Danya Lette

Pentagram ≅ Pentagon

animation javascript math svg graphs


Pentagrams and pentagons are actually a bit spooky…


Pentagons and pentagrams 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.


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



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.