My day started off like any normal day. I sipped on my daily latte and went to work. I logged onto Gold Token, thinking I would make all my backgammon moves, perhaps glance at a chess game or two, and then resume design work on a code generator I've been working on. But a new post in the Puzzles and Riddles discussion board caught my eye.
A man starts from the North Pole and walks 6 miles in a straight line. He then faces East and walks 8 miles in a straight line. He then returns to the North pole in a straight line. How many miles did he walk altogether?
I had no idea what I was getting myself into. A heated discussion began and I learned more about spherical geometry than Reimann's cat (a lesser known feline who was much better off, by the way, than Schrödinger's cat, although she did have a rather warped view of things). Soon into the discussion thread, I spotted a very rare creature -- a Flatlander! His arguments baffled me for quite some time until I finally figured out what he was.
So if you dare, you can view the entire thread as it originally appeared, and my summary arguments revealing the Flatlander's existence to the general public. People did not run screaming.
4 comments:
Reimann's cat? I always felt sorry for Schroedinger's cat!!
Hm... Must google..
LOL!! Awesome..
The confusion seems to be stemming around "what is a straight line" and "what does it mean to walk a straight line on a sphere?"
Screw all of this wrapping maps around globes, get a styrofoam ball, some rubber bands, and some stick-pins for points if you want to make a model.
Here are some related questions:
HOw can you check in a practial way if something is straight- without assuming that you have a ruler, because then the question is "how can you check that the ruler is straight?"
How do you construct something straight- lay out fence posts in a straight line, or draw a straight line?
What symmetries does a straight line have?
Can you write a definition of "straight line?"
****Don't read below til you've thought about these questions***
Lines have reflection symmetry a.k.a. bilateral symmetry- that means we could mmirror an object over the line.
Lines also have half-turn symmetry; rotating 180 degrees about any point on the line.
A reflection through any axis perpendicular to the line will take the line onto itself.
Rigid-motion-along-itself symmetry: Any portion of a straight line can be moved along the line without leaving the line.
Central Symmetry: Hold your hands in front of you with the palms facing each other, your left thumb up and your right thumb down. Your hands have approximate central symmetry about a point midway between the center of the palms.
Anyways, the next step is to figure out what a straight line is on a sphere.
SunGroove, a true straight line in 3D space cannot lie on the surface of a sphere. So the generally accepted definition of a straight line of a sphere's surface is a great circle.
How can you check in a practical way if something is straight?
Get a little device that only has forward propulsion, say the Energizer Bunny. Wind it up and let it go. If it stays on your "line" then you have a straight line. This works for lines on a flat plane, and lines (great circles) on a sphere. (Of course any practical test is subject to measurement flaws; the point here is to have simple thought experiement so we can agree on what a straight line on a sphere is.)
A great circle (geodesic) has:
1) Reflection symmetry
2) Half-turn symmetry
3) A reflection through any axis perpendicular to the line will take the line onto itself
4) Rigid-motion-along-itself symmetry
5) Central symmetry (I think; not sure I understand your explanation)
A great circle also conforms with most people's intuitive definition of a straight line on a sphere's surface (to walk forward without turning left or right).
The Flatlander guy in the thread couldn't get past that a line of latitude (other than the equator, which is a great circle) is not a straight line. He saw that "east" is straight on a 2D map, and he saw that a circle viewed in 3D space from the side is a 2D straight line, and he concluded that all lines of latitude must be straight lines. He concluded that if you "walk east" you're walking in a straight line, when in fact you're not (unless you're on the equator). And he didn't understand that you can "face east" and then walk in a straight line and no longer be on the same line of latitude.
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