For most of the years since Euclid wrote the Elements in 325 B.C., people felt that only one sort of geometry was possible. Planes looked like infinitely large, flat sheets of paper, lines went on forever as straight as the mind could imagine, and a grid of parallel lines could be drawn to make a plane look like graph paper. However, the foundations of this geometry were unfortunately vague.
As discussed in Chapter 1, the concepts of point, line, and plane were not given formal definitions. The individual points, infinite straight lines, and flat planes discussed throughout this book all fit the properties and descriptions of points, lines, and planes, but other objects fit these general descriptions as well. When these basic objects are different, the resulting geometry is different as well.
Similarly, as discussed in Chapter 2, the entire structure of geometric proof rests upon unproved postulates. These postulates lead to the geometry with which we are familiar. However, why should we believe one set of postulates and not a different set? If we start with different postulates, then our theorems will be different as well. Our choices lead to different sorts of geometry, called non-euclidean geometries.
Euclid’s Elements began with only five postulates. We shall go through them with an eye for alternatives.
POSTULATE 1: One and only one straight line can be drawn through any two points. (Chapter 2, Postulate 11)
We further postulate that there exist at least two points, A and B, so there exists a straight line.
POSTULATE 2: A line segment can be extended in either direction indefinitely. (Chapter 1, description of a line)
Traditionally, we suppose that the line through A and B looks like Fig. 19-1(a), but our postulates do not rule out the possibility of Fig. 19-1(b). In Fig. 19-1(b), the line reaches no end in either direction, and thus in some sense can be extended indefinitely. Remember that the original meaning of “geometry” comes from “earth” and “measure.” The straightest line that could be drawn on the earth would wrap around to form a great circle, as discussed in Chapter 17.
Fig. 19-1
POSTULATE 3: A circle can be drawn with any center and radius. (Chapter 2, Postulate 14)
When we believe this postulate, we believe that we have a plane with no end in any direction on which we could find circles with ever-larger radii. It could be that this leads to circles like ripples on the surface of a still pond, as shown in Fig. 19-2(a). However, if our plane were more like the surface of the Earth, our circles might look more like the circles of latitude, shown in Fig. 19-2(b).
Fig. 19-2
POSTULATE 4: All right angles have the same measure.
This postulate enables us to measure angles in degrees. A plane would have to be somehow lumpy or uneven if some right angles could be bigger than others.
POSTULATE 5: Through a given point not on a given line, one and only one line can be drawn parallel to a given line. (Chapter 4, Parallel-Line Postulate)
This postulate is the one which establishes that our plane cannot look like a giant sphere. On a flat plane, given point P not on line 
, only a point C which makes ∠APC ≐ ∠PAB will make 
|| , as shown in Fig. 19-3(a). On a giant sphere, straight lines are great circles which divide the sphere into two equal-sized pieces. Any two such lines will always meet at two spots at opposite points of the sphere (called antipodal points), such as X and Y, as shown in Fig. 19-3(b). Because any two straight lines meet, it is impossible for there to be parallel lines on a sphere.
Fig. 19-3
For about 2000 years, certain mathematicians tried to use the first four postulates to prove the fifth. This challenge was called the fifth postulate problem. The first four seem to come straight from the basic tools of geometry: the straight edge (connecting points and extending straight lines), the compass (drawing circles), and the square (measuring right angles). The fifth, it seemed, was something that ought to be provable. However, no one was able to succeed in proving the fifth postulate without introducing new, equivalent postulates that required belief without proof. (Actually, the fifth postulate stated previously is a simpler alternative to the one Euclid actually used.) If a postulate about parallel lines must be accepted without proof, why must it be the traditional fifth postulate of euclidean geometry?
In the 19th century, mathematicians finally established that there are actually three basic types of planar geometry, not one. The traditional one is called euclidean geometry, which is based on the fifth postulate (or one of the many equivalent postulates). However, we could instead believe one of the two alternate fifth postulates:
POSTULATE 5a: There are no parallel lines.
POSTULATE 5b: Through a given point not on a given line, many different lines can be drawn parallel to a given line.
If we believe Postulate 5a (along with Postulates 1 through 4), then we will end up with what is called elliptic geometry. This is the sort of geometry where a plane is actually shaped like the surface of a sphere and lines are great circles.
