Creating an 𝑛 = 4 Pancake Graph in 3D

We give an explicit, symmetric realization of the pancake graph P4 in three-dimensional space. Starting from the Cayley-graph description of P4 via prefix reversals, we identify canonical polygonal cycles arising from alternating generators and use them to build a layered 3D embedding consisting of four hexagons on the faces of a tetrahedron. We define a concrete coordinate map (equivariant with respect to a natural action of S4) and prove that edges corresponding to flips are represented as straight segments in the model. This geometric viewpoint makes the structure and symmetry of P4 transparent and provides a practical framework for visualization.