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Date:         Sat, 19 Dec 1992 03:40:14 CST
From:         tushar@MATH.UCHICAGO.EDU
Subject:      GEOMETRY OF WINE: NATALIE, YOUR ATTN HERE, PLEASE.

Natalie, kindly read this. It will solve all your wine storage problems
forever.

The geometry of wine requires very careful thought. A look at the simplest
factors involved will assure anyone of the complexity of the question. Just
for the sake of argument, take the ideal world in which bottles and jugs
are both perfectly cylindrical. Take the ideal world in which jugs have no
handles! Why, for that matter take the ideal world in which every stocker
tries to maximize her wine-in-fridge function under the fridge-volume
constraint!! As we shall see, the problem still requires some rather
intensive (though quite interesting) analysis!

If the jugs are perfectly cylindrical, then you can still pack them
hexagonally. If the bottles are taller than the jugs, they will hold
more. On the other hand, they are probably thinner in girth than the
jugs. Aha! Now everything is seemingly in favour of the jugs! No, not
because of their lack of volume - the packing ensures the same propor-
tional wastage of space - but because of the very significant loss of
space due to the thickness of the glass. Do we not all know that glass
and wine cannot occupy the same space?

Thus, even in our simple, idealized world, we are led, willy-nilly, to
considering at least three factors:

1) h, The ratio of the heights of a jug and a bottle
2) t, The ratio of the thicknesses of the jug wall and the bottle wall
3) g, The ratio of the girths of a jug and a bottle

Remember, these factors are independent of each other. Yes, the aesthetics
of our times may well bring it within bounds of near-certainty that there
will never be a wine bottle the shape of an kettle or a jug the shape of a
pipette! But we must not make such assumptions at this point.

Further, just to be clear, we posit that wine does not occupy any space
between bottles (or jugs, as the case may be)! Our world is not idealized
enough to have no gravity!

Well, a rather simple marginal analysis shows that the extrema of space-
saving will always be achieved for all jugs or all bottles! For if there
were a motley conglomerate of jugs and bottles, we could easily replace the
the jugs with bottles (or vice versa!) to achieve a concomitant increase
(or decrease) in the volume of wine (but not necessarily respectively!)

We can thus assume that the race is between jugs and bottles, each with its
own kind, without mixtures! Now a calculation shows the following rather
interesting fact: if x be the ratio if the thickness of the bottle wall to
its radius, then the ratio of the volume jugs will hold to that the bottles
will hold is h((1-(t/g)x)/(1-x))^2! Believe it or not, but that's a hard
fact of life.

Ah, now everything seems to be going against the jug! Well, well, the only
way jugs will win is by having t very close to g and hoping for the best.
We know how the jug doesn't do extremely well on the 'h' test! The aesthetics
of our times go against it! Well, the only other way a jug can hope to win
the race is by aiming for a t/g so large that the fudge factor becomes highly
negative!

A good idea, but unfortunately impractical, even in our ideal world. For it
would require the jug to be so thick-walled compared to its own girth that -
alas! - it would have to have its walls ingrowing into its own walls even to
exist. We cannot allow such pathological jugs to inhabit our world! And
even if they did, one can be sure that no wine could be stored in them, for
as remarked before there is not place enough in one spot for a glass wall
and wine, much less a glass wall, an opposite glass wall and wine!

The jug is doomed to hold less wine, given a fridge. In the real world where
we all live and email, it is worse. Jugs have handles, bottles don't. Bottles
are tall, jugs stocky. And egg-slots, they are the shortest of all, half-high.

In a last desperate attempt, the jug might try to make x so small that the
fudge contributes nothing at all. But there are three fundamental difficulties
there: One, given the sensibilities of our times the jug cannot possibly
dictate the appearance of the wine-bottle. Two, even if it could, the wine
bottle would have to have such thin crust that merely the act of holding the
bottle in hand would suffice to break it! A gentle puff of breath on the
bottle would weaken it enough to cause wine to ooze out! Well, how could the
bottle be strong enough to weather the changes of temperature as accompany the
bottle when it's taken out of the refrigerator? Common-sense dicates that a
bottle of such a description would be a bottle impossible to obtain (and a
bottle not wished for, for that matter!)

But far more important is the THIRD point, for it is true even if the first
two were somehow refuted! For even if the first two were somehow refuted
the jug would stand no chance of winning the race, because of the inability of
the jug to overcome its 'h' handicap. Notice that we are out of the domain of
the jug's fudge-factors. This is purely a case of tallest wins. Unfortunately
the jug loses! One can state, with some confidence, that the dimensions of
the jug do not measure up to the highest standards of height. It's curtains for
the jug, for has anyone seen one taller than one's arm? What sort of jug would
the jug be if it resembled an ion exchange column?

Thus, it is mathematical truth that forces jugs to hold less fluid in them
when packed in a fridge. Logical reasoning holds sway over the world, as
usual!

tushar
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