What do kale, coral, and potato chips (crisps) have in common? It turns out that the surfaces of these objects have an unusual type of geometry. Recently Clara and Phil were talking about such objects, in particular their surfaces. It reminded Clara of her days as a student. Quickly the conversation turned to math.
Clara: There was a moment in my education that changed how I view mathematics. I was taking a Modern Geometry course, and the instructor started a section on non-Euclidean geometries. Up until this point, I was used to hearing the term geometry, singular, not geometries, plural. As far as I was concerned, Euclidean geometry (that is, geometry on a flat surface) was the only geometry, or the only way to study points, lines, shapes, and space. Throughout the semester we explored something totally different. These non-Euclidean geometries could be used to prove seemingly outlandish theorems, claiming things like rectangles do not exist and the interior angles of a triangle add up to something other than 180°. As an example, the surface of some coral can be used as a model of a geometry known as hyperbolic geometry.
Phil: What is hyperbolic geometry? Why was it even developed?
Clara: It was largely developed in the 1800s by mathematicians Bolyai and Lobachevsky. They, like many others, had an issue with Euclid’s 5th postulate. Well, Euclid called it a postulate, but in the context of an axiomatic system, it’s like an axiom.
If you want a quick refresher of Euclid’s postulates, which are the building blocks of Euclidean geometry, this list provides a summary:
You can connect any two points with a line.
You can extend any line indefinitely.
You can describe a circle with a center point and a radius.
Right angles are congruent.
If two lines intersect a third line so that the sum of the inner angles on one side of the third line is less than two right angles, then you can extend the first two lines so that they intersect on that side.
Alternatively, sing along with TheSingingNerd:
Phil: Mmm... Euclid's 5th postulate is wordy and reads more like a theorem.
Clara: I agree, however mathematicians had not been able to prove it as a theorem resulting from only the first 4 postulates. Bolyai and Lobachevsky's main contribution to geometry was to use Euclid's first 4 postulates to define hyperbolic geometry, but not the 5th. Here is a great video by Uncommon Nonsense explaining this point in the context of hyperbolic geometry :
Phil: How can you model hyperbolic geometry? On a coordinate plane?
Clara: Hyperbolic geometry was initially modeled with a crazy looking thing called a pseudosphere.
There were several other mathematical models, like the Poincare disk model, that were used to prove the geometry was consistent. Hyperbolic surfaces, or surfaces that can be used to model hyperbolic geometry, occur naturally in coral reefs and leaves (particularly lettuce, kale and the like). They also occur not-so-naturally in potato chips/crisps.
Phil: Neat! So, not all geometries are modeled on a coordinate plane. How about modeling on the surface of a sphere, such as on the surface of the Earth? In fact, how do you even define a line on a curved surface?
Clara: Well, the surface of the sphere is used to model spherical geometry, another type of non-Euclidean geometry. A line is defined to be the shortest path between two points (also known as a geodesic). In spherical geometry, these lines always lie on great circles, which are the circles on the surface of the sphere that have the same radius as the sphere itself.
Here's something fun: You may remember learning that in Euclidean geometry, the ratio of circumference to diameter of a circle is a constant, regardless of the size of the circle. This constant is called "Pi". It turns out that this ratio is no longer constant on the surface of a sphere.
Phil: First I guess we need to be clear what we mean and define a circle on the sphere as the set of points equidistant to a fixed point, with distance measured as the shortest path on the surface.
Clara: Perfect. Here’s a visual of a sphere (say the surface of the Earth), with a circle labelled C outlining a "polar cap", with its center at the north pole. This circle is at a latitude 30 degrees away from the north pole. The curve labeled R in the diagram is actually the distance on the sphere to the north pole.
Let's use the surface of the Earth as an example. We'll take the radius of the Earth to be 6,371km. Using trigonometry, we can work out that
Since R lies on a great circle, it must be a fraction 30/360 of the circumference of a great circle, which works out to be approximately
We can then easily calculate the ratio,
Phil: Oh, so perhaps the readers can try it when the circle C is at a latitude 45 degrees away from the north pole, as in the following diagram:
Clara: Yes! In that case we get
You see, the ratio is not constant in this case, as it is on the plane. It is a different geometry. Can you reproduce this calculation? If you need help, check out this explanation by dangarbo10:
Phil: This is quite incredible that on the plane circles behave differently than on the surface of a sphere. I wonder what happens on a hyperbolic surface.
Clara: That's a great thing for our readers to explore! Another fun thing to research is real-life applications of non-Euclidean geometry. For example, check out Dr. Mark Liu's article about print-making using hyperbolic and spherical geometry.
Did you have fun? Let us know.