  In this article we study a cake cutting problem. More precisely, we study
symmetric fair division algorithms, that is to say we study algorithms where
the order of the players do not influence the value obtained by each player. In
the first part of the article, we give a symmetric and envy-free fair division
algorithm. More precisely, we show how to get a symmetric and envy-free fair
division algorithm from an envy-free division algorithm. In the second part, we
give a proportional and symmetric fair division algorithm with a complexity in
O(n 3) in the Robertson-Webb model of complexity. This algorithm is based on
Kuhn's algorithm. Furthermore, our study has led us to introduce a new notion:
aristotelian fair division. This notion is an interpretation of Aristotle's
principle: give equal shares to equal people. We conclude this article with a
discussion and some questions about the Robertson-Webb model of computation.
