Pseudo-deltoidal icositetrahedron

As the faces of the pseudorhombicuboctahedron are regular, the vertices of the pseudo-deltoidal icositetrahedron are regular.[1] But due to the twist, these 26 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).

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

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 2-isohedral); these 24 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)

Pseudo- and actual deltoidal icositetrahedron Pseudo- and actual rhombicuboctahedron

Pseudo- and actual deltoidal icositetrahedron Pseudo- and actual great deltoidal icositetrahedron

Pseudo-deltoidal icositetrahedron as die