Frank Morgan

Abstract: The version of the shortest “Steiner” network problem in which you minimize length plus number of Steiner points has an interesting analog in manifolds with density. Continue reading ‘Networks in Manifolds with Density’ »

What is a surface? Different technical definitions serve different purposes. Here we’ll focus on two-dimensional surfaces S in R³ or R⁴. Continue reading ‘Surfaces, currents, and varifolds’ »

Revised to “Five or Six” from “Eight” November 7, 2010.

In response to frequent questions, I now recommend shuffling the bridge deck just five or better six times and then preferably dealing the cards back and forth instead of cyclically. The recent article by Conger and Howald** supersedes the revolutionary 1992 paper of Bayer and Diaconis* in showing how the randomness of a shuffled deck is enhanced by dealing out the cards, even more so if the cards are dealt back and forth (West North East South South East North West) instead of the usual repeated cycle (West North East South West North East South). Their following table shows the remaining order after n shuffles for the undealt deck, for the bridge hands dealt cyclically as usual, and for the bridge hands dealt back and forth

n                 5      6      7       8       9      10

undealt       92%  61%  33%  17%   8.5%  4.3%

cyclic deal   23%   7%   3%     2%     1%

back&forth  31%   3%   1%

Dealing back and forth has the added advantage of being a bit faster than dealing cyclically as usual. Some questions about the accuracy of the mathematical model remain. Continue reading ‘Five or Six (not Eight) Shuffles’ »

My senior seminar on “The Big Questions” asked me for a succinct explanation of Gödel’s Incompleteness Theorem, a radical result which I’m beginning to think is a simple one. Continue reading ‘Gödel’s Incompleteness Theorem’ »

At our faculty seminar today, I mentioned a problem I had discussed a bit with colleagues Ed Burger and Tom Garrity, which moves the isoperimetric problem from geometry to numbers.

Continue reading ‘Isoperimetric Sequences’ »

Guest post by Jack Wadden ’11 from my Discrete Mathematics 251 class. For more on this topic google “paths in Pascal’s triangle,” e.g. Baez.

I was thinking today about Pascal’s triangle and how amazing it is that the binomial theorem actually works and why each number corresponds to a combination. It turns out that if you think of Pascal’s triangle as an upside down binary tree, this relationship is obvious. Continue reading ‘Pascal’s Triangle’ »

Guest post by Ville Satopaa ’11 from my Discrete Mathematics 251 class.

We all know of the famous Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, …

in which each term is the sum of the previous two.

When tends to infinity, the ratio between the term and the term gets closer and closer to something that we call the Golden Ratio, denoted by :

This ratio seems to define beauty to some extent. In fact, it turns out that a person with a beautiful face has nose, eye position, the length of chin and many other measurements of the face in the Golden Ratio. Here are some measurements that should follow the Golden Ratio:

1. length of the face / width of the face

2. width of the mouth / width of the nose

3. width of the eyebrows / the distance between the pupils.

4. outside distance between eyes / hairline to pupil

5. nose tip to chin / mouth to the chin

Well, are faces with ratios close to the Golden Ratio actually beautiful? Fortunately, it is easy to test this. Let’s pick a photo of an attractive person looking directly at the camera, so that it is easy to make the measurements.

Using this picture of Meghan Fox from exposay.com I measured the following ratios (the picture pasted to this file is of difference size):

1. 7.1 / 4.3 = 1.65116
2. 1.6 / 1.0 = 1.6
3. 3.3 / 1.9 = 1.73684
4. 2.9 / 2.1 = 1.38095
5. 2.0 / 1.3 = 1.53846
The mean ratio turns out to be , and the mean difference from the Golden Ratio turns out to be . Therefore her face is overall quite close to the Golden Ratio. However, whether this provides evidence that the Golden Ratio can be used as an estimate of beauty depends on whether you consider Meghan Fox attractive or not. I personally do.

The Prime Number theorem says that the probability P(x) that a large integer x is prime is about 1/log x. At about age 16 Gauss apparently conjectured this estimate after studying tables of primes. Hugh Bray via Greg Martin suggested to me the following heuristic way to approach the same conjecture, which appeared in my Math Chat column on August 19, 1999:

Suppose that there is a nice probability function P(x) that a large integer x is prime. As x increases by the new potential divisor x is prime with probability P(x) and divides future numbers with probability 1/x. Hence P gets multiplied by  or roughly

The general solution to this differential equation is P(x) = 1/log cx.