Tag Archives: Spherical Geometry

Map Projections

The world is not flat (citation needed). This is a very important aspect of our planet Earth; indeed, were our world flat it would not rotate on its axis the same way, would have to have an edge, and would probably crumble into a non-flat ball of rubble from its own sheer weight. We should all, therefore, be grateful that the world we live on is the 3D almost-perfect-sphere that it is. Cartographers tend to be a little less happy about our world’s roundness than others because it presents them with an irksome problem: how do we model a non-flat 3D world on a flat 2D surface? Initially this may seem like a straightforward issue. We could, for example, just draw the earth by what it looks like from space! Well this doesn’t quite work because, among other things, you would only have a 2D projection of the side of the earth facing you. Angles, sizes, and shapes get distorted, especially as we approach the edges of our disc. Perhaps we could imagine we have a globe that we cut a slice through from North Pole to South Pole, which we can spread out flat on a table! This method is unfortunately flawed as well, since the sphere will never lie flat regardless of how many cuts are made. Clearly this problem isn’t quite as straightforward as we initially hoped! Well luckily for us, mathematicians and cartographers love these types of problems, and many have offered many possible solutions, of which we will discuss a few.
The first and probably most familiar solution (called a projection) to this problem is the Mercator projection. This is probably the map that you had hanging on the wall of your elementary school classroom. Gerardus Mercator developed it in 1569. The goal of this particular projection is to maintain direction of rhumb lines (aka paths of constant bearing), which are lines that meet each meridian (lines between the two poles) at the same angle. It was particularly useful for navigation because of these lines. The Mercator projection’s biggest failure is generally that it distorts sizes more and more as we venture away from the equator, causing the poles to have infinite size. For example, Greenland and Africa take up roughly the same size on the Mercator projection when in actuality Africa is nearly fourteen times larger!

The Mercator Projection with red dots showing size distortion. Image: Stefan Kühn, via Wikimedia Commons.

The Mercator can be created by projecting the Earth onto a vertical cylinder with circumference equal to the circumference of the Earth. The next projection, called the Transverse Mercator, is obtained using a horizontal cylinder instead. This projection does not maintain straight rhumb lines like its counterpart and distorts scale, distance, and direction away from the central meridian used.

Transverse Mercator projection. Image: Public domain, via Wikimedia Commons.

Next among the more famous projections is the Robinson. This map features a flat top and bulging sides, with meridians starting and ending equidistant to each other but spreading out as they approach the equator. This projection can be seen as a compromising projection: it loosely preserves size, shape, and distance by not being exact in any one of them in particular. The Robinson, like the Mercator, is frequently used in classroom maps due to providing good guesses for relative shapes and positions and being very easy to understand.

Robinson projection. Image: Strebe, via Wikimedia Commons.

Next up are the Stereographic and Orthographic projections. These projections have existed for thousands of years. They were even used by the ancient Greeks! These two methods are projections of the sphere onto a plane, resembling what it would look like if you were to view the earth from space. The Orthographic projection maps along straight lines perpendicular to the tangent plane of the sphere (think looking through a window from space) while the Stereographic projection maps each point by constructing a line through a predefined point (like the north pole) and drawing where it intersects the tangent plane (think the image in a mirror that the earth is placed on). Thus, the main difference is that Orthographic takes the projection from infinity while the Stereographic takes the projection from a point on the sphere. This means that the Orthographic projection only shows one hemisphere, where the Stereographic can show the entire sphere (except the pole) but in a more distorted way. For both of these projections, directions are true from the center point. With the Orthographic projection, any line going through the center is a great circle.

Orthographic projection. Image: Strebe, via Wikimedia Commons.

Stereographic projection. Image: Strebe, via Wikimedia Commons.

There are many more projections to be found online and in books. Indeed, Wikipedia has a stellar list of some of the many different types. These projections all have different uses and are able to convey information in their own clever way, but they share at least one thing in common: they take quite a bit of ingenuity and creativity to come up with and they reflect a deep love and understanding of math in their creators.

Transition from Euclidean to Non-Euclidean Geometry

Euclidean geometry is the geometry that everyone learns and uses throughout Middle School and High School. In general, geometry is the study of figures, such as points, lines and circles in space. Euclidean geometry is specifically any geometry in which all of Euclid’s postulates and axioms hold. Axioms and postulates are the beginning of reasoning, they are simple statements that are believed to be true without proof. Assuming Euclid’s axioms and postulates found in his book Elements, the rest of Euclid’s classical geometry could be deduced. However, Euclid’s fifth postulate, the parallel postulate, was disconcerting because it was lengthy compared to the rest and not necessarily self evident. Many other ancient mathematicians were dissatisfied with Euclid’s fifth postulate. They thought that it was presumptuous and tried to prove it using lesser axioms or replace it altogether with something they thought to be more intuitive. But their proofs always included an assumption equivalent to the parallel postulate, so for centuries the postulate was assumed to be true.

