersect. This is what we have in the following Figure:


where "straight" black color hyperbolic intersect at the point p although not parallel to each other are parallel to all the "right" hyperbolic magenta This shows something important: in a hyperbolic geometry, through a point p outside a given line can be drawn not once but an infinite number of parallel lines to the given line . Contrast this with what happens in Euclidean geometry, where through a point p outside a given line only possible to draw a single line parallel to the line given. Contrast also with what happens in spherical geometry (elliptical) where through an external point p to a given line is not possible to draw any line parallel to the given line. And here also occurs another curious fact. We can see that two lines are not parallel, as the three hyperbolic lines of black that intersect at the point p , however, may be all parallel to another straight line and the magenta color hyperbolic. This is in direct contrast to the old theorem in Euclidean geometry tells us that "two parallel lines to a third are parallel to each other."

The way in which defined the circle of investment, it would seem that in order to build the inverse image of an object located outside the disk would need to have a calculator handy to be able to be calculating each point image of the object within the disk. However, such is not necessary, by a theorem that says: The inverse point P 'to a point outside P with respect to a radio disc investment r can be constructed using only a compass . No formula, no algebra, no arithmetic, only required a compass. This, incidentally, is about the spirit of the geometry of ancient Greece Euclid's times where everything took place using only a straightedge and compass (and we're talking about a non-graduated rule, used only for the drawing of straight lines and not to measure). This construction was carried out in the following way: Take a point P foreign investment to a record which we find, using only a compass, the image P ' its part within the disc. For this, making center at point P , first we draw an arc that passes through the point O investment disk to disk intersecting points and T S . Then, using these points as centers, draw two circles of radius r (the radius of the circle of investment). That's it! point P ' where the curves intersect is precisely the image of point P exterior, as shown in the figure below:


Taking several points of an external object to the disk, locating in the manner just noted the images of the points within the disc, and connecting the interior points, we obtain the inverse image of an object located outside the disk. This procedure is so simple and fun, sure to be readers mine who want to carry it out if you have a compass on hand. The proof of the theorem is carried out first showing in the picture above the isosceles triangles OTP ' and OTP are similar triangles. To do this, just to prove that the OTP angle 'equals the angle TPO. By construction, the large isosceles triangle OTP angle equals the angle POT. But the angle POT equals the angle of an isosceles triangle P'OT boy, which in turn must be equal to the angle of OP'T small triangle, which shows they are similar triangles. However, if both are similar triangles, then the following proportionality between the corresponding sides must be true:

OP / OT = OT / OP '

is:

OP • OP' = OT • OT = r ²

that is precisely the relation that defines investment, which demonstrated that our procedure is correct geometric construction.

There is another even easier to find geometrically (without a calculator) within a disk image investment A ' At one point outside the disk . To do this, first draw a line from point A to the center of the disc O . After that, draw a straight line from point A to the edge of disk that is tangent to the edge of the disc , touching at point B . If we join O point to point B, we see that it forms a triangle, the triangle OAB . All we have to do is trace the height of the triangle to its base to find the hypotenuse of the triangle point A ' corresponding to the image of the A' as shown in the figure below:


We can show that this construction method is correctly seeing that when plotting the height A'B within the triangle OAB it is split into two triangles similar, triangle and triangle OAB OA'B . They are similar because both have three internal angles equal (they have in common the angle AOB and also have a common angle, so that the other two corresponding angles must also be equal.) As semjantes figures, this we can write the following proportionality relating their sides:

OA / OB = OB / OA '

OA • OA' = OB ²

OA • OA '= r ²

Y This is precisely the relation that defines the circle of investment.

There is a very easy way to build within the circle of investment that would be a hyperbolic line , the equivalent of the line representing the shortest distance between two points for an inhabitant of the universe hyperbolic, or put another way, the route followed by a light beam launched investment within the disc. This line is an arc. But not just any arc of a circle serve to that purpose. You must be an arc that intersects the edge of the disc at right angles. The requirement that every ray of light thrown into the circle follow a route that they distribute it perpendicular to the edge of the disc allows us to develop a simple way to map a route that a hyperbolic two points A and B on the edge any disc. To do this, we draw two lines tangent to the disk at points A and B, perpendicular to the radius of the disk, which should be extended to the side which will be at a given point P as shown in the figure below:


This done, is done with a compass center at P and draw an arc within the disk that points A and B, which is the desired hyperbolic line.

