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The Voronoi tessellation of a homogeneous Poisson point process in the lowerhalf-plane gives rise to a family of vertical elongated cells in the upperhalf-plane. The set of edges of these cells is ruled by a Markovian branchingmechanism which is asymptotically described by two sequences of iid variableswhich are respectively Beta and exponentially distributed. This leads to aprecise description of the scaling limit of a so-called typical cell. The limitobject is a random apeirogon that we name menhir in reference to the Gallichuge stones. We also deduce from the aforementioned branching mechanism thatthe number of vertices of a cell of height $\lambda$ is asymptotically equal to$\frac45\log\lambda$.

Elongated Poisson-Voronoi cells in an empty half-plane