Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks)

by Banks, Robert B.

  Of course, all the water forming these vast ice sheets came from the oceans. During the ice age, volumes of frozen ocean water were so gigantic that, according to Gross (1985) and Byalko (1987), the level of the ocean was approximately 120 to 150 meters below the present level. Most of what we now call the relatively shallow “continental shelf” was dry land.

  There is a fascinating anthropological-geographical feature related to the great ice age of twenty thousand years ago. The Bering Strait is a relatively narrow (88 kilometers) passage connecting the Pacific and Arctic Oceans, separating Siberia from Alaska. Even now the maximum depth of the Strait is only 50 meters. During the ice age there was certainly a land bridge, perhaps 100 meters above sea level, connecting Asia and North America.

  Anthropologists believe that tribes of people from Northern Asia crossed this bridge, over time periods of hundreds to thousands of years, and slowly migrated eastward and southward throughout North, Central, and South America. In this regard, in their very interesting book dealing with the Mayan civilization of Mexico, Schele and Freidel (1990) indicate that “…archaeological history begins with evidence of the first people moving into the Yucatán Peninsula about eleven thousand years ago.” The “Bering bridge” hypothesis is certainly plausible and an interesting thing to think about.

  Back to the ice-covered circular land mass we built at the North Pole. It is estimated that the present volume of ice in the earth's glaciers and ice caps is about 30 × 1015 m3. Although this amount represents only about 2.1% of all the water in the world, these ice caps and glaciers contain nearly 80% of the earth's fresh water.

  According to Oerlemans and van der Veen (1984), if all this ice were to melt, the level of the oceans would rise by approximately 72 meters. These researchers estimate that the current rate of melting is sufficient to cause an ocean level rise of about 0.5 centimeters per year or about 50 centimeters per century. At this rate, it would take about 15,000 years for all the ice to melt. It is highly probable that the melting rate will increase if greenhouse warming becomes a serious problem.

  In addition to the melting of ice, global warming would have another significant effect on raising ocean levels: thermal expansion of sea water. For example, if the temperature of all ocean water were suddenly increased from 15°C to 20°C, the level would rise by about 3.5 meters because of thermal expansion of the water.

  There is certainly not complete agreement among scientists regarding the long-term rate of glacier and ice cap melting, so we set that matter aside. However, there is no question concerning the enormous annual production of ice. It has been estimated by Oerlemans and van der Veen that about 3.0 × 1012 m3 of ice are produced annually, mostly in Antarctica (80%), a great deal in Greenland (15%), and the remainder in the Canadian and Russian Arctic islands.

  Approximately 10% of this annual ice production is lost to the oceans by outright melting during the warmer seasons. However, nearly 90% of the total—and virtually all of Antarctica's ice—falls into the ocean by calving: the breaking off of large detached pieces. This is the process by which enormous icebergs are created.

  The annual production and subsequent thawing of 3.0 trillion cubic meters of ice each year involves a great deal of frozen and then melted water. The corresponding melting rate is fifteen times larger than the flow over Niagara Falls and five times more than the discharge of the Mississippi River. It would fill Lake Erie in two months. It is double the total consumptive use of water in the entire world.

  It is important to remember that all this water from glaciers and icebergs, before it mixes with ocean water, is fresh water. What a dreadful shame that we are unable to use it.

  FIG. 21.3

  Hypsometric curve of the oceans. (From Sverdrup et al. 1942.)

  What Happens If We Dump All the Land into the Ocean?

  The hypsometric (or bathometric) curve of the oceans is shown in figure 21.3. As indicated previously, the world's land mass covers an area Ac = 1.49 × 1014 m2 (29.2% of the total) and the world's oceans extend over an area A0 = 3.61 × 1014 m2 (70.8%).

  As shown in the figure, the hypsometric curve features a continental shelf to a depth of about 135 meters. This area was above sea level during the ice age. The shelf is followed by a zone called the continental slope to a depth of around 3,000 meters. From this depth to approximately 6,000 meters is a vast abyssal plain or deep sea platform.

  Finally, the hypsometric curve displays a relatively small region containing trenches with very large depths. The greatest known depth is 11,035 meters in the Marianas Trench south of Guam in the Pacific Ocean.

  The average depth of the oceans is 0 = 3,740 m. Consequently, the volume of the oceans is V0 = 1,350 × 1015 m3. The average elevation of the earth's land mass is c = 840 m and so the volume is Vc = 125 × 1015 m3.

  Recall that we had moved the entire above-sea-level land mass to the North Pole to construct an axially symmetric mountain conforming exactly to the hypsometric curve of the earth's continents. The volume of the land mass was Vc = 125 × 1015 m3. Along with the land, we moved all of the earth's glaciers, ice caps, lakes, and rivers with a total volume Vw = 37.5 × 1015 m3. So the entire amount moved was V = 162.5 × 1015 m3.

  After our mountain construction project is completed, the decision is made to dump the entire mass—dirt, rock, sand, ice, everything—into the ocean. What happens?

