Geodesic Dome Green houses

Does anybody know what advantage these type of Greenhouses have over more traditional models? At least in terms of growing capacity.

If made from primarily clear (poly or glass panels) one will have many more CUBIC feet of grow space for the same footprint.  This allows for more creative use of space.  But the primary structural advantage is the cost of material per square foot of space; and the rigidity to the elements with minimal construction cost.  

As far as their growing ability I don't know of anything special about geodesic over a traditional structure.  There are some good web resources that will help you calculate cubic foot capacities of domes to compare to other structures.

Dome calculator

I built a geodesic for its wind-resistance (though it has yet to be tested!). The idea being that at no point is there a big flat face to the wind, and in the windy hilltop location where it lives a geodesic shape would allow the wind to pass over the structure more easily.

A geodesic structure allows you to build the biggest space with the least materials... however everything is still sold in rectangular sizes, so I think you have to buy just as much, but end up with more waste.

HeartWater farm in southern Utah has/had (not sure if they are still going) 2 greenhouses made from geodesic domes. They definitely worked well, they had vents which auto opened based on temps using a wax inside a cylinder, so no electronics to break but it was still automatic. One vent at the top where a whole panel would lift and I think a pair of panels at the base as well. I'm not sure how old they were when I visited but they were in good shape. I'm also not sure of the material used for the panels, it wasn't glass. If you are ever near Bryce Canyon their farm is maybe 30 minutes from there and they were looking for someone to take over the 20 acres since their kids weren't interested.

Be aware that different latitudes lead toward to different green house geometry.
A pure dome would work best at the equator.  The further North or South one goes the more you want the sun facing wall as the solar gain and the opposite wall as thermal mass.
I'm at 60*N and the best green houses here are glazed on the SE, S and SW, pure thermal mass on all other sides.