Re: How to Make Cisterns

On 5/3/2025 1:08 AM, David Entwistle wrote:

Adapted from "Amusements in Mathematics" by Henry Ernest Dudeney.
 Given a large sheet of zinc, measuring (before cutting) one metre square,
you are asked to cut out square pieces (all of the same size) from the
four corners of this sheet. You are then to fold up the sides of the
resulting shape, solder the edges and make a cistern. The cistern is open
at the top. You can assume you have the appropriate tools and skill to
carry out the task.
 The puzzle is what size should the cut out pieces be, such that the
cistern will hold the greatest possible quantity of water?
 I guess a follow on question could be; is it possible to cut out four non-
square pieces (all of the same size and shape), producing a cistern with
tapering sides, which has greater volume. I haven't looked at that yet...

My thoughts about the follow-on puzzle is, "What if the four identical non-square pieces are fractal in nature?".  I imagined a first step of cutting kite-shaped corners using two straight-cuts at each corner (resulting in a truncated pyramid-shaped cistern).  The next refinement would be to cut curved cuts instead of straight-cuts and bending the metal in a curve to allow the edges to be soldered together.  The cut-out portion would resemble a kite-shape with two curved sides.  The kite-shapes could taper towards the center of the square so that the volume's cross-section could look like semicircles from two directions and square from above (but, I'm not sure that this is optimal).  Since a portion of the original square is still being cut out and not used, there may be more room for improvement.  What if the curved portions are not smooth, but zig-zag, creating fine "fingers" along a curve.  The fingers on one side could be soldered to complimentary fingers on the other side.  By adding finer and finer levels of zig-zags to the curve (fractal-like), less and less of the material is wasted. In the limit, no material would be wasted.  The final volume could approach the maximal volume for the given surface area.  The Banach–Tarski paradox makes me think that one might even be able to get more volume.
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Carl G.
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