Is optimization of torus possible? I'm a high school student needing to write an maths essay with 20 pages of length. I have an idea of doing an optimization problem with a torus. Either to minimise the surface area with a fixed volume, or to maximise the volume with a fixed surface area. But I'm not sure if such maximum or minimum exists since r can shrink infinitely. Could this idea work, or if not, is there any similar things about the Torus that I can do? Thank you!
 A: Unfortunately this idea will definitely not work. The volume is proportional to $Rr^2$ and the area is proportional to $Rr$. So for a constant volume you have $r$ proportional to $R^{-\frac12}$ and the area therefore proportional to $\sqrt R$. So the bigger and thinner the torus, the greater the area, without limit. “Greatest volume for a given area” is equally frustrating in the opposite direction.
If you are keen on tori, what about the Seven-Colour Theorem?
The Five-Colour Theorem says that on a plane or a sphere (they are the same thing) you can colour a map with just 5 different colours. The proof is ingenious but not beyond your abilities, and as a bonus it is almost certainly outside what you have been taught. It involves:

*

*The observation that every map must have at least one country that has fewer than 6 neighbours - from which it’s easy to deduce that you can colour any map with just 6 different colours.

*A clever trick which takes you from “6 colours” to “5 colours”.

Where this comes in with tori is that this theorem is not true on the torus. It is possible to draw, on a torus, a map which requires 7 colours.
The interesting part of this paper will then be “How does this proof break down on a torus?”
There are actually two breakages when you go from sphere to torus - corresponding to 1 and 2 as I numbered them above. I shan’t say more. You will have fun identifying and explaining them.
Yes, I know that the four colours are actually enough on the sphere or plane. However, the jump from “5 colours” to “4 colours” (a) is graduate-level stuff, (b) involves no new concepts, and (c) is exceptionally ugly. So for the purpose of this paper, “5 colours are enough” is (a) true, (b) beautiful, and (c) the perfect jumping-off point for your torus work.
In addition to your 20 pages, you will be able to include an edible appendix - a bagel, donut or onion ring with a map drawn on it and coloured with icing or food colouring in 7 distinct colours - all mutually touching.
A: The formulas for the volume and surface area of a torus with cross-section radius $r$, and torus radius (distance between centre of torus and centre of cross-section) $R$, are:
$A=4\pi^2rR$
$V=2\pi^2r^2R$
Note that we require $0<r<R$, otherwise the torus does not exist (unless you allow a spindle torus, in which case you would use different formulas for the area and volume).
From the area formula, we get $R=\dfrac{A}{4\pi^2r}>r$ so $r<\frac{\sqrt{A}}{2\pi}$. Plugging this into the volume formula, we get $V=\frac{Ar}{2}$ where again $r<\frac{\sqrt{A}}{2\pi}$. So for fixed $A$, the maximum volume is $\frac{A\sqrt{A}}{4\pi}$.
From the volume formula, we get $R=\dfrac{V}{2\pi^2 r^2}>r$ so $r<\left(\frac{V}{2\pi^2}\right)^{1/3}$. Plugging this into the area formula, we get $A=\frac{2V}{r}$ where again $r<\left(\frac{V}{2\pi^2}\right)^{1/3}$. So for fixed $V$, the minimum area is $2V\left(\frac{V}{2\pi^2}\right)^{-1/3}$.
So far, your questions are not enough for a 20 page paper. (If your paper is an IB math IA, the guideline is 12-20 pages.)
