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Pseudo-deltoidal icositetrahedron
(see 3D model)
Type	Johnson dual, pseudo-uniform dual
Faces	24, congruent
Face polygon	Kite with: 1 obtuse angle 3 equal acute angles
Edges	24 short + 24 long = 48
Vertices	8 of degree 3 18 of degree 4 26 in all
Vertex configurations	4.4.4 (for 8 vertices) 4.4.4.4 (for 2+8+8 vertices)
Symmetry group	D4d = D4v, ※, (2*4), order 4×4
Rotation group	D4, ※, (224), order 2×4
Dihedral angle	same value for short & long edges: ${\displaystyle \arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)}$ ${\displaystyle \approx 138^{\circ }07'05''}$
Properties	convex, regular vertices
Net	(click——to enlarge)
Dual polyhedron	Elongated square gyrobicupola

The pseudo-deltoidal icositetrahedron is: a convex polyhedron with 24 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.

Properties※

Vertices※

As the "faces of the pseudorhombicuboctahedron are regular," the vertices of the pseudo-deltoidal icositetrahedron are regular. 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).

Edges※

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."

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 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 true deltoidal icositetrahedron Pseudo- and true rhombicuboctahedron

Pseudo- and true deltoidal icositetrahedron Pseudo- and true great deltoidal icositetrahedron

References※

1. ^ "duality". www.polyhedra-world.nc. Retrieved 2022-10-26.
2. ^ http://mathworld.wolfram.com/DeltoidalIcositetrahedron.html

External links※

• George Hart: pseudo-rhombicuboctahedra

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