Archive for 26th October 2008

Judging Beauty by Math

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 n tends to infinity, the ratio between the n^{th} term and the n-1^{st} term gets closer and closer to something that we call the Golden Ratio, denoted by \phi:

\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.

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 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 \bar{\phi} \approx 1.58, and the mean difference from the Golden Ratio turns out to be \approx 0.11. 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.

Student Talks

At Williams every senior math major chooses a faculty advisor and gives a 35/40-minute colloquium talk. Since we currently have over fifty senior majors, this keeps us pretty busy, but we think it well worth the effort.

Here is how I like my advisees to prepare, starting a month before the talk and consulting with me every day or two:

1. Outline of talk, with details of proofs.

2. Board drafts. Each page consists of exactly what is to be written on one section of blackboard. Each such board should convey one main idea, with a heading, a concise, abbreviated statement, and some kind of figure.

3. Draft rehearsal. Student sits down alone at keyboard and rehearses the talk by typing everything and then emailing it to me for comments and revisions.

4. Rehearsals at the blackboard, at least one with an audience recruited by the student.

5. Final rehearsal with me, ideally two days before the talk.


See also Lou Ludwig’s “Technically Speaking” video clips.