Even if the sun wobbles/rotates, if the mass is centered in the sun, it's part of the sun. We orbit it, it orbits about the galaxy which is zipping about in our cluster and so on and so on.

It's not so much a detailed question of right or wrong on the orbit, but when theologians say that the universe is a layered backdrop of colored glass that spins about the earth, I'm fairly sure they're off on their conceptualization.

I do buy into Sean's idea, though. All hail Sean, avatar of the Flying Spaghetti Monster!

The universe revolves around Jesus!  Everyone who isn't going to burn in hell knows that!

I'm Jewish. Do I get a free pass?

and here is where thoumsins illustration becomes pertinent. The center of mass isn't always in the Sun and many times its outside of it. Although it always is very close (astronomically speaking) to it

Promises, promises.

So the first thing, I think the science writer just made a writing mistake.   Of course the earth doesn't revolve around the sun every 28 days.

For the second thing, you can say anything is the center of the universe you want.   Everything is relative.  But that doesn't mean it makes the mathematical modelling easier.  In this case, sure, we can say the sun revolves around the earth.  I mean every 365.25 days it makes a full circle around the earth, right?   The problem comes when trying to model the other 8 planets and their relationship to the earth.  It's not so much that the sun doesn't revolve around the earth--it's that the other 8 planets don't.  Oh I'm sorry, did I say 8?  I meant 7.  Even that is relative.  But anyway, if you model the earth as the center, modelling the sun's path around the earth is easy--it's the other 7 planets, Pluto, the asteroid belt, the comets--everything else BUT the sun and moon you have to worry about, because they follow very weird parabolic trajectories around the earth.  BUT--if you model the sun as the center, everything becomes very easy:  all the planets follow very simple oblong-shaped paths around the sun.   And from there, it's easy to calculate the projected distance from the earth to any other planet at any given time.   Likewise, you could project Jupiter's moons relative to the earth if you wanted to, but you don't want to.  It's far-and-away easier to model Jupiter's moons as revolving around Jupiter and then calculate their distance to earth from there.

So there is no hard rule that says the sun absolutely has to be the center of everything.  It's just that's what's most easily modelled.  Also this discovery that the planets all follow these arcs around the sun has ramifications when theorizing about the origins (and future) of the solar system, such as it indicates that maybe the sun was here first and the other planets got caught up in orbit--not the other way around.

Nothing... but since you ask... look at your animation... same if the sun rotate around the center of mass, in some way, the planet seam to rotate around the sun and not the reverse... in fact, planet and star don't rotate at the same speed... change the speed of the planet and it will explain the season...

Not really... the sun rotate around the marked point on the center of mass curve... by example, for 1945, the center of the sun rotate around the point marked "1945"... you need to see the curve "center of mass" like a extra dimension called "time"... well, if someday i have enough free time, i will maybe make a 3D animation of it... 

Mainly append when both of the massive planet ( jupiter and saturn ) align in their perihelion orbit ( more near the sun in a elliptical orbit )