Is a structure made up of two polyhedrons connected by a common vertex or edge still called a polyhedron? This link(< states that two separate polyhedrons joined in this manner cannot be called polyhedrons. But mathematician Hessel once pointed out these structures as being polyhedrons that do not follow Euler's theorem. The two statements seem to be contradictory to one another. Is it tolerable to call these joined structures as polyhedrons?

It just depends on which definition of the term "polyhedron" you adhere. In a very loose sense those surely are thingies with poly (many) hedrons (seets = faces). Same for any other type of speciality and awkwardness, e.g. completely coincident faces, edges, or vertices, etc. Several authors, just for the reason not having to deal with all such sort of anoying stuff, simply exclude such cases by definition from their usage of a "polyhedron" (or, more general: "polytope"). - Therefore quite general: you always would have to look up according definitions, when comparing deduced statements.

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