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Hello,

In conformal geometry there is a 15-parameter symmetry group.

I have an rough conceptual understanding of the 3 spatial translations, the 1 temporal translation, the 3 rotations, the 3 Lorentz "boosts", and the 1 dilation transformation.

I am having trouble conceptualizing the remaining 4 "special conformal transformations", which apear to be combinations of translations, rotations and possibly something referred to as "inversion"

If there are any math aficionados out there who can give me a more intuitive, conceptual, visual understanding of what is going on physically with these special conformal transformations, I would be eternally grateful.

Maybe one approach might be to start with a homogeneous sphere and say: "Ok, when we subject the sphere to a SCT, these physical processes happen to the sphere.

Also, why are there specifically 4 SCTs?

Any help would be welcome and the more diferent perspectives, the better.

Yours in science
Knecht
www.amherst.edu/~rloldershaw

Uh, Lorentz boost is a very cool name!

(Whisper in my ears what it is anybody )

Minkowski Space is a way to treat space and time on equal footing. Conformal Space is a way to treat n-planes and n-lines on equal footing.

So Cobformal Minkowski Space says thateverything you can do with a line (for instance, rotating around it), you can also do with a circle and everything yiu can do with a plane (reflecting in it) you can do with a sphere and you can use all those transformations equally on space and time.
With mild complications because space and time have different signs which basically trades geometry of circles with that of hyperbolas.

that graphic also shows a crystal structure in minkowski space, which is pretty awesome. It's a space-time-crystal! (in one space and one time dimension)