For whatever reason, I was thinking about water heaters and considering how you would best wrap a coil of wire around it to keep the contents evenly warm.
This brought my mind to a math question.
Is it always possible to wrap a loop around a solid so that the following are both true?
a) There are no points, only lines, that we trace if we describe all locations where we are equidistant from the loop at two or more different locations, measuring from the closest points. (We call these locations of “coolness”.)
b) All lines of coolness are of even length.
c) The lines of coolness are all of the same coolness. I.e. there is not a warmer line and a cooler line, they are both equidistant from the coil.
Note: It is permissable for lines of coolness to cross and merge. These junctions are not considered “points”. However, all branches need to be of even length or for any two branches added together in length there is another two off the same junction which will add to the same length. If two junctions connect to one another then the connecting lines are split in the middle for the previous calculation.
For example, if we wrap a sphere with a coil fashioned onto a perfect circle around the middle of the sphere, two points will be formed that are the furthest from the coil. That is not allowed.
If, instead, we wrap the coil around like the line on a tennis ball then we will have to equal lines of coolness.
If we wrap a jellybean (two half spheres capping a curved cylinder) so that the coil traces the longest arch and the smallest, then we’ll get two equal coolness lines down the sides of the cylinder. Whereas, if we run the coil down the sides of the cylinder, then we’ll have a shorter coolness line in the smaller arch and a longer one in the longer.
You will have two lines of coolness but they will not be an even temperature along their length. Does that violate “c”? Look at the shape of the pieces needed to cover the ball:
The line of coolness runs from the center point of one rounded end to the center point of the other end, but the line is much closer to the heating coils as it passes through the “waist”.
You don’t actually want your water heater to have evenly spaced heating coils :). I know that’s not the question, but electric water heaters are actually designed so that the cold water comes in at the bottom and the hot water goes out the top, with as little mixing as possible while the hot water is drawn out. That maintains the water temperature longer, rather than it getting cooler and cooler the moment you start using it. It also means that once the hot water is sucked out the top, it goes cold pretty fast. Since most normal electric water heaters just have a top heating element and a bottom element, this leads to the bottom element doing much more work as that’s where it gets cold first. A single heating coil wrapped around the whole thing would overheat the top and underheat the bottom as it’s being used, even if it eventually evens out.
Anyway, this made me think about the High Flux Isotope Reactor at Oak Ridge. The core is a cylinder, basically just a nuclear water heater. The fuel elements are arranged around the center in curved involute shaped plates that make it look from the top like a squirrel cage fan. Inside a Nuclear Reactor - YouTube The involute shaped fuel plates allow for an even spacing when stacked around a circle, yielding the best heat transfer efficiency. I’m not sure if that would apply to this exercise, but it seems at least tangentially related.
Also, electric water heater coils extend into the interior of the tank, perpendicular to the walls. If they were on the outside, they would waste about half of the energy used radiating away from the tank.
related.