  The total area of the world is A = 5.10 × 1014 m2. So the enormous land mass that we dump into the ocean, along with all the ice and fresh water, raises the ocean level by an amount h = V/A = (125 + 37.5)(1015)/(5.10)(1014) = 318 m.

  However, this dumping operation still leaves an area, formerly covered by land, where the depth is only 318 meters, and a much larger area where the depth is much more than this. So, in an effort to be neat and orderly, we launch a massive project of underwater bulldozing to level off the entire bottom of the oceans.

  The outcome is that our earth is now a perfectly smooth sphere covered entirely by an ocean with a constant depth of h = V/A = (1,350 + 37.5)(1015)/(5.10)(1014) = 2,720 m. All is serene.

  22

  Cartography: How to Flatten Spheres

  It is said that Columbus must surely have been an economist, a stockbroker, or something like that to have successfully raised the funds for the journey of his three small ships across the Atlantic back in 1492. But even though Columbus did receive fairly strong financial support from the Spanish crown, he definitely was not an economist. In fact, he was a very competent seaman and, more importantly, he possessed considerable knowledge of and experience in the principles of navigation and geography.

  The fact that Columbus mistook the vast land mass we now call the Americas for south or east Asia was not entirely his fault. By the late fifteenth century, when he made his journeys, there were quite accurate maps of most of Europe and the Middle East and fairly good maps of much of Africa and Asia. However, the New World simply did not exist!

  Indeed, much of the geography of the Old World had been known since the time of the great astronomer Claudius Ptolemy of Alexandria (87–150). Even before that, the Babylonians, Persians, and Egyptians had produced maps of the then-known world and maps of the skies. By the fifth century, the Greeks had laid the foundations of what we now call cartography: the science and art of maps.

  Over the centuries, a great many people made contributions to the development of maps and globes of the world and the heavens. Two names especially stand out in the long history of cartography. One is Ptolemy and the other is Gerhard Kremer (1512–1594); we know him better by his latinized name, Gerardus Mercator. Though born in Holland, he spent nearly all of his long life in Germany. In 1569 his famous Great World Map was published. Mercator's map, or more precisely, the Mercator projection of the world, is familiar to all of us; we shall return to it shortly.

  There are numerous books devoted to the history of maps and cartography; those of Bagrow (1985) and Brown (1949) are recommended. Maps of the Heavens is the title of a beautiful
book by Snyder (1984) dealing with cartography of the skies.

  How Map Projections Are Classified

  Suppose we have a globe or some other spherical object on which we have a map of the world, or indeed any kind of configuration or pattern of lines. Suppose also that we want to transform this three-dimensional sphere to a two-dimensional plane in order to produce a map of the world or pattern of lines on a nice sheet of flat paper.

  Basically there are two methods to accomplish this. The first method is to place the sphere—the globe—under a powerful hydraulic press, like those used in steel mills, and flatten the heck out of it. We discard this procedure because it is not scientific. The second method is to utilize various techniques to transform the sphere mathematically onto a plane.

  It turns out that it is impossible to transform a sphere to a plane without some kind of distortion in the map you are making. If you want to retain some features of the map in the transformation from sphere to plane (e.g., angles or distances), then you must sacrifice other features (e.g., areas or directions).

  There are three types of surfaces we can use to transform or project a three-dimensional sphere onto a two-dimensional plane. These are illustrated in figure 22.1. The first is a plane surface itself; this is called the azimuthal projection. The second is the conical projection. In this case, after the sphere's pattern is projected onto the cone, a cut is made along the side of the cone and the surface is spread out to provide a flat map. The third is the cylindrical projection. Again, after the globe's pattern is projected, the cylinder is cut and laid flat.

  FIG. 22.1

  Basic types of map projections.

  As shown in the figure, these three basic geometries, the plane, cone, and cylinder, are tangent to the sphere at a certain point or along a certain curve. However, each of the three surfaces could actually intersect the sphere at a specified place; these are called secant projections. Although there are many applications for secant maps, we shall not consider them here.

  From the preceding, we note that the first way to classify map projections is according to the type of plane surface onto which the sphere is projected: azimuthal, conical, or cylindrical.

  A second way to classify projections is according to the projection source. For example, suppose we want to construct an azimuthal projection. That is, as shown in figure 22.2, we want to project point P of the sphere onto the plane. We could select the center of the sphere, point A, as the projection source and obtain point P1 on the plane. This is called a gnomonic projection. Alternatively, the projection source could be at point B, located directly opposite the point of tangency T. This is called the stereographic projection; it produces point P2 on the plane. Finally, we could move the projection source all the way to infinity at point C to generate point P3 on the map. This is the orthographic projection.

  FIG. 22.2

  Types of projection sources, (a) APP1. gnomonic. (b) BPP2: stereographic. (c) CPP3: orthographic.

  There is a third way by which map projections can be classified. This classification specifies which properties we want to preserve as we transform from a sphere to a plane. There are four such properties:

  Equal angles. This property, usually called the conformality property, assures that any angle on the sphere is transformed without change onto the plane. For example, the meridians (lines of constant longitude) and the parallels (lines of constant latitude), which are perpendicular on the sphere, are also perpendicular on the plane.

