In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron.

Johannes Kepler in his Harmonices Mundi applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. However, Louis Poinsot in 1809 rediscovered two more, the great icosahedron and great dodecahedron. This was proved by Augustin-Louis Cauchy in 1812 that there are only four regular star polyhedrons, known as the Kepler–Poinsot polyhedron.

Brückner (1900) extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the complete stellation. Wheeler (1924) published a list of twenty stellation forms (twenty-two including reflective copies), also including the complete stellation. 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. 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.

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 stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. 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.

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.

The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio

$${\displaystyle {\sqrt {{\frac {3}{2}}\left(3+{\sqrt {5}}\right)}}\,:\,{\sqrt {{\frac {1}{2}}\left(25+11{\sqrt {5}}\right)}}\,:\,{\sqrt {{\frac {1}{2}}\left(97+43{\sqrt {5}}\right)}}\,.}$$

Convex hulls of each sphere of vertices

Inner	Middle	Outer	All three
20 vertices	12 vertices	60 vertices	92 vertices
Dodecahedron	Icosahedron	Nonuniform truncated icosahedron	Complete icosahedron

When regarded as a three-dimensional solid object with edge lengths ${\displaystyle a}$, ${\displaystyle \varphi a}$, ${\displaystyle \varphi ^{2}a}$ and ${\displaystyle \varphi ^{2}a{\sqrt {2}}}$ (where ${\displaystyle \varphi }$ is the golden ratio) the complete icosahedron has surface area

$${\displaystyle S={\frac {1}{20}}(13211+{\sqrt {174306161}})a^{2}\,,}$$

and volume

$${\displaystyle V=(210+90{\sqrt {5}})a^{3}\,.}$$