Question
I read that a torus can shrink so that it intersects with itself: a 2D cross-section would look like 2 circles merging, and a 3D shape shows the
central opening becoming a spindle. If a torus shrinks enough, so that it completely intersects itself, then does it become a sphere?
When I tried to solve the toroidal surface area & volume equations for a complete intersection, the results weren't equivalent to the spherical
equations. Am I simply doing something wrong, or did I misunderstand the shrinking process?
Anthony
Answer
I'm not sure what you mean by "shrink". Does shrink mean the outside diameter from one side of the torus to the other? Does it mean the inside diameter of the donut hole? Does it mean the diameter of the meat (annular region) of the donut? Intuitively and visually "shrink" to me simply means getting smaller, in which case a small torus is congruent with a larger one, the donut hole is still a hole, the hollow annulus is still a hollow annulus. I assume you are using the concept of limits to solve this rather than derivatives? I recommend that you rephrase your question and give the method of your solution. As an engineer I tend to solve these things differently than a mathematician. Also, you said that you "heard that a torus becomes a sphere", you may have successfully disproved your original source! Don't discount the possibility that your solution is correct! Lastly, consider whether the original source was working in a Euclidean or non-Euclidean space.
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