Honors Project: Portraits of the Earth – A Mathematical Look at Maps
The Pennsylvania State University - Dr. Juan Gil
A look at spherical maps, Gaussian curvature, and impossible triangles
January 2021 - May 2021
This project is an honors-credit grade for a Calculus and Vector Analysis course. I used mathematical concepts I learned in class to understand a given reading material concerning Gaussian curvature. This understanding was used to give a 15-minute talk to my professor on the subject.
Background
Through the Schreyer Honors College, there are a certain number of honors credits I must complete each semester. For Spring 2021, I decided to take Calculus and Vector Analysis (MATH 230) for honors credit. The project I had to complete relates to the book Portraits of Earth – A Mathematical Look at Maps by T.G. Freeman. A group of two other students and I read a few chapters out of the book that covers topics such as distances and shortest paths on a sphere; angles, triangles, and area on a sphere; and curvature of surfaces. Each of us took a chapter, read it, and gave a 15-minute talk on it to our professor and the rest of the group.
My Involvement
The chapter that I was assigned was entitled Curvature of Surfaces. In the previous two chapters/talks, the following topics were discussed:
- How to measure the shortest distance between two points on a sphere (great circles)
- Creating impossible triangles using spherical triangles, where angles sums can be greater or less than 180 degrees
- The relationship between angle sum and surface area for spherical triangles
- The idea that the above concept proves that no 2D map of a sphere is ideal
My chapter discusses how these concepts can be used to prove the final statement of the above list: that a PERFECT 2D map of a sphere (such as Earth) is IMPOSSIBLE.
In my talk, I drew a half-circle up on the board. Consider that r is the distance between two points on a sphere, a is the angle between the two points, and R is the radius of the sphere they lie on. A circle can be formed by all the points on the sphere that are a distance r away from the north pole (top), as seen in the image above.
To find the circumference of this circle, b must be found. Using some math, it can be found that:
sin(a) = b/R
Arc length gives the relation:
r = Ra
This means that:
a = r/R
Substituting in the first equation gives:
sin(r/R) = b/R
b = Rsin(r/R)
This means the circumference can be found by:
C = 2πRsin(r/R)
The value Rsin(r/R) is less than r, which can be proven by looking at a graph. It can also be eyeballed by considering, looking at the above image, that the arc of r is longer than b.
Now consider a map of the north pole of this sphere, with a scale factor M. This means that all of the points that are distance r on the globe would be a distance Mr away. All distances away from the north pole would create a circle of radius Mr. The circumference of this circle is 2πMr. Since M is simply a scale factor, we can say that it equals 1 for simplicity. This gives a circumference of 2πr.
The circumferences of these two circles do not equal. The spherical circumference is less than 2πr while the 2D circumference is exactly 2πr. Thus, a 2D map of a spherical object, no matter the scale, is impossible.
I also gave some insight into Gaussian curvature. I discussed the statement that “not all curved surfaces are curved alike”. If a circle is drawn on a curved surface, its Gaussian curvature will determine its circumference. If a surface has positive Gaussian curvature, the circumference will be greater than 2πr. Positive curvature can be thought of as a sphere, which has a curvature that pushes outward. If a surface has negative Gaussian curvature, the circumference will be less than 2πr. Negative curvature can be thought of a saddle, which has a section where curvature pulls inward into the surface.
