Interesting topics for (very) short talks Part of the requirements for my Honours is that I record a short 4-7 minute digital talk, which is then distributed to all the other students and staff at my university’s mathematics department. The video doesn't have to be related to my thesis, and it should ideally be accessible to all the other Honours students. I have already chosen a topic (a Galois theoretic proof of the Fundamental Theorem of Algebra), but it got me interested in learning about other results that have a pithy/clever proof. It doesn't even need to be a proof, just a subject which would make for an excellent short mathematics video.
 A: I understand from the question ("talks accessible to everyone") that the audience will be broad. For inspiration, you might look at Math talks that blow your mind.
A: I would suggest the 3Blue1Brown video on The essence of calculus, from minute 2 to 7. It explains how to find the formula for the area of a disk, as an preliminary to integrals. Not research level anymore, though.
A: I suggest fractals as a topic that can be understood relatively easily and leads to some incredibly beautiful mathematics. You can define the Mandelbrot set without using any more advanced topics than quadratic functions and complex sequence convergence and then there's any number of videos on the internet (of varying lengths) to show the complexity arising from the very simple problem setup.
A: I made some fancy animated diagrams of platonic solids, showing the way they nest inside each other, duality relations, axes of symmetry and so on.  I think that's a good basis for a short talk.  I made some videos myself, but I'm sure it's possible to do better.  If you wanted to modify the diagrams, the code is available (but not well-documented).
A: I have a fascination for 'simple' topics (which often turn out to be not so simple), so here a couple of ideas:

*

*What is a number (set theory or even category theory?)

*What is continuity (intuition of $\epsilon-\delta$)

*Cantor's diagonal argument

There is a huge amount to say about each of these, of course, but I think the essence of each can be boiled down to a short talk. Example - natural numbers:
You have a bag of apples and a bag of oranges, and for each apple you have an orange: they are in a sense the same 'size', if you will. That is the essence of a number: the property that all sets of the same size possess. This can then lead on to relations, functions, equivalence etc etc.