If we believe Postulate 5b, we will end up with an even stranger geometry called hyperbolic geometry.
Just for comparison, here are several properties of euclidean geometry:
Planes are infinite in area and flat.
Lines are infinite in length.
The angles of triangles always sum to 180°.
There are similar triangles of different sizes.
The circumference of a circle with radius r is C = 2πr2.
The area of a circle with radius r is A = πr2.
The properties of elliptic geometry are
Planes are shaped like spheres, finite in area.
Lines are great circles, finite in length.
The angles of a triangle add up to more than 180°. For an example, take two longitude circles that meet at right angles at the North and South Poles and add the equator. These will cut the globe into 8 triangles where every angle is 90°. Thus, in elliptic geometry it is possible to have a triangle with an angle sum of 270°, as shown in Fig. 19-4. Smaller triangles will have smaller angle sums, but always more than 180°.
Fig. 19-4
The size of a triangle on a sphere is related to the sum of all its angles. For example, every triangle with an angle sum of 270° will have area equal to one eighth of the whole sphere, just like the triangles in Fig. 19-4. Smaller triangles will have smaller angle sums. Because of this, it is impossible to have two triangles of different sizes that have the same angle measures. Thus, there cannot be similar triangles of different sizes in elliptic space.
A circle in elliptic space is defined as usual: the set of all points a given distance from a point. Because distances are measured along lines which curve around great circles on a sphere, this means that a larger radius does not always lead to a larger circle. Once the radius exceeds one quarter of a great circle’s circumference, circles actually begin to shrink, as illustrated in Fig. 19-5. The distance from the North Pole N to point A is one eighth of the circumference of the sphere, from N to B is one quarter of the circumference, and from N to C is three quarters of the circumference. However, the circumference of the circle around N with radius NB (the equator) is greater than the circumference of the latitudinal circle around N with radius NC. A larger radius does not guarantee a larger circumference. In fact, the circumference of a circle with radius r in elliptic space is less than 2πr.
Fig. 19-5
Similarly, the area of a circle with radius r in elliptic space is A < πr2. If a circle has a very small radius, then it will be almost flat and have area close to πr2 and circumference near 2πr (though slightly less in both cases). The larger a circle’s radius becomes, the further it will be from having the area and circumference of a circle in Euclidean space with the same radius.
Technically, elliptic space violates the first postulate. Between the North and South Poles, any of the longitudinal great circles counts as a straight line, thus, there are many straight lines between two points. A trick to overcome this is to call the North and South Poles together as one single point. If each pair of antipodal spots on a globe is viewed as a single point, then the resulting geometry is called projective space, which has all of the above properties in common with elliptic space.
Euclidean geometry is based on flat planes. A flat object is said to have zero curvature. Elliptic geometry is based on planes that are positively curved, like the surface of a sphere. Hyperbolic geometry is based on planes that are negatively curved and frilly like certain kinds of seaweed. The three kinds of curvature are illustrated in Fig. 19-6.
Fig. 19-6
Just as with elliptic space, any tiny piece of hyperbolic space is almost flat and will have properties close to those of euclidean space. As pieces get larger, however, they contain more of the warps, bends, and frills that are symptomatic of this space. The word “hyperbolic,” by the way, has roots meaning “excessive” and “exaggerated.”
The reason why hyperbolic space is the one that satisfies Postulate 5b can be seen in Fig. 19-7. Because of all the frilly waves of space through which a line can travel, there is more than one line through point P that will never cross . Note that these do not have the usual property of parallel lines; they are not every-where equidistant.
Fig. 19-7
In hyperbolic space
Planes are negatively curved and infinite in area.
Lines are infinite in length.
The angles of a triangle add up to less than 180°. One such example is illustrated in Fig. 19-8.
Fig. 19-8
Just as with elliptic space, the larger a triangle is, the further its angle sum will be from 180°. A tiny triangle will have angles that add up to just less than 180°. The largest triangles in hyperbolic space have angle sums of almost zero. Because of this, there are no similar triangles of different sizes in hyperbolic space.
Finally, a circle has more circumference and encompasses more area in hyperbolic space than it would in the euclidean plane. Thus, a circle with radius r in hyperbolic space has circumference C > 2πr and area A > πr2.
