The higher categorical parity cube is a broken pentagon, such as the one that appeared in operad theory. It is simply the diagram

The top face really looks like

if we use trees to label choices of bracketings. Carl Brannen’s idempotents for lepton masses are also labelled by the parity cube, which describes three directions in space as well as three generations. It is nice to know that this agrees with the number of quark generations that we get by calculating an Euler characteristic for a gluon orbifold modelled by twistors.

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## CarlBrannen said,

February 19, 2007 @ 12:57 am

Of course I don’t know anything about these things. From the way they’re labeled, I would think that they seem more similar to the idempotent structure of 3×3 complex matrices. I.e. eight states, top is 1, bottom is 0, three primitive states and three states given by sums of pairs of different primitives.

The cube that gives the elementary particles is from a more complicated idempotent structure, one that has 1 on the top (not shown), then 8 primitives (shown as the corners of a cube), then 28 given by sums of different primitives (not shown), and etc. through the rest of the Pascal triangle for (1+x)^8.

Meanwhile, my Java simulation of Stern-Gerlach filters is slowly progressing. With a little luck I should have results in a day or two.

## Kea said,

February 19, 2007 @ 1:41 am

Yes, it’s the 3×3 complex matrices that I’m thinking of here. Good luck with the simulation.

## a quantum diaries survivor said,

February 20, 2007 @ 12:28 pm

Hi Kea,

as much as I would love to understand what you sometimes start talking about here, my head usually explodes in the fashion of the aliens in “Mars Attacks!”…

Darn. They should warn would-be physicists that there is really no chance that they will understand things once they get a degree. They need seven.

So how do you get the number of generations, for the rest of us ?

Cheers,

T.

## Kea said,

February 20, 2007 @ 8:23 pm

So how do you get the number of generations, for the rest of us?Tommaso, that’s what this whole blog is about. We are getting there slowly. Interestingly, at my last conference talk there was a guy who has worked on the existence of a fourth (sterile) neutrino. We await the experimental conclusions. Only one of us can be right.