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a_century_of_surprises.docxA Century of Surprises
The nineteenth century saw a critical examination of Euclidean geometry, especially the parallel postulate which Euclid took for granted. It says, essentially, that through a given point P not on a given line L, there exists exactly one line parallel to L. Any other line through P will, if extended far enough, meet L. Mathematicians sought a proof of this postulate for 2,000 years even though Euclid presented it as a self-evident idea not requiring proof. They did this because to them it was not self-evident, but instead, they thought, a consequence of previous results and axioms which were self-evident.
After failing to prove the parallel postulate, mathematicians wondered if there was a consistent “alternative” geometry in which the parallel postulate failed. To their amazement, they found two! The secret was to look at curved surfaces. You see, the plane is flat – it has no curvature. (Actually, its curvature is 0.)
Consider the surface of a giant sphere like Earth (approximately). To do geometry, we need a concept analogous to the straight lines of plane geometry. What do straight lines in the plane do? Firstly, the line segment PQ yields the shortest distance between points P and Q. Secondly, a bicyclist traveling from P to Q in a straight line will not have to turn his handlebars to the right or left. His motto will be “straight ahead.” Similarly, a motorcyclist driving along the equator between two points will be traveling the shortest distance between them and will appear to be traveling straight ahead, even though the equator is curved. Like his planar counterpoint on the bicycle, our motorcyclist will not have to turn his handlebars to the left or right. The same would hold true if he were to travel along a meridian, which is sometimes called a longitude line. (Longitude lines pass through the North and South Poles.)
Meridians and the equator have in common that they are the intersections of the earth with giant planes passing through the center of Earth. In the case of the equator, the plane is (approximately) horizontal, while for the meridians, the planes are (approximately) vertical. Of course, there are infinitely many other planes passing through the center of Earth which determine many other so-called “great circles” which are neither horizontal nor vertical. Given two points, such as New York City and London, the shortest route is not a latitude line but rather an arc of the great circle formed by intersecting Earth with a plane passing through New York, London, and the center of Earth. This plane is unique since three non-collinear points in space determine a plane, in a manner analogous to the way two points in the plane determine a line.
Geometers call a curve on a surface which yields the shortest distance between any two points on it a geodesic curve, or just a geodesic for short. This enables us to do geometry on curved surfaces. Imagine a triangle on Earth with one vertex at the North Pole and two others on the equator at a distance of the circumference of the earth. All three angles of this triangle are 90 °, so the angle sum is 270 °!! In fact the angle sum of any spherical triangle is larger than 180 ° and the excess, it was shown, is proportional to its area.
In this geometry, there is no such thing as parallelism. Two great circles must meet in twoantipodal points, that is, two endpoints of a line passing through the center of the sphere.
An even stranger geometry is needed for surfaces like a saddle. Imagine a saddle on a camel placed between its neck and its hump. (See Figure 11-1 .) The camel driver is sitting at the bottom of a U-shaped curve determined by the neck and hump. But he is also sitting on an upside down U-shaped curve determined by his legs which wrap partially around the body of the camel. On this kind of surface, parallel geodesics actually diverge! They get farther apart, for example, if they go around different sides of its neck. The stranger part is that through a point P not on a given line L on the surface, there are infinitely many parallel lines. Furthermore, angle sums of triangles on these saddlelike surfaces are less than 180 °.
Figure 11-1
These two geometries prepared mathematicians and physicists for an even more bizarre geometry required by Albert Einstein 1 (1879-1955), whose theory of relativity, in the first half of the twentieth century, would shock the world and alter our conception of the physical universe.
Another interesting development was in the making – the algebra of vectors. A vector is best viewed as an arrow. It has magnitude (length) and direction. It is an excellent candidate for representing velocity or force. After all, a speeding car has a numerical speed, like 60 mph – but it has a direction, too, like northeast. So we can represent the velocity vector by drawing an arrow of length 60 pointing in the northeast direction, as in Figure 11-2 .
Figure 11-2
Letting bold letters such as u and v represent vectors, mathematicians and physicists wondered how to do algebra with them, that is, how to manipulate them in equations as if they were numbers. The simplest operation is addition, so what is u + v ? Picture yourself in a sailboat (reading this book), and suppose the wind pushes you due east at 8 mph while the current pushes you due north at 6 mph. The sum of these vectors should reflect your actual velocity, including both magnitude and direction. Since the two velocities (wind and current) act independently, it was realized that the two vectors could be added consecutively, that is, one after the other, as is shown in Figure 11-3 , where the tail of the second vector v is placed at the head of the first vector u. The sum is a vector w whose tail is the tail of the first vector and whose head is the head of the second.
Figure 11-3
The magnitude of w is easy to find here since the three vectors form a right triangle. The Pythagorean Theorem tells us that the length of w is . The speed of the sailboat is 10 mph. The boat is not traveling exactly northeast because the angle between vectors uand w is not 45 °. The exact angle may be computed using trigonometry. It will be a bit less than 45 ° since v is shorter than u.
How do we evaluate sums of three or more vectors? The same way. Place them consecutively so that the tail of each vector coincides with the head of the previous one. The sum will be a vector whose tail is the tail of the first vector and whose head is the head of the last vector. Figure 11-4 demonstrates this for the sum s of three vectors.
Figure 11-4
Needless to say, all of this applies to forces impinging simultaneously on an object or even an atomic particle. Furthermore, these vectors do not have to lie in the same plane. Fortunately for scientists, the entire analysis can be done in three-dimensional space. Otherwise, vectors wouldn’t model the real world.
A vector can be described with the use of components, as follows. Place the vector in x, y, z space with its tail at the origin. The coordinates of the head are then taken as the components of the vector. We use the notation [a, b, c] here to distinguish vectors from points, that is, to distinguish components from coordinates. Mathematicians were delighted to discover that the geometric instructions for addition given above simplify greatly to a mere adding of respective components.
Thus if u = [a, b, c] and v = [d, e, f], then u + v = [a + d, b + e, c + f]. This also answered the question, how do we multiply a vector by a number? What would 2 × u be? It seems that it should correspond to u + u, or [a, b, c]+[a, b, c] = [2a, 2b, 2c]. This suggests that we have the right to distribute a multiplying number to each component of the vector.
Let’s summarize these amazing facts.
Here is an example to make this more concrete.
Having a penchant for generalization, mathematicians of the nineteenth century conjured up an n-dimensional world, called R n, in which points have n coordinates and vectors have n components! The above laws carry over quite easily to these n-dimensional vectors and yield an interesting theory which most find impossible to visualize. After all, R2 has two axes which are mutually perpendicular (meet at right angles). R3 has three axes which are mutually perpendicular. One adds the z-axis to the existing set of axes in the plane to get the three-dimensional scheme of R3. Now what? How do we add a new axis so that it will be perpendicular to the x-, y-, and z-axis? This is where imagination takes over. We imagine a new dimension that somehow transcends space and heads off into a fictitious world invisible to no mathematicians.
All of this might have seemed like a game to its founders until Einstein 2 showed that the universe is four dimensional. Time is the fourth dimension and must be taken into account when computing distance, velocity, force, weight, and even length! He posited that large massive objects (like our sun) curve the four-dimensional space around them and cause other objects to follow curved trajectories around them – hence the elliptic trajectory of Earth around the sun. Einstein 3 correctly predicted that light bends in a gravitational field. This was verified during a solar eclipse at a time when Mercury was on the other side of the sun and normally invisible to us. The eclipse, however, rendered it visible and it seemed to be in a slightly different location precisely accounted for by the bending of light in the gravitational field of the sun.
