In this 6th meeting, we looked at visualizing networks. Cobbled together also using some older slides like

Cladni figures on a graph

If we look at the Kirchhoff matrix on a graph we can look at eigenvectors and draw level sets of the Eigenfunctions.

120 Cell, a Platonic Solid

There are exactly 6 Platonic solids in 4 dimensions. Animated here in 2009 for a mathcircle talk See the handout [PDF].

Discrete Calculus 2025

Discrete Sard

The Discrete Sard theorem is some magic. We know in that in the continuum a function on a manifold has the property that for almost all values the level set f=c is a submanifold or empty. In the discrete this is also true. If c is not in f(V), then f=c is a submanifold or empty (a result from 2015). One can generalize it to higher dimensions. An elegant formulation is that if f is a map from the vertex set V of a q-manifold to a finite set {0,...,k} then the set of all simplices for which f reaches {0,...,k} then this forms a q-k manifold if it is not empty. A very general result is from 2024 Manifolds from partitions. It can even be taught.

Natural Graphs

This was from a project about Orbital networks work done with Montasser Ghachem in 2013. These networks are generated by simple maps on finite rings.

Internet in 1977 had genus 12

We looked at the internet from the very beginning to now.

Discrete Klein bottle

Manifolds can be visualized as networks. Every computer graphic rendering of a manifold of course is a network.

The 4 color theorem

Instead of talking about topological maps and having to explain what we mean with two countries to have a common border, mathematicians more than 100 years ago used graph theory instead avoiding so notions from the continuum. I myself started to get interested in the 4 color theorem in 2014 in this project with Jenny Nitishinksaya and remained interested in the topic, like Coloring discrete manifolds (an update is here). An example of a youtube presentation is here 2021 It is also a nice topic for teaching like in this 2015 extension school class.