In this work, we present the manifolds that can be obtained by gluing together in pairs the faces of an octahedron. Since the resulting gluings are not always manifolds, we introduce the notion of a truncated octahedron and consider the gluings of its hexagonal faces. We first focus on analogous gluings of polygons into compact surfaces without boundary. The resulting manifolds obtained from the truncated octahedron are separated according to their boundary, and for each boundary we present one gluing into the manifold with the selected boundary.
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