The shape can't exist: if you imagine a square face in 2D you have 4 edges and 4 vertices. Adding a further vertex out of the plane (introducing a 3rd dimension) you MUST have 4 more edges to join the original 4 vertices to the novel vertex. This sums to 8 edges as a minimum.
This is of course assuming Euclidean geometry.
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square * * * * * A square? It has 4 edges and one face!
square
The geometric shape that has 1 square face and 4 triangular faces is a pyramid. A pyramid also has five vertices and eight edges. The square acts as the base for the shape on which it sits.
A pyramid has 8 edges in total.
square-based pyramid