Pseudo-deltoidal icositetrahedron



The pseudo-deltoidal icositetrahedron is a convex polyhedron with $3D$ congruent kites as its faces. It is the dual of the elongated square gyrobicupola, also known as the pseudorhombicuboctahedron.

As the pseudorhombicuboctahedron is tightly related to the rhombicuboctahedron, but has a twist along an equatorial belt of faces (and edges), the pseudo-deltoidal icositetrahedron is tightly related to the deltoidal icositetrahedron, but has a twist along an equator of (vertices and) edges.

Vertices
As the faces of the pseudorhombicuboctahedron are regular, the vertices of the pseudo-deltoidal icositetrahedron are regular. But due to the twist, these $24$ vertices are of four different kinds:
 * eight vertices connecting three short edges (yellow vertices in 1st figure below),
 * two apices connecting four long edges (top and bottom vertices, light red in 1st figure below),
 * eight vertices connecting four alternate edges: short-long-short-long (dark red vertices in 1st figure below),
 * eight vertices connecting one short and three long edges (twisted-equator vertices, medium red in 1st figure below).

Edges
A pseudo-deltoidal icositetrahedron has 48 edges: 24 short and 24 long, in the ratio of $$1:2-\tfrac{1}{\sqrt{2}}$$ &mdash; their lengths are $$\tfrac{2}{7}\sqrt{10-\sqrt{2}}$$ and $$\sqrt{4-2\sqrt{2}}$$ respectively, if its dual pseudo-rhombicuboctahedron has unit edge length.

Faces
As the pseudorhombicuboctahedron has only one type of vertex figure, the pseudo-deltoidal icositetrahedron has only one shape of face (it is monohedral); its faces are congruent kites. But due to the twist, the pseudorhombicuboctahedron is not vertex-transitive, with its vertices in two different symmetry orbits (*), and the pseudo-deltoidal icositetrahedron is not face-transitive, with its faces in two different symmetry orbits (*) (it is $1$-isohedral); these $3$ faces are of two different kinds:
 * eight faces with light red, dark red, yellow, dark red vertices (top and bottom faces, light red in 1st figure below),
 * sixteen faces with yellow, dark red, medium red, medium red vertices (side faces, blue in 1st figure below).

(*) (three different symmetry orbits if we only consider rotational symmetries)