I got involved in the 'drill bit' issue both here and in the metalworker's newsgroup . That went on for weeks, with about twenty vocal supports for each position. There are points to be made on BOTH sides (and perhaps a few MORE sides as well). The real answer is lost in the mists of time somewhere back a few centuries ago.
Shall we restart the "Switch" vs. "Turnout" mess while we're at it?
I agree it FUNCTIONS as you state, but that's NOT how it's built! The two parallel 'through' routes use DIFFERENT sets of rails. Except when using it as a pure crossing (one route) the rails used are not the same. That's why it's topologically different.
Now we're into topology. Whee! That's where a donut and a coffee cup have the SAME shape, but different functions.
The funniest story I've heard about those has to do with the big old steam engines that used to power shops of all types, with long ceiling mounted lineshafts running all over the place. Usually the engine was in a separate room, or even a separate building, with a great long FLAT leather belt connecting the engine's flywheel to the main lineshaft. The belt might be from one to two FEET wide. Sometimes these shafts would be
50 feet or more apart, and you could NOT see both ends at once.
Sometimes such belts were assembled with a half twist in one side to even out wear and reduce tracking problems. Thus they were effectively a mobius strip. Farm traction engines also often use this method when driving threshing machines and such.
A favorite trick, as the story goes, was to assign some greenhorn employee the task of painting the OUTSIDE of the belt bright yellow to make it more visible for safety reasons. The worker was also instructed NOT to get any paint on the INSIDE of the belt, or the paint would make the belt slip on the pulleys. Then the joker's would go away and wait for the fun to begin.
IF the story has any truth, I don't suppose management was real thrilled with the gunked-up belt! These could not have been cheap.
A more plausible version would have the employee applying belt dressing to the inside of the belt, with instructions not to get it on the OUTSIDE.
Either way, it's funny to think about the poor fellow's misfortune.
Clifford Stoll makes and sells Klein bottles on his web site:
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If you remember some time back, he was the Berkley astronomer who got transferred to the university's computer department and given a project to discover why two computer charging programs gave different results - and found hackers whose time was accounted for on one but not the other. And tracked them electronicly to Berlin where they were working for the Soviets, in one direction and to various military sites in the other via Berkley as a gateway.
It was a brilliant piece of work from somebody who wasn't very computer literate at the time.
The case was famous at the time. He wrote a very readable book about it - The Cuckoo's Egg.
I'll have to think about that one. At first rush I want to disagree with you and say that a torus does not have a closed end, while a coffee cup most demonstrably does. However, I will look into it and reply definatively later.
Just think of the possibilities. A donut that can serve as its own coffee cup. The potential for making a fortune boggles the mind.
In the Navy, we used to use three-ply carbon sets which came on long rolls (paper, carbon paper, paper, carbon paper, paper). The "newbie" trick was to unwrap one turn of the top layer of paper, trim it off even with the others, then hand it to the new guy and say, "The factory must have messed up. Get these layers back in the right order." It was a rare rookie who didn't start by unrolling the entire thing down the passageway.
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