A complete graph of $n$ nodes is a graph where every node is connected to every other node. It is known that one cannot draw a complete graph of 5 nodes on a piece of paper (plane) without any crossing edges. However it is possible to draw this graph on a donut (torus) without any crossing edges. How can you do it? Bonus question 1: how can you draw a complete graph of 6 nodes on a torus without any crossing edges? Bonus question 2: what about a complete graph of 7 nodes?
Here is a picture of the 5 graph:
A flat version (perodic boundaries) is easier to digest and reveals the fundamental symmetries:
and of the 6 graph:
and the 7 graph:
Flat version (perodic boundaries). We can see that this construction maxes out at 7. Please note how simple and clear this representation is. It gives a good intuition of how the torus offers us an additional back alley and why we can neatly make six connections per node.
Start with this 'snowflake' pattern:
So, we can do it like this:
This can be placed on the side of the torus: the lines that run off the left edge go all the way around and end up on the right side. The lines that run off the top edge go through the hole in the middle and end up on the bottom. And so all of the connections are established.
There's another way to look at it that's based off:
the 7-coloring of the plane. If we take this pattern:
here, and repeat it over and over, every color touches every other color. If we draw lines from the center of each region to each one that touches it, we're done! Each of the 7 colors here touches all of the other 7, and so the graph of their centers will be the complete graph on 7 vertices.
The only minor issue is that we have infinitely many vertices labelled with each number from 1 to 7, rather than one of each.
All that's left is to make all of those colors the same region. To do this, we can use the topology of the torus! Take the graph, and cut out a section that we want to repeat forever:
Now, look at this square we can get by distorting it:
If we stretch this square out into a really long rectangle...
...we can wrap this around the outside edge of the donut, and then stretch the bottom and top sides to connect through the hole. This means that there are only 7 regions, and they all touch each other -- exactly what we wanted!
Since the surface of a donut (or toroid) is topologically equivalent to a rectangular space with both pairs of edges wrapping, we can represent a toroidal embedding of a graph as such (which is also easier to see than a drawing on an actual toroid).
$K_5$ is easy:
$K_6$ is a little bit harder:
$K_7$ requires serious thought, because
we have 7 nodes and 21 edges to draw, which requires us to draw a triangulation of the toroidal surface.
I ended up drawing the following:
This shows four copies of the hexagonal subgraph to make the wrapping edges easier to follow. If you imagine the entire figure moving horizontally and vertically and overlap the hexagons, you can see that no edges cross any other edge.