Pentagrams and pentagons are actually a bit spooky…

## Complementary

Penta*gons* and penta*grams* are complements:

They have the same vertex set but exactly opposite edge sets. That is, every pair of vertices that are connected in one graph are *not* connected in the other.

## Isomorphic

However, pentagons and pentagrams are also isomorphic.

In other words, they are technically *the same graph* (bearing in mind that a graph is an abstract object and not to be confused with a representation of a graph).

To illustrate: if you labelled the vertices of the pentagon {A, B, C, D, E} clockwise, then the edge set of the pentagon would be {AB, BC, CD, DE, EA}. It is possible to affix those *same* labels to the pentagram such that the edge set is also {AB, BC, CD, DE, EA}. Reading them off clockwise, the pentagram’s labels might be, for instance, {A, D, B, E, C}.

So, in a sense, this means that they are

## Equal&Opposite.

Hmm.

But, really, since “they” are the same graph, there is no “they”. It is technically a single graph that falls in the class of “self-complementary” graphs. So, there aren’t even two graphs to be equal and opposite to each other.

## More info

For more info on graph theory and on that animation, check out this post.