Since the only difference between hyperbolic geometry and Euclidean geometry is the fifth postulate, the parallel postulate, this means that any show that takes place in Euclidean geometry that does not make use of the fifth postulate is a demonstration equally valid in hyperbolic geometry . This immediately allows us to incorporate many demonstrations of Euclidean geometry within of hyperbolic geometry. Similarly, this means that all theorems in Euclidean geometry, making use of the fifth postulate in geometry will be false hiporbólica and have to be modified (as well as with elliptic geometry). Since a geometry built using only the first four principles will be equally valid in the three worlds (Euclidean, elliptic, hyperbolic), this type of geometry is known as absolute geometry .

Lets look at some of hyperbolic geometry theorems whose proof can be carried out by procedures similar to those used in Euclidean geometry: Euclidean geometry

: "In any triangle, the sum of the interior angles equals two right angles (180 degrees)." hyperbolic geometry: In any triangle, the sum of the interior angles is less two right angles. "

Euclidean geometry:" The sum of the angles of a quadrilateral is equal to four right angles (360 degrees). hyperbolic geometry: "The sum of the angles of a hyperbolic quadrilateral is less four right angles (360 degrees).

Euclidean geometry: "The medians of a triangle (defined as the median line joining a vertex of the triangle with the midpoint of the side of the triangle which is drawn) are concurrent (to be found in one place). "hyperbolic geometry :" The medians of a hyperbolic triangle are concurrent. "(The theorem remains unchanged in both geometries.)



Euclidean geometry:" The altitudes of a triangle (defined as the perpendicular height drawn toward the base of a triangle, base triangle on which side seems to rest) are concurrent. hyperbolic geometry: "The heights of a hyperbolic triangle are concurrent." (The theorem remains unchanged in both geometries.)



geometry Euclidean : "The bisectors of a triangle (defined as the perpendicular bisector drawn a line segment at its midpoint) are concurrent. hyperbolic geometry:" The bisectors of a triangle are concurrent. "(The theorem remains unchanged both geometries.)




Euclidean geometry: "The bisectors of a triangle (defined as the bisector of an angle from one vertex divides the angle into two equal parts) are concurrent." hyperbolic geometry: "The hyperbolic bisectors of a triangle are concurrent." (The theorem remains unchanged in both geometries.) Below is the hyperbolic angle bisector:




Euclidean geometry: "The point at which they are the bisectors of a triangle is equidistant from the three sides and therefore can inscribe a circle within the triangle using that point as center. hyperbolic geometry: "The point at which they are the bisectors of a hyperbolic triangle is equidistant from three sides and therefore can inscribe a circle inside the triangle using this point as center. "(The theorem remains unchanged in both geometries.)




now will review a definition that had previously given to introduce the geometry of Saccheri. The entry" The First non-Euclidean geometry "talked briefly about defect of a triangle defined as it lacks the sum of the angles of a triangle Saccheriano to be 180 degrees. This definition of a triangle default on Saccheri's geometry is of utmost importance to what is discussed below.

In Euclidean plane geometry, when talking about the area of \u200b\u200ba triangle or a square or a parallelogram, we are talking about a number used to represent the number of "flat" contained within a geometric figure . In our Saccheriana or hyperbolic geometry, the area of \u200b\u200ba triangle should also be a single number that can be measured in some way content inside the triangle, the plane contained within the triangle. But this number can not be in square units (square meters, square centimeters, etc.) Because there are no squares in hyperbolic geometry. Although there are several definitions that we can attempt to measure the "hyperbolic area" inside a triangle, all have serious defects except one, which extends the concept of an infinitesimal straight ds to the universe Euclidean hyperbolic modified to take into account the "stretching" of space that occurs as we get closer and closer to a boundary line or breaking point in a hyperbolic plane, a definition which, if the distance to a boundary line is measured by a shaft and then the infinitesimal hyperbolic area is given not by a product of the infinitesimal dx and dy as usual in the Euclidean case but as the product of infinitesimals dx / and and dy / and (see Supplement # 2 related areas to obtain another model using hyperbolic plane hyperbolic plane known as the Poincare upper-half ). It can be shown by application elements direct calculus that this definition of infinitesimal hyperbolic area leads directly to the following result for the area of \u200b\u200ba hyperbolic triangle with interior angles p , q and r :

S = π - (p + q + r)

The presence of π number in this relationship, which we take as a measure of 180 degrees angle expressed in radians, is that in the process of integration required to obtain of this formula from area infinitesimal hyperbolic elements appear naturally the number π . This formula may seem suspiciously similar to the definition of δ default of a hyperbolic triangle. And in fact, is the same! As the formal expression can be obtained from the procedures of calculus, the expression can be expressed as a theorem: Theorem

: The area of \u200b\u200ba hyperbolic triangle is equal to the defect of the triangle δ.

