It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron.

It can be viewed as an irregular, simple, and star polyhedron.

[4] Wheeler (1924) published a list of twenty stellation forms (twenty-two including reflective copies), also including the complete stellation.

[5] H. S. M. Coxeter, P. du Val, H. T. Flather and J. F. Petrie in their 1938 book The Fifty Nine Icosahedra stated a set of stellation rules for the regular icosahedron and gave a systematic enumeration of the fifty-nine stellations which conform to those rules.

[6] The complete stellation is referenced as the eighth in the book.

In Wenninger's book Polyhedron Models, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W42.

[7] In 1995, Andrew Hume named it in his Netlib polyhedral database as the echidnahedron after the echidna or spiny anteater, a small mammal that is covered with coarse hair and spines and which curls up in a ball to protect itself.

The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation.

The Du Val symbol of the complete stellation is H, because it includes all cells in the stellation diagram up to and including the outermost "h" layer.

[9] As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram.

These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2.

When regarded as a three-dimensional solid object with edge lengths

is the golden ratio) the complete icosahedron has surface area[11]

Each face is an irregular 9/4 star polygon, or enneagram.

[9] Since three faces meet at each vertex it has 20 × 9 / 3 = 60 vertices (these are the outermost layer of visible vertices and form the tips of the "spines") and 20 × 9 / 2 = 90 edges (each edge of the star polyhedron includes and connects two of the 180 visible edges).

When regarded as a star icosahedron, the complete stellation is a noble polyhedron, because it is both isohedral (face-transitive) and isogonal (vertex-transitive).