Area of a sphere

Question

I want to know if the following GIF is accurate or not.

I saw that we cannot flatten a sphere without a deformation (world map problem).

And the GIF is actually "rolling" the sphere on a plane, projecting it so does this implies that the GIF is just here to help understand or is it false ?

I know that we could integrate to calculate the area but I wanted to know if this GIF was accurate !

Here is the GIF : https://m.imgur.com/gallery/5RE0Twe

i thought u just integrate using surface area of solid of revolution on a semicircle — Saketh Malyala, Jun 07 '17 at 18:26
The sphere inherits a Riemannian metric from $\mathbb{R}^3$ which then induces a volume form which can be integrated over the sphere. — Daniel Schepler, Jun 07 '17 at 18:32
The gif is flattening the sphere over a plane and it can't be done so it's not 100% correct ! — Rom, Jun 07 '17 at 18:33
What does this question have to do with topology? Area is not a topological property. — William Elliot, Jun 07 '17 at 19:29
" Topology studies the properties of space that are preserved under continuous deformations, this means streching and bending ". Well, I want to know is this space propriety called area is kept under the deformation :) Try to answer the topic and try not to contest a specific tag, that's useless. — Rom, Jun 07 '17 at 19:55
Agreed, splitting hairs over tags is rather useless. However for future reference, cutting the sphere in half isn't a continuous transformation, so it isn't really in the domain of topology — CulDeVu, Jun 07 '17 at 21:36

score 3 · Accepted Answer · answered Jun 07 '17 at 21:48

Answering in a more calculus-oriented way, because it's obvious that's what OP is asking for.

It is true that you can't flatten a sphere onto a flat plane with a finite collection of pieces, but there's nothing stopping you from using an infinite number of pieces. The first part of the gif, before connecting all the little pieces, you're correct in that not being completely accurate. However, after cutting the sphere an infinite number of times, that geometric trick can be done.

The real issue with the gif, however, is that it gives no intuition as to why the area is related to sine, instead of some other wave-like curve. If it helps you remember, well the by all means use that animation as you see fit. However, if you want a deeper explanation, these gifs won't help you.

score 0 · Answer 2 · answered Jun 07 '17 at 21:20

Topology concerns about continuous maps, as you said the continuous map can make the figure change, so does it's area. For example, the map $$f(x)=2x$$ maps an interval $[0,1]$ to $[0,2]$. "The area" of $[0,1]$ is 1 and "the area" of $[0,2]$ is 2. So continuous maps do change the area. I hope this expression can convince you facts from topology of a figure tell nothing about its area.

Then if you want to know how to compute the area of a figure, for example sphere, then I suggest you take a look at a course in analysis and integral to see how. Or easier, here how to find surface area of a sphere

Area of a sphere

2 Answers2