Quotes from the book Bucky Works, by J. Baldwin, p 112, chapter Why Domes?
All domes share certain advantages, whether or not they are geodesic. Their compound-curved shape is inherently strong, giving a self-supporting clear span with no columns. Domes are resource and energy-efficient because, of all possible shapes, a sphere contains the most volume with the least surface. This holds true for dormal slices of a sphere as well. A dome has a circular footprint. Of all possible shapes, a circle encloses the most area within the least perimeter. Thus, for a given amount of material, a dome encloses more floor area and interior volume than any other shape.
The minimal surface presents the least area through which to gain or lose heat. Field experience has shown that home-size domes use about 1/3 less heating fuel than an equally well-insulated conventional home of the same floor area, built of the same materials. .. When you double the exterior dimensions of a dome (or any other object), the skin area rises by a factor of four while the volume rises by a factor of eight. ... Larger domes are more energy efficient because less percent of the contained air is near or touching the skin where most heat loss or gain occurs. Doubling the size of a dome doubles its thermal efficiency. Bucky suggested that the huge mass of air contained in a big dome would make insulation superfluous, especially if the dome had a double skin. ..
The favorable surface-to-volume ration is not the only reason for a domes remarkable thermal performance; interior and exterior aerodynamics play a part too. ... Except for the calculation of wind loads, most architects do not consider aerodynamics at all. Bucky, however, found aerodynamics to be critical in the design of energy-efficient buildings. He discovered in the 4D experiments, a buildings heat loss is in direct proportion to its aerodynamic drag. .. Unlike most buildings, domes are streamlined. Wind slides smoothly over and around them, generating minimal eddies and vortices to disturb the insulating boundary layer of air that clings to the exterior of any object. ..
The domes heat loss is further reduced by the concave interior. .. A heating device located somewhere in the central updraft will distribute heat evenly throughout the dome. The concave interior also bestows a less-expected thermal advantage: self-cooling.