what is the most difficult or most interesting geometry question that you have encountered?

Question

it will have to be very difficult and/or very interesting. and answers to the question are also needed. thanks!

i need a specific question.

Answer 1

One of the most interesting is the Blaschke - Lebesgue theorem in three dimensions. Let me take a moment to explain it.

In two dimensions, a circle is an object that clearly has constant width (i.e., the diameter all the way around is constant). But it is not the only planar figure of constant width. There is also the Reuleaux Triangle, which looks like a triange with curved edges. If you have seen the rotating part of a Wankel engine, that is a Reuleaux Triangle. It has the remarkable property that if you have a drill bit shaped like one, you can drill holes in wood that are almost square!

About 100 years ago it was proven that, in the plane, among all of the convex objects of constant width, the Reuleaux triangle has the smallest volume. My masters' thesis advisor has done a more modern proof, and a summary appears here:

wthttp://www.mathphysics.com/harrell/pub/LightestCoin.pdf

Now let's jump to three dimensions. The sphere certainly has constant width, but it is not the only body of constant width in three dimensions. There are many others (rotate the Reuleaux triangle on an axis to get one of them).

The question is, which convex object of constant width in three dimensional space has the smallest volume?

Another way of asking the problem is: what is the lightest possible roller bearing?

This is the Blaschke Lebesgue Theorem in three dimensions.

Despite substantial efforts by many mathematicians, this problem has remained unsolved for nearly 100 years.

Follow-up:

You said you need answers to the question. I do wish I had one! Well, we know that the sphere and the rotated Reuleaux Triangle have too much volume to be the right answer. The common conjecture (and it is only a guess, not an answer that can be proven) so far is what are called trhe Meissner bodies (there are two of them, with the same volume). Here is a link to more on Meissner bodies.

http://www.lama.univ-savoie.fr/~lachand/Spheroforms.html

Answer 2

A reel of audio tape has an inside diameter of 1" and an outside diameter of 3.75 inches. The tape is 1/2 mil thick. What is the length of the tape on the reel?
Geometry plus calculus.

Answer 3

Discretization of continuous hyperspaces for terrain mapping.