I need some fun questions with answers in differential geometry ()

[Brian_Dehority]

TL;DR

Im throwing bachelor party for my brother, who loves differential geometry. And need fun questions that hell enjoy.

I am throwing a bachelor party for my brother, who is currently getting his PhD in Math at columbia, and as you might expect, he is not very much of a party animal. I want to throw him a party he’ll enjoy, so I came up with scavenger hunt in the woods, where every step in the scavenger hunt is a question in a subject he loves, one of which is differential geometry. But I know nothing of the subject.

So my question is: what are some quick, fairly easy problems (or even trivia) in this area that would work?

 

[andrewkirk]

1. What is a south-pointing chariot. Who made it and when?

Spoiler: Click to see answer

See the wiki article on it at https://en.wikipedia.org/wiki/South-pointing_chariot

It was made in China in the third century CE. It was designed to have a pointer on it that always pointed to whatever compass point it was originally pointing at, as it rolled around a large plain, no matter what turns were made, as long as the wheels all remained in contact with thr ground and rolled without slipping. It used gears and differentials connected to the wheels to accomplish that.

2. If a mechanically perfect south-pointing chariot traveled from Moscow to Paris and then on to Rome, and it was pointing South when it started, would it still be pointing South when it reached Rome.? Why or why not?

Spoiler: Click to see answer

No. Even if mechanically perfect, it would become progressively less accurate as it covered long distances, because it cannot take account of the curvature of the Earth.

3. Imagine you start anywhere on the equator and travel North for a distance (the 'initial distance'), and then make a ninety degree left turn.
(a) will your new direction be West, North of West or South of West?

Spoiler: Click to see answer

South of West

(b) will you ever reach the equator again?

Spoiler: Click to see answer

Yes

(c) If you do reach the equator, in which of the following situations will your route back to the equator be shortest:
(I) if the initial distance is 5000km
(II) if the initial distance is 6000km
(III) if the initial distance is 7000km

Spoiler: Click to see answer

Answer (III) because your direction will be more Southward than in the other two cases. If you had traveled all the way to the North Pole, you would be pointing due South after the left turn.

(d) Bonus question, more difficult: if when you reach the equator you turn and go along to the equator until you reach the place where you started, under which of scenarios I-III will the length of your total journey be shortest? [I don't know the answer to this one. I'd need a pencil and a large piece of paper to work it out. But maybe your brother can do it!

4. Is the number of straight lines that go through a point P on a hyperbolic surface and never meet (ie intersect with, or 'cross') a line L on the surface:
(a) one
(b) zero
(c) seven
(d) twelve
(e) infinite

Spoiler: Click to see answer

Infinite. that is, there are infinitely many lines through P that never meet L

 

[berkeman]

1. What is a south-pointing chariot. Who made it and when?

It sounds like he needs the answers too. Maybe you could use Spoiler tags to hide them in case anybody wants to post their try at the answers...

 

[haushofer]

Ask him about Euler characterisics of spheres and doughnuts, although that's more topology. :P