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This is not an answer to you question and I am sure you question (once it is formulated correctly) does not have a complete answer.

The embeddings of a collection of triangles with a common vertex is the same as an embedding of a graph in $S^2$.

The same way, the embedding of a collection of tetrahedra with a common vertex can be reformulated as embedding spherical polyhedral space into $S^3$. The embeddings into $\mathbb R^{3,1}$ correspond to embeddings of hyperbolic polyhedral space into Lobachevsky 3-space.

I think the most interesting theorem about this is Alexandrov's embedding theorem --- the theorem usually formulated for embedding of polyhedral spaces in to $\mathbb R^3$, but it was also proved for embedding of spherical polyhedral spaces to $S^3$ as well as for embedding hyperbolic polyhedral space into Lobachevsky 3-space.

2 added 208 characters in body

This is not an answer to you question and I am sure you question does not have a complete answer.

The embeddings of a collection of triangles with a common vertex is the same as an embedding of a graph in $S^2$.

The same way, the embedding of a collection of tetrahedra with a common vertex can be reformulated as embedding spherical polyhedral space into $S^3$. The embeddings into $\mathbb R^{3,1}$ correspond to embeddings of hyperbolic polyhedral space into Lobachevsky 3-space.

I think the most interesting theorem about this is Alexandrov's embedding theorem --- the theorem usually formulated for embedding of polyhedral spaces in to $\mathbb R^3$, but it was also proved for embedding of spherical polyhedra polyhedral spaces to $S^3$.S^3$as well as for embedding hyperbolic polyhedral space into Lobachevsky 3-space. 1 This is not an answer to you question and I am sure you question does not have a complete answer. The embeddings of a collection of triangles with a common vertex is the same as an embedding of a graph in$S^2$. The same way, the embedding of a collection of tetrahedra with a common vertex can be reformulated as embedding spherical polyhedral space into$S^3$. I think the most interesting theorem about this is Alexandrov's embedding theorem --- the theorem usually formulated for embedding of polyhedral spaces in to$\mathbb R^3$, but it was also proved for embedding of spherical polyhedra to$S^3\$.