From this theorem we can deduce immediately that two triangles have the same area hyperbolic if and only if they have the same sum of internal angles . We set a

consistent system of units by introducing a proportionality constant k we define the hyperbolic units that will be used as a measurement system (meters hyperbolic hyperbolic yards, etc.), so that the area of \u200b\u200ba hyperbolic triangle ABC be given by:

S = k • δ (ABC )

Thus, whereas in Euclidean geometry to measure the area inside a hyperbolic triangle we use a specific dimension for a area in square units, in hyperbolic geometry have to use, as well as the proportionality constant k gives us the equivalent of our "hyperbolic metric, the sum of the interior angles of a triangle, which is not needed in Euclidean geometry. But this is not the only difference. There is another difference that may be shocking to many who try to assimilate the first time. To simplify the following discussion, we select a unit system (hyperbolic) such that the constant k is (arbitrarily) equal to 10, so that the area of \u200b\u200ba hyperbolic triangle ABC is given by:

S = 10 • δ (ABC )

In Euclidean geometry, the area of \u200b\u200ba triangle is given by the usual formula of the product of the base times the height divided by two (bh / 2), with which to gradually increase the size of the triangle the number that we express the Euclidean triangle area also increases without any upper limit, we have a triangle of one hundred square meters, a hundred thousand square meters, or a thousand trillion miles square area, there is an upper limit for this number. But in hyperbolic geometry, as the sum of internal angles of any triangle is always less than π (180) degrees and certainly can not be zero, the areas of all hyperbolic triangles are limited to have a value (assuming a unit system in which the constant k be 10) between 31,415 units hyperbolic ( π multiplied by 10) and zero! This triangle can contain no more than 31.45 units hyperbolic area. A hundred units hyperbolic triangle can not exist in this universe. However, this apparently puzzling results from that we have not adjusted our thinking to the new reality of hyperbolic universe. The higher the defect of a triangle, certainly more area that contains hyperbolic, which can only be achieved with triangles interior angles whose sum is increasingly different from the usual 180 degrees Euclidean geometry, which can only be achieved with triangles hyperbolic ever larger. As the sum of the angles of a triangle is approaching the limit of 180 degrees, the hyperbolic triangle size will grow disproportionately to reach astronomic dimensions, without ever reaching 180 degrees accurate. In contrast, the smaller is a hyperbolic triangle, the less it must be for This failing, which means that the internal sum of the angles will be closer and closer to the 180 degree limit Euclidean geometry. In short, the smaller is contained in a hyperbolic universe, their properties will be more and more like the properties found in Euclidean geometry .

S number as defined gives us the right properties we would expect to have for a definition of area (hyperbolic). In Euclidean geometry, if a triangle is subdivided by a cross in two parts, then the area of \u200b\u200bthe triangle will be greater than any of the parties that was subdivided. And as already demonstrated in the entry "The First Non-Euclidean Geometry" the absence of a hyperbolic triangle is equal to the sum of the two subtriangles defects corresponding to a cross inside the triangle, we need the concept of fault is equivalent to the concept we have of the area in the Euclidean plane. Thus, we can make the following comparison between Euclidean geometry and hyperbolic geometry, Euclidean geometry

: "If the three angles of a triangle ABC are equal, respectively, at three angles of a triangle A'B'C ;, then those triangles are similar triangles. hyperbolic geometry: "If the three angles of a triangle ABC are equal, respectively, at three angles of a triangle A'B'C;, then those triangles are equal . This conclusion is interesting. In hyperbolic geometry there is a triangle similar to a given triangle is not equal to it! In other words, there are no similar triangles, only the triangles equal. In this way, as well as with spherical geometry, even in the hyperbolic geometry of similar figures exist, it only occurs in geometry flat, Euclidean geometry, where two triangles with internal angles equal may have correspondingly different areas.

