Question

Show that the surface of a sphere can’t be represented as a plane?

Answer 1

Hint: In this question we have to show that the surface of a sphere cannot be represented as a plane. A plain is a two-dimensional surface which has two axes $x$ and $y$. A sphere is a three-dimensional shape which has its coordinates in three-dimension. We will use the property from the Egregium theorem by Gauss to prove that a sphere cannot be represented on a two-dimensional plane.

Complete step-by-step answer:
We know that sphere is a three-dimensional shape. We can put a tangent plane in every point of the surface of the sphere but globally, we cannot do that.
A similar three-dimensional object is a cone which is different from a sphere is a cone. In a cone we can cut through the circle of base and then generate a two-dimensional shape which covers the entire surface of the cone. We can do the same thing with a cylinder.
Now based on the Egregium theorem by Gauss, we know that a surface can only be ruled if and only if its curvature is zero.
So, to show that a circle cannot be ruled we have to prove that its curvature is not zero. We know the curvature of a sphere is $K=\dfrac{1}{{{R}^{2}}}$, where $K$ is the curvature of the circle and $R$ is the radius of the circle. Now at any given time, the radius will be positive and greater than zero therefore, the curvature cannot be zero. Hence proved that the sphere can’t be ruled in a plane.

Note: It is to be noted that for this reason we have to approximate the sphere on a two-dimensional plane with other surfaces which can be ruled such as cones and cylinders. The real-life application of ruling the sphere on a plain by approximation is making the world map on a paper. Since the world map is an approximation, a globe is more accurate than a map.