> For instance, SU(5) groups quarks and antiquarks together with leptons and antileptons into “fiveplets,” which are like the indistinguishable sides of a regular pentagon.
The idea is to put the 5 particles in 5 places that are undistinguishable.
For that you need to use the vertex corners of an hyper-tetraedrum ( https://en.wikipedia.org/wiki/5-cell ). Don't get confused by the bad drawings, if you have one of them in 4 dimensions, you can put each point at the same distance of all the other points.
If you use a regular pentagon, then you must select an order for each of the particles/vertex. If you select one, some are more close than the others.
(An alternative is to use a pentagon, but consider not only the rotations and flip, but also the operations that mix the vertex/particles in any order. But then the nice identification with the symmetry of the geometric figure is gone. You can use a square with the central point.)
In some sense, yes. I remember the class were my physics professor explained why the SU(5) could be The Group of the universe and why it failed (IIRC the predictions were 1% off with the experiment, a good try, but enough to be discarded.) You could feel how sad he was that SU(5) has to be discarded. :(
In some sense, no. The SU(5) group includes all the symmetries of the hyper-dodecahedron, were you can rotate it in the 4-dimmensional space to exchange one of the vertex/particles with another. But it also includes more strange things, like half mixing two particles.
a -> (a+b) / srqt(2)
b -> (a-b) / sqrt(2)
This is more difficult to explain. Technically, it's a rotation in the plane x-y of 45 degrees. And you can rotate another angles, for example in some particles decays the important rotation is 13 degrees. And you can mix three or four or five particles. All of this is more difficult to imagine, but it's easy to write analytically.
The use of geometric shapes like the hyper-dodecahedron is more a nice visualization technique. It's easier to explain than the details of the SU(5) group and it provides a good intuition, even after studding the theory with more details. So I prefer to ignore this technical detail, specially for a popular science article.
> For instance, SU(5) groups quarks and antiquarks together with leptons and antileptons into “fiveplets,” which are like the indistinguishable sides of a regular pentagon.
The idea is to put the 5 particles in 5 places that are undistinguishable.
For that you need to use the vertex corners of an hyper-tetraedrum ( https://en.wikipedia.org/wiki/5-cell ). Don't get confused by the bad drawings, if you have one of them in 4 dimensions, you can put each point at the same distance of all the other points.
If you use a regular pentagon, then you must select an order for each of the particles/vertex. If you select one, some are more close than the others.
(An alternative is to use a pentagon, but consider not only the rotations and flip, but also the operations that mix the vertex/particles in any order. But then the nice identification with the symmetry of the geometric figure is gone. You can use a square with the central point.)