# Noneuclidean Geometry

Luckily, no! Measurements of this type can be carried out, not with rockets, but with natural objects already placed at a good distance and equidistant to each other: the warm spots of the fossil radiation. This method has recently been used the Boomerang, Maxima, and WMAP experiments to set limits on the curvature of cosmic space.

Strangely, Gauss, Lobachevsky, and Bolyai did not think to clarify the consequences of the inverse hypothesis, in which the sum of the angles in a triangle is greater than degrees. It had been known, however, since antiquity that, on a sphere, the sum of the angles for a triangle made up of three arcs of great circles the equivalent of straight lines is always greater than degrees. This spherical trigonometry had been the subject of innumerable treatments, at first useful for astronomy, and then for geography and navigation.

The celestial vault in fact offers a spherical concavity, and if one wants to mentally trace a straight line between two stars, one must use an arc of a celestial great circle centered on the Earth. In the same way, if one desires to travel from one point of the terrestrial surface to another one which is very distant, the only authorized paths are situated on the convex surface: their is no question of digging a direct tunnel through the interior of the planet. The shortest possible path is thus an arc of a great circle, a fact well known to navigators.

One could have thus noticed that spherical trigonometry, in two dimensions, led to a coherent geometry, although it was not Euclidean. Why the resistance against passing to higher dimension? Probably because this implies a physical space which is finite but without frontier, encircled by straight lines. Now, up to this point, geometry had not been the science of space, but the science of figures in space.

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As a homogeneous, indefinite expanse of three dimensions, physical space constituted a neutral reality which gave rise to no speculation. This picture had been established in the eighteenth century by the philosophy of Kant, when he made Euclidean space a category of reason.

## Non-Euclidean geometry

The story of this lecture is an astonishing anecdote which shows how the sciences, just like universal history, sometimes progress unexpectedly through small accidents and fortuitous circumstances. The rules asked that each candidate propose, some time in advance, three subjects, among which the professors chose that which seemed most promising. Traditionally, it so happened that the professors always selected one of the two first subjects on the list. Riemann had therefore carefully prepared his two first subjects and slightly neglected the last, which in fact concerned the foundations of geometry.

However, he had not accounted for the presence in the jury of the great Gauss who, although a septuagenarian, remained the dominant figure on the faculty. Naturally, Gauss felt that the third subject was the most interesting. Panicked at the prospect of this seminar, on which his entire career depended, Riemann threw himself madly into the preparation of his lecture and subsequently fell ill, to the point that he missed the date of presentation and only recovered his health in April.

He then prepared his lecture for seven weeks, an amount of time that Gauss had requested as an adjournment for medical reasons. Riemann finally delivered his oral dissertation on June 10, But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value.

If we prolong all the geodesics starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i. Riemann also proposed to extend geometry to spaces having an arbitrary number of dimensions.

For this, geometry must be understood as the study of possible varieties. In the most general sense, a variety is nothing more than a collection of objects from a set. Geometry then becomes the study of the rules and conditions placed on the objects from the set in order to determine the distances between elements.

For example, in a two-dimensional space, each point is define in a unique way by an ordered pair of real numbers x 1 ,x 2 : its coordinates.

### What Is Euclidean Geometry?

In threedimensional space, each point is defined by a triplet of numbers x 1 ,x 2 ,x 3. In an abstract n-dimensional space, each point has a unique address given by an n-tuplet x 1 ,x 2 ,x 3 ,…. The rules for determining the distances between elements in the set are summed up by the metric, a sort of generalization of the Pythagorean theorem whose coefficients determine the curvature of the space.

Riemann is not only the creator of the most profound lecture in the history of geometry, but in he accomplished an exploit of the same order, with a completely innovative article of eight pages, on number theory. Perhaps he would have performed a third exploit, if he had not died of tuberculosis at thirty-nine years of age. The way was now definitively opened for Clifford, Einstein, Weyl, and others to model certain natural phenomena like gravitation in terms of non-Euclidean geometry. Rosenfeld, B. Gavronsky, Berkeley: Univ. Euclid, Elements , T. Heath transl. Clifford, Nature vol. Your email address will not be published.

## Non-Euclidean geometry

And what are to we to make of Kant's assurance that space has to be Euclidean, a synthetic a priori fact? If one has a prior background in Euclidean geometry, it takes a little while to be comfortable with the idea that space does not have to be Euclidean and that other geometries are quite possible. In this chapter , we will give an illustration of what it is like to do geometry in a space governed by an alternative to Euclid's fifth postulate. One of most important by-products of the efforts to derive Euclid's fifth postulate were simpler, alternative formulations of the postulate that could be used in place of Euclid's original.

Many were found, including:. There exists a pair of coplanar straight lines, everywhere equidistant from one another. There exists a pair of similar, non-congruent triangles. If in a quadrilateral a pair of opposite sides are equal and if the angles adjacent to the third side are right angles, then the other two angles are also right angles. Saccheri There is no upper bound to the area of a triangle.

Non-Euclidean Level Design (Portal 2 Version)

Of all the reformulations, one proves to be most useful. It was stated by an 18th century mathematician and physicist, Playfair. His postulate, equivalent to Euclid's fifth, was:. ONE Through any given point can be drawn exactly one straight line parallel to a given line. This formulation made it easy to state what the alternatives were. Through any given point MORE than one straight line can be drawn parallel to a given line.

Through any given point NO straight lines can be drawn parallel to a given line. Once again the import of the alternative postulate is hard to draw since the screen is a Euclidean surface.

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In the figure, the line through the point is a straight line. The postulate tells us that no matter which straight line we pick through the point, the outcome is the same. It is not parallel to the line not on the point. If extended it will eventually meet the other line. Let us join the explorers of the nineteenth century and take the first steps into the new space of these odd geometries.

Let us explore the space of 5 NONE. In a Euclidean geometry, they could not possibly meet. However this is not a Euclidean geometry, so odd things will happen. This is just the beginning This construction could be continued with points D, E, F, At each we erect a perpendicular, which will intersect the others at the same point O.

So we can certainly make it as big as a right angle. Let us say that this happens with a base AG. And we can keep extending the base to G' until we have a second right angle at GOG'. And we can extend to G" so that we have a third right angle at G'OG''. And finally we can extend the base to G''' so we have a fourth right angle at G''OG'''.

We have arrived at something remarkable in this figure. It is not just that all these lines are straight lines. It is more.

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The angle AOG''' is four right angles. How can that be? Rather than tell you right away, let me give you a clue. We don't need to draw all the lines as straight in the figure. We just need to remember which are straight--in this case all of them. So we can redraw the figure as:. Think once again what it means for the angle AOG''' to be four right angles. Consider the line OA as it sweeps around O. But if a radial arms sweeps four right angles, it has returned to its starting point. Or the point G''' just is the same point as A.

So the figure is more correctly drawn as:. Notice what has happened. So the straight line on which points A and B lie is actually a straight line that wraps back onto itself. Now recall that there was nothing special about this line. It follows that all straight lines in the new geometry wrap back onto themselves. Since these straight lines fill all of space, it follows that that this space wraps back onto itself in every direction.

This last figure has more surprises. To begin, recall that all the lines in it are straight.