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Essentially, I am trying to find the specific name of the convex polyhedron or icosahedron shape in the gif below. The 3d software (Blender) reports 92 vertices, 180 faces and 540 edges, although there are 270 unique edges. Every face is a triangle, composite surface patterns include a number of hexagons and pentagons.

The convex shape is of icosahedron 3v origin, as the pentagon frequency (or edge quantity from one pentagon to another) is 3 edges (see link). However, the frequency (and 3v) naming scheme seems to be non-standard.

The snub dodecahedron matches the convex shape specified (see gif below) very closely, except that the regular pentagons (within the snub dodecahedron) have a central vertex, thus increased vertices, edges and triangle faces.

The pentakis dodecahedron matches the icosahedron 2v. The icosahedron is the 1v variant. Perhaps there's a pattern in naming convention I'm missing to help discover the name of the shape in question.

"dome" model I am attempting to identify

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    $\begingroup$ Who said this thing should have a name in the first place? Besides, 540 can't be true, as $92-540+180\ne2$. $\endgroup$ – Ivan Neretin May 29 '19 at 8:06
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    $\begingroup$ @IvanNeretin See my answer. 540 edges is the sum of the edge counts of each face, which is therefore twice the number of actual edges. $\endgroup$ – Parcly Taxel May 29 '19 at 8:08
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The solid is topologically equivalent to the polyhedron obtained by performing a kis operation on a truncated icosahedron – adding a shallow pyramid on each original face. As such, it is a kis-truncated icosahedron.

It is also a geodesic polyhedron, formed by subdividing each icosahedral face into nine triangles at frequency $3$, and in Wenninger's notation described at the Wikipedia link is $\{3,5+\}_{3,0}$.

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  • $\begingroup$ Thanks @Parcly Taxel - that identity has helped me recreate the model. For posterity, it can be created using antiprism software with the command: geodesic -c 1 -f 3 ico | antiview to produce the same model. $\endgroup$ – pds May 29 '19 at 12:01

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