A request for a suggestion for a mathematics talk aimed at first year and second year undergraduate students in science

I am to give a mathematics talk in a couple of weeks and I am writing to request suggestions for possible topics of the talk. The relevant information is as follows:

(1) The audience for the talk consists of first and second year students in science; some of the students might be first or second year students in mathematics. In particular, I cannot expect that any member of the audience is particularly knowledgeable in mathematics; I think at a minimum I can assume a knowledge of the elements of calculus. However, I would prefer the talk to not be directly based on calculus.

(2) The audience consists of very intelligent students. Nevertheless, I probably should not expect the audience to have to do a great deal of thinking while listening to the talk.

(3) The talk is to be 15 minutes in length. In particular, I probably have to focus on one theme during the talk.

I would like to talk about a mathematics topic that fits the following description:

(a) The talk appeals to an intelligent person who is not particularly knowledgeable in mathematics but is not too trivial that it does not appeal to a mathematics student.

(b) The talk illustrates a beautiful piece of mathematics.

(c) The talk is solely based on mathematics.

Thank you very much in advance for all suggestions for the topic of the talk! I will certainly acknowledge you in the talk if I use your suggestion for the topic of the talk.

I think that the periodicity of Fibonacci numbers modulo $m$ is a topic that can be introduced to your desired audience and about which non-trivial, appealing things can be said within 15 minutes (you might want to mention some of the more advanced, more appealing things at the end without proof).

Specifically, I imagine you could introduce the Fibonacci numbers (but I think they have seeped into popular knowledge to a significant extent), modular arithmetic, observe the periodicity in a few example cases, prove that the period is always even, and prove that the period modulo $m$ is less than or equal to $m^2-1$ (which is just the pigeonhole principle). You could then mention without proof that in fact the period modulo $m$ is less than or equal to $6m$, and various observations about the period modulo a prime number.

If I am right, your talk is going to be for our conference in a few weeks. I can tell you a little bit about the first year audience: Most of them are not well acquainted with mathematics and few actually will understand its true power - including me. Perhaps those in second year who have done the rigorous analysis course will be able to appreciate more of what you are going to talk about.

Most of the students, at least those in first year are now doing the standard calculus/linear algebra course offered. I think at this point they have not seen how analysis and linear algebra come together beautifully and perhaps your talk can be based on showing how different fields of mathematics come together (at the moment most students - at least in my year - think that analysis and linear algebra are not related at all).

For me I would suggest a talk on non-Euclidean geometry. Several reasons are:

$\textbf{(1)}$ I think students would be fascinated about the fact that the sum of the angles in a triangle being 180 degrees holds only for Euclidean geometry. Besides, how are so called "curved" surfaces not to impress anybody? :D

$\textbf{(2)}$ You can show a lot of pretty pictures by M. C. Escher (he has a work called "circle limit" which is the Poincaré model of hyperbolic geometry, you can explain why the "devils" go smaller towards the edge) that can fascinate people - it is way easier for people at this level to imagine "beauty" like this than to explain say what Maschke's Theorem is. The latter would require having to explain a lot of technical language first.

$\textbf{(3)}$ Students of physics I understand have learned at least special relativity. You can mention Minkowski spacetime and how this is related to the hyperboloid model of hyperbolic geometry. In particular maybe you can link this to why the inner product used in class has signature $(1,1,1-1)$.

$\textbf{(4)}$ I think one can present a topic on this while expecting minimal prerequisites from the audience. You can have a look at Needham's book as mentioned above - there is a chapter in there on hyperbolic geometry. Proofs were neat and elegant, many of them using so called symmetry arguments like proving why every automorphism of the unit disk to itself has the form $e^{i\theta}$ times some linear fractional transformation which I don't remember.

I hope this helps!