  Equal areas. This property stipulates that every small region on the sphere has the same area after it is transformed onto the plane.

  Equal distances. This property requires that distances from the center of a map projection be the same on the sphere and the plane.

  True directions. This property means that directions from the center of a map projection must be identical on the surface and the plane.

  As indicated above, it is not possible to preserve all these properties in a particular transformation. There must be tradeoffs. For example, the flag of the United Nations displays an azimuthal stereographic conformal (equal-angle) projection of the world with the point of tangency at the North Pole. The other properties are not preserved on this projection. Table 22.1 gives a summary of the various ways by which map projections are classified.

  TABLE 22.1

  Over the past two thousand years or so, many hundreds of different kinds of map projections have been devised. Of this very large number, perhaps fifty kinds find some type of present-day application and maybe a dozen or so can be described as extremely useful. Not surprisingly, mankind's relatively recent ventures into space exploration have produced renewed interest and great progress in mapping not only the earth but also the moon and nearby planets.

  We are going to take a close look at only two of the many different kinds of projections. If you want to study cartography and map projection in detail, numerous books are available; some of the best are those of Raisz (1962), Snyder (1987), Snyder and Voxland (1989), and Snyder (1993).

  In your studies, it will be very helpful if you know something about spherical trigonometry and solid analytic geometry. If you would like to become knowledgeable in cartography, you should study calculus and differential equations and also an interesting branch of mathematics called differential geometry.

  The two projections we are going to examine in some detail are the Mercator projection and the Lambert azimuthal equal-area projection. We start with the first of these.

  The Mercator Projection and Loxodromes

  This projection was devised by the great Flemish cartographer Gerardus Mercator (1512–1594) and first appeared in 1569. It is developed onto a cylinder in such a way that angles are preserved (i.e., it is conformal). Without doubt, this is the most famous of all map projections. It has the great advantage that paths on a sphere that hold constant angle to the meridians (lines of constant longitude) transform onto a plane as straight lines. This feature is extremely useful in navigation. However, the Mercator projection has the great disadvantage that regions at high latitudes (at, say, 50° or more) are considerably enlarged and distorted. Indeed, the poles are infinitely large in area.

  We begin our studies involving the Mercator projection with a geometrical analysis of a small area on the surface of a sphere. The following mathematical relationships are obtained:

  where λ is the longitude, ø is the latitude, and θ is the angle that a certain path makes with a meridian. An incremental distance along the path is ds and R is the radius of the earth.

  We rewrite the first of equations (22.1) in the form

  The lower limits on the integrals indicate that when λ = λ0, ø = ø0. If we assume that θ is constant, that is, the path is always at the same angle with the meridians, then (22.2) can be integrated to give

  where the angles are expressed in radians. This equation is called the rhumb line or loxodrome. It transforms a straight line on the plane of a Mercator projection to a curved path on a sphere.

  The second of equations (22.1) provides the expression

  This simple equation gives the length S of the loxodrome between the point located at (λ0, ø0) and any other point (λ, ø).

  In a moment, we return to our analysis of the Mercator projection and the Lambert azimuthal equal-area projection. But first, we need to take a close look at the following very remarkable phenomenon.

  The Strange Behavior of the Mysterious Honking Bird

  This is an appropriate place to describe and analyze the unbelievably strange behavior of the fascinating (and mythical) “honking bird.” This extremely interesting winged creature is hatched on or very near the earth's equator. Then, when it reaches a certain age, it mysteriously begins an east-northeastward migration, which ultimately terminates at the North Pole, although a small fraction of these birds prefer to fly to the South Pole. Whether heading toward the north or the south, the honking bird nevertheless flies on a path that always holds a constant bearing with the meridians.
Typically, the bearing angle θ is around 80°.

  At this point we pause to let our mathematics catch up with us. To simplify our equation, we let ø0 = 0 in (22.3). Then we solve for ø to obtain

  This expression provides the value of ø (latitude) for any value of λ (longitude).

  In addition, from (22.4), we have, with ø0 = 0,

  This expression gives the total length of the loxodrome, S, between the equator and any latitude ø.

  Back to our honking bird: We select the value θ = 80° as the bird's “true heading” from north. The source of its flight is on the equator (ø0 = 0) near its well-known (but mythical) breeding ground along the northeastern shore of Lake Victoria in Africa (λ0 = 35°E).

  Remembering that ø and λ must be expressed in radians, we use equation (22.5) to compute the bird's position, ø = ƒ(λ).

  At this point it is suggested that you get your globe and world atlas. It will be helpful if you have them for the following analysis. With θ = 80°, our honking bird is flying in an east-northeast direction. Hence, from (22.5), with λ0 = 35°E, you easily calculate that when the bird is at longitude λ = 125°E (i.e., one-quarter of the way around the world), it is at latitude ø = 16°N. Your globe indicates that this location is east of the island of Luzon in the Philippines. Likewise, when the bird is at λ = 215°E (i.e., 145°W; halfway around the earth), then ø = 30°N; this is northeast of Hawaii.