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View into Brooklyn (above). View from Brooklyn (below).
Just below on the left is a wonderful and fairly ancient genuine paradox that results from the special properties of some mathematical surfaces and volumes. Some mathematical surfaces can go on and on getting longer and much longer (or wider and much wider) without any end at all! Some such surfaces may have no limits whatever on their dimensions (here it is the length of the slowly narrowing tube). In this case, the horn will go on and on and on, getting narrower and narrower, extending without end. At the same time, the three-dimensional volume (length x cross-sectional area) defined by such an ever-narrowing tube can have a very well-defined limit.
As a parenthetical note, I ask the reader to consider that if the horn were to lie horizontally, its cross-sectional area would be the product of its width and its height. So, as the length continues to extend away from the horn's bell, both the width and height are always shrinking. Thus their multiplicative product at any point (in other words, the lessening area), diminishes by the squared power of the shrinkage of the width or height when either is considered alone. So the area declines to almost nothing (approaching, but never quite reaching, zero), and the length goes on and on.
Although it is not at all rigorous, it might help to visualize things this way: As the horn becomes longer, the cross-sectional area essentially vanishes; the horn becomes almost, but not quite, one-dimensional. But there is no upper limit to how long the horn can be. It simply extends. Another way to get a feel for this might be to remember that π is a solid, exact number, as exact as the distance around a circle and the distance across the same circle are exact, but that its decimal expansion goes on forever.
Think of π's decimal expansion running endlessly inside the ever narrowing horn
— but never exceeding the value of pi.
And so I thought it would be nice to point this out, although in our actual quantized world no one will encounter surfaces like this.
However, in 2004 some very astute mathematical physicists (Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then) offered a view of the entire 4-dimensional universe which we live in that has a 3-dimensional projection that closely resembles the 'Horn of Torricelli'. In their approach (take a deep breath: it may have been prefigured or added to by others, but it's not yet 'disproved'...), the cosmos as a whole, seen in one way, has a shape very much like that of half of something called a 'pseudosphere'. That is, it's rather like the shape of the a common trumpet bell or the old Victrola 78 rpm record player horn depicted in early RCA advertisements which showed a dog listening for "His Master's Voice". The point is that (in our three spatial dimensions) the pseudosphere, the shiny trumpet bell, the old Victrola horn, and Torricelli's Horn all of them have what is called a "negative measure of curvature". What that means is that the trumpet bell, for example, bends 'up' or outward more and more as an ant (let's say) walks farther from the trumpeter's lips along the top of the trumpet, but at the same time the bell bends 'down and around' to make a circle and meet itself when the (newly bathed, certainly) ant walks at right angles around the trumpet and stays the same distance from the trumpeter's lips.
Victrola 
After the colon is a link to what is called the Picard topology or Picard Horn model (speculative) of the whole universe (impossibly seen from 'outside' as a projection where time increases from left to right): Big Bang glow hints at funnel-shaped Universe. Another link to Picard topology: Picard Horn. Finally, a more popularized link describing what "boffins" (a wonderful form of life found here and there in the UK) perhaps think of it all: Boffins trumpet horn shaped universe. Called Picard. No really.
Do I myself think the universe is shaped (in a 3-D map of a 4-D entity) like the better half of a Victrola? Well, possibly not. But what about this: (If multiply connected, then each event in spacetime may be represented by more than one set of coordinates) and where we also perhaps interfere with our own pasts or probable futures? | |
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Oh, and thank you for not dividing by
zero !!
| |   Philosophiana (Piazza Armerina), Sicily. | |
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