The Hoffman–Singleton graph
It's time for the twice-monthly Visual Insight
post! This time it's a picture by +Félix de la Fuente
, an architect and dedicated amateur mathematician in Barcelona who is in love with discrete geometry, polytopes and combinatorics.
He drew the Hoffman–Singleton graph
by connecting 5 pentagons to 5 pentagrams. The picture at left shows the pentagons on the outside and the pentagrams on the inside. The picture at right shows how one of pentagons is connected to all 5 pentagrams. At Visual Insight
you can see the whole construction and the final result:http://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/
The resulting graph has 252,000 symmetries! These symmetries form a group called PΣU(3,F₂₅), which I explain in the post.
For now let me just say that this group is built using the field with 25 elements, which is called F₂₅
. To get this field, you can take the integers mod 5 and throw in a square root of some number that doesn't already have a square root. As a result, F₂₅ has an operation that acts a lot like complex conjugation, and this is used to define PΣU(3,F₂₅).
All this is nice... but it's not surprising that if we take 5 pentagons and 5 pentagrams and connect them up in a highly symmetrical way, we get a graph whose symmetries are connected to algebra involving the numbers 5 and 25.
I'm more tantalized by the mysterious connection between the Hoffman–Singleton graph and the Fano plane! The Fano plane has 7 points and 7 lines; it's not very 'five-ish'. And yet, you can build the Hoffman–Singleton graph starting from ideas involving the Fano plane. Why???