Fallen Slices of a Sphere
*If you wanna copy one (or more) of these applets, here are the links:*
Fallen Slices of a Circle
Fallen Slices of a Sphere
Aproximation (Lower Sum) of "Fallen Slices of a Sphere"
Other Deformations of the Sphere
Imagine a circle above a line. If we make several vertical cuts in the circle and let the pieces fall towards the line, what curve is formed?
Fallen Slices of a Circle
Well, what if we do that with a sphere? That is, slice it ad infinitum so that each slice is a circle. What shape will the solid take when all the pieces fall?
Fallen Slices of a Sphere
A very interesting thing (besides the shape) is that, due to the transformation applied to the sphere, the volume of the solid remain unchanged, but the surface area does not.
(Note: you can move the points to get a better fit).
Aproximation (Lower Sum) of "Fallen Slices of a Sphere"
This property can be observed using calculus, but you can conclude, more informaly, that since the shape of each slice (that is, each circle) does not change, its area is the same and, therefore, the volume (the "sum of infinite pieces of areas") of the solid formed is the same as that of the original sphere.
This sort of argument (Cavalieri's principle) was used by Archimedes to conclude that the sum between the volume of the double cone and the volume of the sphere is equal to the volume of the cylinder.
We can think of some more transformations. For example, if the slices of the sphere fall on a cylinder instead of a flat surface, what solid will be formed? What if we slice the "Fallen Slices of a Sphere" (with vertical cuts perpendicular to the cuts made in the sphere) and let the pieces fall back into the plane? The answers are the solids on the left (red) and on the right (blue), respectively, in the next applet (can you find the functions?).
Other Deformations of the Sphere
Again, both transformations conserve the volume but not the surface area of the solid. Note that all the transformations presented here are the result of the difference between the functions that define the two hemispheres of the sphere and the function that defines the lower half of a horizontal cylinder.
Finally, it's a cool fact that if we slice the sphere vertically in one direction, drop it in one plane, slice vertically in another direction perpendicular to the first and again drop it (blue solid on the right), the final result is different from that obtained by slicing in both directions and only after dropping (solid of revolution of the "Fallen Slices of a Circle").