This triangle version of the associahedron is given coordinates of homogeneous degree $2$. That is, it lies in the plane

$x + y + z = 2$.

The other ternary functions have varying degree, according to the partition of the total number $27$, where the six point triangle uses $6/27$ points. Note that the degree $3$ case might resemble the seven points of a Fano plane. However, the six outside points are the mixed functions, like $012$, and the centre point is the function $111$, so now six lines are required to cover the outside points. The point $111$ also lies on a line $y + z = 2$, when $x = 1$ ,and also the line $x + y = 2$. Thus the hexagon actually has a self intersection at $111$.

7 years ago

Can this hexagon be thinked of as a Fermi surface?

ReplyDeleteHi Ulla. In a vague sense, yes I guess so!

ReplyDeleteI have thought of this a few days now, and must ask further. It may be a silly question, but I need more flesh on the bones. This is extremely interesting for me as a biologist.

ReplyDeleteWhen space expands as for ions, or say the Higgs field that gets energy, then this Fermi surface will 'jump' and get exited as a spike. Then it is the hole that do spikes? It can also be seen as impulses? In this way the Higgs field would oscillate?

When kaons oscillate they form dimuons, that is extended hadrons as a BE-condensate. Can this hyper-determinant also be extended to hadronic scale and even ions? Then maybe the massivation would be an open-up of closed strings, induced by energetic oscillations of the field? /Ulla.

PS. I must post as anonymous, because I don't succeed in posting on google-account.