Perhaps one of the most dramatic differences between Euclidean geometry and hyperbolic geometry displayed by the Pythagorean theorem tells us that in a right triangle with sides , b with hypotenuse c the square of the hypotenuse equals the sum of the squares of the legs to and b :

c ² = a ² + b ²

has been shown that the Pythagorean theorem is equivalent the fifth postulate. The demonstration was conducted by Scott Brodie and can be found at the following address:

http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml

Since the Pythagorean theorem is equivalent the fifth postulate, the Pythagorean formula is no longer valid in hyperbolic geometry. His counterpart in this geometry is as follows:

cosh (c) = cosh (a) • cosh (b)

Again, appears another hyperbolic trigonometric formula, the hyperbolic cosine cosh. This expression tells us that if we draw a triangle in a hyperbolic plane, the hyperbolic cosine of the hypotenuse equals the sum of the hyperbolic cosine of the legs, where the hyperbolic cosine a variable x is defined as follows:


Using Maclaurin series expansion for the hyperbolic cosine:




no problem we can deduce that when the triangle drawn in the hyperbolic plane becomes very small, which is equivalent to assign to , b and c very small values, the hyperbolic Pythagorean theorem approximates the usual form of this theorem in Euclidean geometry.

In three dimensions, we can construct a hyperbolic geometry on a surface known as the hyperbolic paraboloid , which is shaped like a "saddle" as used by riders. In the following figure we have a triangle drawn on the surface of a hyperbolic paraboloid and a couple of lines "parallel" as you can see but never touch is becoming more and more will be separated with respect to the perpendicular drawn that line "parallel" where these reached their highest close (the perpendicular which is divided internally to the two hyperbolic lines in two right angles, 90 degrees):




Below is a graphic summary of the three geometries in which we have the same triangle drawn on each:




hyperbolic geometry we have built using the Poincaré hyperbolic disk is not the only possible. You can build many other hyperbolic non-Euclidean geometries, including highlights Klein hyperbolic geometry.

Establishing a rigid line of thinking in the way used by Euclid, now known as the axiomatic method , starting with a set of precise definitions and a set of axioms or postulates as "truth" is so obvious it requires no demonstration ( an axiom is something that in any way can not be proved, can not be obtained from the most basic principles), and then using these axioms to carry out the proof of all theorems that can be obtained from them, we have a rigorous mechanical, formal, which can extend beyond the geometry to other areas which includes the same math . This was precisely what made the Italian mathematician Giuseppe Peano (the same that started many of the notation used by "modern mathematics" better known as set theory), who much the way that Euclid laid the groundwork for axiomatic development of arithmetic through various postulates yours today known as the Peano axioms , influencing greatly about his greatest pupil, Bertrand Russell, who in turn tried to free mathematics forever the possibility of internal inconsistencies with his monumental work Principia Mathematica . The Peano axioms that define exactly the set of natural numbers (positive integers), as they were written in Latin for the first time, read as follows (compare the axiomatic form by Euclid used):


Peano axioms


1. 1 is a number.

2. The immediate successor of a number also is a number.

3. 1 is not the immediate successor of any number.

4. Two different numbers are not as immediate successor

5. All property belonging to 1 and the immediate successor of any number that also has that property belongs to all numbers.


Indeed, they seem to be true so "obvious" that there seems no other way to deduct more basic truths. As a professional mathematicians take a lot like his art formalism in mathematics textbooks found the Peano postulates set forth as follows (same, but more "sophisticated")


1. 1 is a natural number. That is, the set of natural numbers is not empty.

2. If to is a natural number, then to + 1 is also a natural number, called the successor to .

3. 1 is not a successor of any natural number. It is the first element of the set.

4. If there are two natural numbers and to b such that their successors are different, then to and b are different natural numbers.

5. Induction Axiom : if a set of natural numbers containing the 1 and the successors of each of its elements then contains all natural numbers.


Since the axiomatic method, since the time Aristotle, is essentially the way he works that branch of human knowledge known as the logical , is attributed to mathematicians as Peano the distinction of being dethroned as the fundamental branch of mathematics knowledge, reducing them to mere rigorous application logic on a set of axioms. And as happened with the axiomatization of geometry, the axioms of arithmetic showed that, depending on the axioms that are used as a starting point, you can build several alternative arithmetic, such as arithmetic and Robinson Pressburger arithmetic. If the same Euclid not developed the elliptical geometry and hyperbolic geometry was considered because no other alternatives other than his fifth postulate as having done so he would have discovered what is now known as non-Euclidean geometries, taking into account the fact that he sat the axiomatic basis to develop alternative geometries. Either way, his example was well used by all his successors, as its axiomatic method is what is used today for the publication of cutting-edge work in advanced areas of contemporary mathematics. And it is on this that we all have a debt of gratitude Euclid.
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