Spherical Models
Can you flatten a sphere? The answer is NO, you can not. This is why all map projections are inaccurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.
However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.
One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.
The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.
So what does this flattened sphere approximated by conical strips look like? Check the image below.
So what does this flattened sphere approximated by conical strips look like? Check the image below.
It’s an interesting puzzle to put together and has possibilities for engineering by manufacturing it out of flexibly materials.
Here’s the image below to print it out, if you have the patience and interest to want to try it yourself.
Here’s the image below to print it out, if you have the patience and interest to want to try it yourself.