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In 4-dimensional geometry, the dodecahedral pyramid is bounded by one dodecahedron on the base and 12 pentagonal pyramid cells which meet at the apex. Since a dodecahedron has a circumradius divided by edge length greater than one, so the pentagonal pyramids requires tall isosceles triangle faces.
The dual to the dodecahedral pyramid is a icosahedral pyramid, seen as an icosahedral base, and 12 regular tetrahedral meeting at an apex.
References
Dodecahedral pyramid Wikipedia(Text) CC BY-SA