Some Interesting Theorems and Conjectures

Following from a discussion of the Mathematics and Science forum called Function Space, I thought I would add a collection of the interesting topics being discussed there in this article. There is many interesting conjectures and theorems in Mathematics, however, my favorites are always the one with the strange names.

I’ve simply copied and pasted some of the theorems.

The ideas which will be discussed are the following:

• Sausage Conjecture
• McNugget Numbers*
• Hairy Ball Theorem
• Infinite Monkey Theorem
• Ham Sandwich Theorem

Sausage Conjecture:

The Sausage Conjecture states that for any n dimensions greater than or equal to 5, if we were to arrange hyperspheres which have a convex hull of minimal content, the arrangement would always resemble a sausage. This is the best method for packing hyperspheres, or so the conjecture states.

The Sausage Conjecture is derived from the Penny Packing Problem, which asks what is the most optimal packing method for x non-overlapping n-dimensional spheres? The Sausage Conjecture then answers with the sausage shape. The spheres are arranged in a long line.

McNugget Numbers*:

Hairy Ball Theorem:

Given a 2-sphere, which is your standard sphere, that we’re well acquainted with from our Euclidean Geometry classes, the theorem states that a hairy ball can never have all flat hairs.

Theorem: For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0.

Infinite Monkey Theorem:

The theorem states if a immortal monkey is given infinite amount of time, and able to randomly type characters from a keyboard, then there is a chance that it will eventually type any given text. For example, a popular choice is the complete works of William Shakespeare. The probability is very small obviously, but doesn’t mean that’s it’s impossible.

The proof for the theorem relies upon statistically independent events, and the product of the probabilities of those events.

For instance, a standard keyboard has 104 keys available to randomly select, and thus there is a 1/104 chance of pressing any given key. Our given text will be something like, “Computer”, then the probability of typing the word “Computer” is (1/104)8, which is a very small probability but is mathematically plausible.

Ham Sandwich Theorem:

Theorem: Given n measurable “objects” in n-dimensional space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane.

References:

Unusual Terms in Mathematics – FunctionSpace