Number of faces on a sphere?

Answer 1

A sphere is continuous with no edges, so a "face" doesn't really apply. It has a surface, and if you consider that a "face" then it has one. If you are looking inside and out, then two. If you do not consider a surface a face, then it has no faces.

I am not sure how you would explain it with Euler's formula.. I am curious as how you would explain it in this manner. Euler's formula shows that e^ix traces out the unit circle in the complex number plane as x ranges through the real numbers. Not sure what this has to do with faces of a sphere... Euler characteristic perhaps?

I am still not sure what this tells me about the number of faces that a sphere has. This basically proves the Euler Characteristic. You can map any polyhedron inside the sphere and get the same result. Are you implying that since any number of F can be mapped, that a sphere has infinite number of faces? That is pretty neat if that is the conclusion.

So in regards to answering the question, you can use your method and explain it to your students. I think they can handle it if you do it properly. Something of the nature:

Imagine a shape/polyhedron inside a sphere. Now map out or project the vertices's and edges of the polyhedron onto the sphere. It is like having the shadows of the edges and vertices's being mapped onto the wall of the sphere. The mapped image still has vertices's and edges and faces. They are the same amount of faces as the original polyhedron. Now imagine any polyhedron or shape inside the sphere. You can do the same with as many sided and faced shape as you want. Thinking this way you can view the sphere having infinite faces because you can map out any shape/polyhedron this way, no matter the number of faces.

Then they will have this embedded into their memory and think that a sphere will have infinite faces. If they ever major in math and take topology... they will learn that a sphere has no faces.Hope this helps.