Flat Shapes in a Curved World
Its common knowledge that the earth is a sphere, or something closely resembling it. This was discovered as early as the 5th century BC when the greeks mentioned it in their work. For thousands of years, experiments have been based on this truth. An example is the size of the earth which one of the first values was discovered by measuring light from the sun in a small portion and then scaling it to the whole world. But what if we didn’t have the sun and its light to tell us that the world was round? No planets or stars to base our understanding around. How would we know? This is the same problem when trying to find if space is curved or flat. Since there is nothing outside of space for us to base our understanding on we have to rely on what we can see and study inside our space
The scientists who discovered the shape of the earth and its size by using the sun and other planets and stars studied it extrinsically. This is where manifolds and their characteristics become useful again. Gauss proposes two methods method for investigating curved spaces externally by studying their extrinsic characteristics. The first is the Normals. Any point of a curved surface can be drawn perpendicularly. This perpendicular line can then be mapped to a corresponding point on a sphere. With this method, a cylinder would map to a straight line around a sphere and would therefore not be “curved“. A crude example of this technique is shown below.
The second method is by using osculating circles by intersecting a curved surface and finding its curve. This also renders the cylinder curveless.
Both of these methods require something outside of the manifold to determine its curvature, but that it’s always possible.
Let’s scaled down a couple of dimensions and imagine a two-dimensional shape living on a sphere. The two-dimensional shape would only be able to use Intrinsic Characteristics to discover the shape of its world. There would be no way to “map“ to another shape or observe the world from an outside perspective. So how would the shape know that it is in a curved world? The first method would be via the Pythagorean theorem. Normal euclidean geometry and laws to not hold on a sphere. In the theorem, square a + square b = square c, there would be no way to know which square was a, b, or c. The second method would be to examine directions. If a square was pointed directly west, walked forward, and made three turns while continuing to face the “same direction,“ at the square’s final stopping point it would no longer be facing west, instead of closer to the north. This is again because normal Euclidean geometry does not map to a curved surface.
Even though there are other limitations for scientists in discovering the true shape of the universe, these rules are still useful in characterizing manifolds. The unique properties of curved surfaces such as direction, distance, and displacements all make their study unique and separate from planes. Gauss’s extrinsic and intrinsic rules become useful in separating different curves and helping us understand the world around us.