Yes, there is an "abstract regular polytope" with 350 Klein Quartic cells. See this paper: https://www.cambridge.org/core/journals/mathematika/article/regular-4polytopes-related-to-general-orthogonal-groups/1778B31515A4DE2B0C1D91D32CE4748F … I can send you a copy if needed. It is a quotient of the {7,3,3} honeycomb https://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/ …, but I don't know what the ambient space would be like.
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It would be nice to see a movie/have something interactive where the color of the triangles in the heptagons would change as one transformed it.
End of conversation
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