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A circle can be partitioned into $n\in\mathbb{N}$ congruent 1-spherical line segments similar to the regular polygons.

A sphere can be partitioned into $n\in\{4,6,20\}$ congruent 2-spherical equilateral triangles similar to the tetra-, octa-, and icosahedron.

Is that the end of the story, or is it possible to partition a glome into congruent 3-spherical tetrahedrons for some $n$s?

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The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.

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