Visualizing K4

Cyclic groups are fairly easy to visualize and understand iteration. But K4, while easy for me to understand intuitively and computationally, was very hard to visualize. Until I used colour to signify “motion” in K4.

In the below, each of the elements is coloured distinctly, with the identity coloured black. Adding an element is indicated by an edge of the same colour: to add an element to another, start from the first and travel along the colour of the second.

The graphs are undirected — this is precisely because each element is of order two, adding an element again takes you back where you started; adding an element is a “toggle” operation.

This tetrahedral image highlights the symmetry of the three non-identity elements.

Each of the non-identities has the same relationship to the identity — you travel between the identity and the element by adding the element, represented by the interior edges coloured the same as the element.

These elments form a complementary triple. The exterior edges connect elments of the remaining two colours; and the four triangles in the graph are each composed of one edge of each colour. This latter feature illustrates that adding each element in succession takes you back where you started.

It displays each non-identity element in opposition to the complement pair by putting each element across from the same coloured edge (representing going between the complement pair by adding the element) and by having the two edges of each colour orthogonal to each other, the one between the identity and the element and the other between its complement pair.

This crossed diamond image distinguishes an element but exposes the regularity of addition. For two colours, the edges are parallel, making it obvious that they indicate motion in the same direction, that is adding the same element.

The distinguished element is easily seen to be the combination of the other pair and commutativity is more immediately apparent. It is perhaps more obvious that adding an element twice takes you back where you started.

It links the non-distinguished elments through the distinguished elements colour making it clear that the distinguished elment acts as a “toggle” between the two, and by crossing the other distinguished line shows that it acts as a toggle on itself as well.