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

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• Brian_Dehority
In summary, my brother is throwing a party for himself and he needs some quick, easy problems in differential geometry to make it interesting.
Brian_Dehority
TL;DR Summary
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?

1. What is a south-pointing chariot. Who made it and when?
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?
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?
South of West
(b) will you ever reach the equator again?
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
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
Infinite. that is, there are infinitely many lines through P that never meet L

Last edited:
dsaun777, Spinnor and berkeman
andrewkirk said:
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...

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

## 1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in a space, and how they are affected by changes in that space.

## 2. What are some real-world applications of differential geometry?

Differential geometry has many practical applications, such as in physics, engineering, computer graphics, and even in the study of the shape of the universe.

## 3. What are some basic concepts in differential geometry?

Some fundamental concepts in differential geometry include curves, surfaces, tangent spaces, curvature, and geodesics.

## 4. How is differential geometry related to other branches of mathematics?

Differential geometry is closely related to other branches of mathematics, such as topology, calculus, and linear algebra. It also has connections to physics, particularly in the study of general relativity.

## 5. What are some famous theorems in differential geometry?

Some well-known theorems in differential geometry include the Gauss-Bonnet theorem, the Fundamental Theorem of Curves, and the Fundamental Theorem of Surfaces.

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