Problem 31 - Astronomy Journey to the Cosmic Frontier

Question

Suppose you made a triangle in your backyard. You used a stretched string to make three geodesics to form the sides of the triangle. You then used a protractor and found that the sum of the angles of the triangle was \(180^{\circ} .\) You know that the surface of the Earth is curved, so why didn't your triangle contain more than \(180^{\circ}\) ?

Step-by-Step Solution

Answer

The triangle is small enough for Earth's curvature to be negligible, approximating it as flat.

Step 1: Understanding Geodesics

Geodesics are the shortest paths between two points on a curved surface. On the Earth's surface, geodesics are generally segments of great circles. However, when we make small shapes, like your backyard triangle, the curvature of the Earth has negligible effect.

Step 2: Local Flatness Approximation

The backyard triangle is likely small in comparison to the curvature of the Earth. For small shapes or distances on the Earth's surface, it can be approximated as a flat plane. Thus, the rules of Euclidean geometry apply.

Step 3: Applying Euclidean Geometry

In Euclidean geometry, the sum of the angles of a triangle is always \(180^{\circ}\). Since the triangle is small enough to be considered flat, it adheres to Euclidean principles.

Key Concepts

Geodesics

Geodesics are fascinating paths that represent the shortest route between two points on a curved surface. Think of them as the straight lines of the spherical world, much like how we draw lines on a map. When dealing with the Earth, these geodesics are not straight lines in a traditional sense but rather parts of what are called great circles.

• On a globe, the equator and lines of longitude are great circles.
• When moving from one point to another along a geodesic, you're essentially following the most direct path possible on that curve.
• For large-scale travel, like flights, geodesics are essential to minimize distance.

However, smaller scales, like your backyard triangle, bring us to the point of local flatness approximation, which makes the concept of geodesics less impactful.

Great Circles

Great circles are intuitive yet essential concepts in understanding navigation and geometry on spheres. Imagine slicing through the Earth with a plane that passes through its center. The intersection between this plane and the Earth's surface forms a great circle.

• These circles carry the largest possible radius on a sphere.
• They serve as natural pathways for geodesics, illustrating the shortest path between points across the sphere.
• An interesting property of great circles is that they divide the sphere into two equal halves.

So, when we talk about geodesics on a global scale, we're really talking about arcs of these great circles. However, the bigger picture shifts slightly when we shrink our frame of reference to something like a tiny triangle in your backyard.

Local Flatness Approximation

The local flatness approximation is a helpful concept when dealing with small-scale geometry on a curved surface. Imagine trying to flatten a piece of orange peel; the curvature would disappear on a very tiny section.

• This approach allows us to treat a curved surface as if it were flat, but only over a small area.
• By simplifying the curved Earth to a flat plane, we can apply Euclidean geometry to small shapes like your backyard triangle.
• This approximation hinges on the scale. If the object is tiny compared to the Earth's size, local flatness holds true.

Therefore, a small triangle in your backyard can adhere to the Euclidean rule, where the sum of angles in a triangle equals \(180^{\circ}\), just as it would on a flat table. This explains why our everyday experiences of geometry seem flat even though we live on a curved planet.