What does this research mean for the field?

The Pentarkehedron and the Umbrahedron are new convex polyhedra that exhibit a self-referential duality and are confirmed to be distinct from existing polyhedral families. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to introduce a new pair of geometrically dual convex polyhedra based on pentagonal symmetry.

March 17, 20260 citationsOpen Access

The Pentarkehedron and the Umbrahedron: A New Convex Polyhedron Pair from Pentagonal Symmetry and Its Polar Dual

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JSJean Santillana

Key Points

The aim is to introduce a new pair of geometrically dual convex polyhedra based on pentagonal symmetry.
Defined the characteristics of the Pentarkehedron and Umbrahedron.
Compared both polyhedra against twenty standard polyhedral families.
Verified their duality through their face and vertex distributions.
Identified the Pentarkehedron with 15 vertices, 25 edges, and 12 faces.
Identified the Umbrahedron with 12 vertices, 25 edges, and 15 faces.
Both polyhedra confirmed as new with C5v symmetry.

Abstract

We introduce the Pentarkehedron (V=15, E=25, F=12, five triangles and seven pentagons, C5v symmetry) and its polar dual, the Umbrahedron (V=12, E=25, F=15, ten triangles and five quadrilaterals, C5v symmetry), as a new geometrically dual pair of convex polyhedra. Both solids are verified against twenty standard polyhedral families and confirmed new. The pair exhibits a self-referential duality: the face distribution of the Pentarkehedron equals the vertex-valence distribution of the Umbrahedron.

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The Pentarkehedron and the Umbrahedron: A New Convex Polyhedron Pair from Pentagonal Symmetry and Its Polar Dual

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