Final answer:
The Fontenay Abbey's design style falls under the Cistercian order, known for their austerity and simplicity in architectural design, adhering closely to the Rule of St. Benedict.
Step-by-step explanation:
The Fontenay Abbey in Burgundy France, being a 12th-century monastery, represents the design style of the Cistercians. This order was known for their simplicity and austerity in architecture. The Cistercian order emerged as a reform branch of the Benedictines, and they sought to follow the Rule of St. Benedict more closely, especially in terms of its calls for manual labor and self-sufficiency.
While Gothic architectural elements like high vaulted ceilings and stained glass windows became prominent in monastic buildings like the Basilica of Saint Denis, the Cistercians strove for more austere designs. They preferred less decoration and more functional structures, which is evident in the architecture of Fontenay Abbey.
The 12th century also saw the spread of Romanesque and then Gothic architecture across France and Europe following the strengthening of French royalty, giving rise to iconic Gothic cathedrals in cities. The Abbey of Fontenay, however, remains a prime example of Cistercian architecture, with its clear and unadorned geometric forms.