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Torricelli's trumpet is a rather quirky geometric figure invented - or some say "discovered" - by Evangelista Torricelli, who succeeded Galileo as professor of mathematics at Pisa. However, the basic function was well known before Evan's time. It was found almost at once by Rene Descartes after he invented the coordinate system that bears his name.
The slick thing about Torricelli's trumpet is it has a definite, finite volume - with the right units it has a volume equal to π, that is 3.1415926, etc. etc. - but it also has an infinte surface area. Some say that means it's a bucket that can hold π gallons of paint but you can never have enough paint to paint it. CooperToons will show that's hogwash. Although it is true the figure has a finite volume and an infinite surface area you can paint it. Not only can you paint it (even if it has a bell measuring 20 miles in diamter), but you can paint the trumpet with a single drop of paint.
The basic equation for the figure is the function f(x) = 1/x. In other words if x =1, the f(x) = 1. If x = 10, f(x) = 1/10 or 0.1. If x = 1,000,000, then f(x) = 1/1,000,000 or 0.000001. So as x gets big, f(x) gets small.
You then take the figure for 1/x and rotate it around the x axis. At this point you can figure out that we need two more axis, the y axis and the one for f(x), that is a z axis. From the equation for a circle we know that x2 + y2 = 2r. Since r = 1/x then y2 = 2/x - x2 or y = +/- (2/x - x2)1/2
The volume and areas of the figure are calculated from the areas and circumference of the figure at any given value of x. The circumference of each circle is equal to the diamter (or 2 times the radius) times π. The area is equal to πr2 (Which recalls the old chestnut of the hillbilly asking his son what he learned at college. "Well, Paw," the son said, "I learned 'πr2.'" "Pie are squared?" he father said, "Hell, son, pie are round! Cornbread are squared!")
Attempts to rationalize the phenomenon by comparing it to the real world are often correct but usually miss the point. A common explanation why you could paint the trumpet inside is that eventually the diameter is so thin the paint won't flow in because molecules have a finite size. "And so, " the writer happily states, "the paradox is resolved."
With all due respect (he sneered) that doesn't resolve the paradox at all because there is no paradox. It's simply an unusual result common in mathematical proofs that involve infinity. The fact paint wouldn't flow to the end of the trumpet is correct but irrelevant. You can, after all, consider the figure a solid object and go through the same calculations and prove the outside surface is infinite in area the volume of the solid trumpet is finite. The math is the same except you have to add the area of the circle at the end of the trumpet to the total surface area.
Besides you can still paint the whole infinite area of Torricelli's trumpet as long as you define "paint" to mean covering a surface with a finite thickness - however, thin - of a fluid. But you have to paint it just right.
In fact, you can paint the whole infinite area of Torricelli's trumpet with as small amount as you like. In other words, the whole interior of a Torricelli's trumpet which starts with a diameter of 20 miles - and can hold 3,459,253,777,669,599 gallons of paint - can be painted with a single drop.