Abstract.
Let S be a non-degenerate simplex in $\mathbb{R}^{2}$. We prove that S is regular if, for some k $\in$ {1,...,n-2}, all its k-dimensional faces are congruent. On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent.
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Martini, H., Wenzel, W. Simplices with congruent k-faces. J. Geom. 77, 136–139 (2003). https://doi.org/10.1007/s00022-003-1643-9
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DOI: https://doi.org/10.1007/s00022-003-1643-9