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  • $\begingroup$ The pentagons you constructed overlap... so then also the quadrilateral. Maybe they can be all be trimmed the same way, so as to not overlap. Probably so. But I think a picture here would be worth 1000 words. $\endgroup$
    – Yaakov Baruch
    Commented Sep 20, 2023 at 13:59
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    $\begingroup$ Ah, sorry, I misunderstood you pentagons as something larger: 4 adjacent triangles and the the slice of pentagon joining the vertices of the 1st and 4th triangle. So nothing wrong (except my comment). Nice construction by the way. $\endgroup$
    – Yaakov Baruch
    Commented Sep 20, 2023 at 19:31
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    $\begingroup$ So the pentagonal hexecontahedron? $\endgroup$
    – Peter Taylor
    Commented Sep 21, 2023 at 10:15
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    $\begingroup$ A distorted version thereof. The constructed spherical pentagon is not planar. We would not exoect such planarity when constructing a loop around an Archimedean-solid vertex except if the loop were triangular. The pentagonal hexecontahedron with planar faces does not have all its vertices on one sphere. $\endgroup$
    – Oscar Lanzi
    Commented Sep 21, 2023 at 10:40