King Abdulaziz Center for World Culture is a new development in Dhahran, Saudi Arabia designed by Snøhetta architects where five complex geometry buildings, called Pebbles, accommodate different civic and cultural functions. Penetrating the ground level podium are three open-air spaces called Oases, each with a glazed wall leading into the basement areas. Each Oasis façade is fully glazed, up to 15m tall and constituted by faceted double glazed panels tilting in and out of plane, all with different shape. The architectural intent called for a fully transparent design with flush glass. Buro Happold proposed a solution using glass fins as secondary support and a toggle system for retaining the glass. Buckling of the slender fins is resisted by horizontal glass beams. The team was able to build on previous experience in considering buckling and restraint issues for tall the slender fins, but the fins' 15° lateral inclination presented a significant stability challenge. The adopted design connects each fin to one side of the double glazed units, so each set of fins and glazing units acts a stable L-shaped structural unit. Unique connection details were developed to achieve the necessary structural behaviour. Detailing around access doors and movements of supporting structures posed further complications which have had to be resolved in an architecturally sensitive manner. The glass fins and the façade panels have been analysed both individually and in a global model to verify the stresses and deflections were within the limits in all the different conditions. This paper presents the design criteria, the findings and the solutions adopted to achieve the architects' vision.