Let \( U(4) \) be the unitary group formed by the unitary matrices of order 4, and let \( D(4) \) be its subgroup of diagonal matrices. The compact set \( U_4 \) of unistochastic matrices, determined by the squared moduli entries of unitary matrices, is contained in the famous Birkhoff polytope \( B_4 \). It is of great interest, and our main goal, to determine the shape of \( U_4 \), and in particular its boundary. For this purpose, the bicoset space \( D(4)\backslash U(4)/D(4) \) will be viewed as the preimage of \( U_4 \) under the unistochastic map \( \Phi_4\colon U(4)\to U_4 \), defined as \( \Phi_4(U):=U\circ \overline U \), where \( \circ \) denotes the Hadamard or entrywise product. The investigation of the critical loci of this map is crucial for our purposes, since every boundary point of \( U_4 \) is the image of a critical point. Using a standard parametrization of the bicoset space \( D(4)\backslash U(4)/D(4) \), we provide a criterion for a point of the bicoset space to be a critical point. We also present an algorithm to decide the unistochascity of a given bistochastic matrix, and we analyze the unistochascity of the bistochastic matrix obtained multiplying by 1/34 the magic square engraved in Dürer's celebrated Melancolia I.