Centuries passed and the postulate remained unproven; however, development to understand Euclid’s postulate continued into the eighteenth century. Perhaps the most well-known equivalent to the parallel postulate is Playfair’s Axiom, which states “through any point in the plane, there is at most one straight line parallel to a given straight line.” Arguably one of the most influential mathematicians, Carl Friedrich Gauss became interested in proving Euclid’s fifth postulate. After attempting to prove the postulate, he instead took Playfair’s Axiom and altered it, creating a completely new postulate. Gauss’ new postulate stated “Through a given point not on a line, there are at least two lines parallel to the given line through the given point.” With this Gauss had unearthed a completely new space that today is called hyperbolic geometry. However, he chose not to publish any of his results, wishing not to get caught up in any political strife. The work was later published  by Johann Bolyai and Nikolay Lobachevesky, who both had academic ties to Gauss.

Shortly after this discovery another type of Non-Euclidean geometry was discovered by Gauss’ student Georg Friedrich Bernhard Riemann. Riemann looked at what would happen when you altered Playfiar’s Axiom in the opposite direction than Gauss. Riemann’s alternate postulate is stated as follows, “through a given point not on a line, there exist no lines parallel to the line through the given point.” With this, what is known as elliptical or spherical geometry was discovered.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry. Image: Anders Sandberg via Flickr.

Spherical geometry provides a somewhat simpler model then hyperbolic geometry. Anyone who has ever attempted to wrap a basketball in paper understands that there are some discrepancies between the two surfaces. For instance, triangles behave quite differently than they do in Euclidean geometry. In hyperbolic space, the angles of a triangle add up to less than 180 degrees, and in spherical space, they add up to more than 180 degrees. Spherical geometry also has other inconsistencies with Euclid’s initial assumptions other than the parallel postulate. In Leonard Mlodinow’s book Euclid’s Window, the author describes how Riemann’s space was inconsistent with other of Euclid’s postulates. For instance, Euclid’s second postulate states that “any line segment can be extended indefinitely in either direction.” In spherical space this is not true; the lines, or great circles, have a limit to their space, namely two pi times the radius of the sphere. Mlodinow describes how Riemann saw this postulate as “only necessary to guarantee that the lines had no bounds, which is true of the great circles.” Also, Euclid’s first postulate became less clear, “Given any two points, a line segment can be drawn with those points as its endpoints.” This postulate can be used to easily describe whether a point is between two other points. However, on the globe, choosing two points on the equator such as Ecuador and Indonesia it is difficult to say whether a third point, Kenya, is “between” them. The problem is that there are two ways to connect the points, one passing over North America and another passing over Africa.

For much of our day to day lives Euclidean geometry works great, because on a local scale we appear to live on a flat world. I can go to a soccer field and trust that it will take four 90 degree turns to walk around the perimeter, or that the Pythagorean theorem will work to describe the path between opposite corners. But looking at a larger scale, the surface that we live on is spherical and has different properties than the flat plane. It is interesting to see how Gauss and Riemann, going against the grain of conventional mathematics, led to new and vast fields of undiscovered mathematics. To me, this shows how mathematics is just as much an experimental science as physics or engineering. These new discoveries of mathematical spaces made possible Einstein’s physical description of the space in which we live. Mlodinow closes his section on Gauss and Riemann saying, “though thoroughly remodeled, geometry continued to be the window to understanding our universe.” Even though the properties of these new geometries differ from classic Euclidean geometry and may have more or less practical use, they are just as important. From Euclid up until Gauss, mathematics was largely pragmatic, but the discovery of these new geometries highlights how math can be appreciated for its own sake.

References:

Case, William A. Euclidean vs. Non Euclidean Geometries. Web. http://www.radford.edu/~wacase/math%20116%20section%207.4%20new%20v1.pdf

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Mlodinow, Leonard. Euclid’s Window. Touchstone New York, 2001. Print

Weisstein, Eric W. “Non-Euclidean Geometry.” http://mathworld.wolfram.com/Non-EuclideanGeometry.html