Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Almost 100 years later, in the early 1800s, the great German mathematician Carl Friedrich Gauss attacked the same issue of Euclid's fifth postulate. He recognized that although Saccheri had simply dismissed the possibility of having more than one parallel line through a given point, one could construct a completely self-consistent geometry that differed from that of Euclid by using this postulate instead of the original. This new geometry describes a world in which the summit angles of Saccheri's quadrilaterals are acute.
There is a mathematical legend that Gauss, as much experimental scientist as mathematician, sought to determine whether or not this new type of geometry was actually the geometry of the real world. To do this, he supposedly constructed a great triangle using signal fires and mirrors set on mountaintops.
He then measured the angles between these points of fire and compared his measurements to the expected finding of 180 degrees total. Why would he use mountain tops? Why not just draw a large triangle on a flat space of land? As we will see soon, a shape drawn on the surface of a sphere (or near-sphere, such as Earth) is different than a shape drawn in space. By connecting the tops of mountains with rays of light, Gauss was creating a triangle using the minimal-length connections made possible in space. These lines were free to be as straight as possible, without having to bend or conform to the shape of the surface of Earth. The veracity of this story is questionable; nevertheless, it illustrates the difference between lines on a curved surface and lines in a potentially curved space.
A more-radical mathematical idea for its time would be hard to find, but Gauss did not publish his findings in this arena. Some suggest that this was to avoid confrontation with the great philosopher, Kant, who espoused that the human perception of reality is Euclidean. Others suggest that Gauss was afraid that he would lose face with his contemporary mathematicians. For whatever reason, it was not Gauss who first brought these ideas to light.
(It is interesting to note that Gauss did not publish many of his ideas. It is commonly thought that this was because he was a perfectionist and would only make his views known if they were above criticism. To that end, he would not provide the intuitions behind his proofs, preferring instead to give the impression that they came "out of thin air." Eric Temple Bell estimated in 1937 that, were Gauss to have been more forthcoming, mathematics would have been advanced by at least 50 years! (Here is yet another example of why students should be encouraged to show their work!))
• Lobachevsky and Bolyai independently came to the same conclusion as Gauss.
Nicolai Lobachevsky was a Russian who, in 1829, published a version of a geometry in which, instead of just one parallel line, multiple parallel lines were possible through any given point.
If we take Saccheri's quadrilaterals and modify them a bit, we can show that Lobachevsky's idea is equivalent to saying that the three angles of a triangle can add up to less than a Euclidean 180 degrees.)
Lobachevsky showed that this assumption would not lead to any logical contradictions and was, thus, just as valid as Euclid's geometry. Almost at the same time, in 1832, a Hungarian mathematician named János Bolyai published a similar finding after studying what he called "absolute," or neutral, geometry— that is, geometry that uses only the first four of Euclid's postulates.
So, Gauss, Lobachevsky, and Bolyai all independently found that a new and completely self-consistent geometry could be established by letting more than one line through a given point be parallel to a given line that does not include the point. Evidently this was an idea whose time had come. What of the other case, the case in which no parallel lines are possible?
Oddly enough, a geometry that allows no parallel lines is not as strange as the type that Gauss, Lobachevsky, and Bolyai described. We can indeed imagine a world in which lines always cross other lines. Think about the surface of a sphere, such as Earth (more or less). Lines that follow the surface will continue around the sphere and back to their beginning points to create circles. Lines such as this that are the maximum length for a given sphere (that is, lines whose length is equal to the circumference of the sphere) are called "great circles"— the equator is an example. Thus, the shortest distance between any two points on a sphere will be a part of one of these great circles. Any two such lines will always intersect in two places; hence, there can be no parallel lines in this system! People as far back as the Greeks understood this, and they understood that geometry on a sphere is different from that on a plane.
In this type of geometry, also known as "spherical geometry," Saccheri's quadrilaterals would have obtuse summit angles, and the angles of a triangle would add up to more than 180 degrees. In such a system, one has to replace the parallel postulate with a version that admits no parallel lines as well as modify Euclid's first two postulates. The first postulate's restriction that "through any two points, there is only one possible straight line" does not hold true on a sphere.
On the surface of a sphere, we must allow for more than one line through any two points. Likewise, we must modify Euclid's second postulate, because "lines" on the surface of a sphere are really circles, which have no end and no beginning—they cannot extend to infinity. We can circumvent this by requiring simply that lines be unbounded.
With the inclusion of spherical geometry, mathematicians now had three broad types of geometry with which to study and measure shapes and space: hyperbolic (Lobachevsky), spherical (Riemann), and flat (Euclid).
Obviously, the type of world you "live in" and its geometry depend on which flavor" of the fifth postulate you choose. If you choose to allow only one line parallel to a given line through a given point, you are choosing to inhabit Euclid's world of flat geometry. If you choose to allow many parallel lines through the same point, you are in Lobachevsky's world of hyperbolic geometry. If, on the other hand, you choose to allow no parallel lines, you are in the world of spherical geometry.
We can envision Euclid's universe as a flat plane; this is the universe that we learn about in high school, and its concepts make intuitive sense. To get from one point to another point on the plane, we all know that a straight line will be the shortest distance. This connection of minimal length can be generalized into the notion of a "geodesic." A geodesic, in the local view, is simply the shortest distance between two points in whichever geometry you choose to use. When looking at the geometry of an object on a global level, a geodesic is the path that a particle would take were it free from the influence of any forces.
To envision the spherical universe, we earlier used a sphere as a model. We said that the shortest distance between two points on the sphere, a "geodesic" by our new terminology, is a part of a great circle. In other words, a straight line in spherical geometry is actually a curve, when viewed from the Euclidean perspective.
Lobachevsky's universe is a bit harder to visualize. Eugenio Beltrami, an Italian mathematician working in Bologna, Pisa, and Rome, found a shape, analogous to a sphere, but with a surface that obeys Lobachevsky's geometry. Imagine a tractrix (the path something takes when you drag it by a leash horizontally) rotated around its long axis to generate a shape not unlike two trumpets glued bell to bell.
This is called a "pseudosphere," and it can be thought of as the opposite of a sphere—or, if you are feeling adventurous, as a sphere of imaginary radius. The surface of a psuedosphere behaves according to the rules of hyperbolic geometry. The geodesic of a pseudosphere, the minimal-length connection between two points, is again a curve, but not a section of a great circle, as it was in the world of spherical geometry.
In this discussion, we have seen how basic axioms can define a mental world and that, by changing the axioms, we change the characteristic behavior, or reality, of this mental world. Axioms are verbal statements; to visualize the worlds that they create, we need visual models. We saw earlier that one way to create such models is to embed them in some sort of space. This is what we are doing when we look at a sphere or a pseudosphere. This method has proven to be handy, because it preserves all the geometric properties, such as length and angle, that are determined by our axioms.
However, we don't always have the option of looking at spatial models of the sphere and pseudosphere. Consequently, we have developed another method of visualizing these spaces: maps. Maps are handy because they can be drawn on a flat piece of paper. Unlike our spatial embeddings, however, maps necessarily distort the picture in some way. They can, nevertheless, be of great help as we try to understand all of these strange geometries, and so it is to maps that we next turn our attention.