User sudeep kamath - MathOverflow

Proof without words for surface area of a sphere

2012-11-18T22:48:08Z

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the cyloid in to three parts and show that each part has an area equal to that of the wheel.

I have always wondered if there is a similar proof for why the surface area of a sphere is equal to the area of a circle with radius twice that of the sphere. Is there a nice way to see this?

I would also like to know if one can show without doing any calculus that the length of the arch of the cycloid is 8 times the radius of the wheel. (Note: $\pi$ does not show up here, so this sounds tricky!)

Locating a submatrix within a matrix

2010-10-20T23:58:18Z

Given an $m\times n$ 0-1 matrix A, I am interested in an efficient algorithm to locate all copies of a given $p\times q$ 0-1 submatrix B within it, where a permutation of rows and columns is allowed, i.e. find all collections of row indices $r_1, r_2,\ldots, r_p$ and column indices $c_1, c_2,\ldots, c_q$ (with $r_i$'s and $c_j$'s not necessarily in increasing order) so that A restricted to those rows and columns in that particular order yields B.

Any references will be useful.

Thanks.

Comment by Sudeep Kamath

2012-11-20T01:48:57Z

I like your proof. The proof for area under a cycloid also required some simple geometrical calculations which I am willing to still consider as a "proof without words". It is still intriguing though that it is not easy to find a natural proof mapping the area of a circle with twice the radius of the sphere to the surface area of the sphere.

Comment by Sudeep Kamath

2010-10-21T18:49:08Z

Thanks Tsuyoshi. Do we know anything of the problem if B is relatively sparse?