An "Icosahedron" twistypuzzle is, overall, a regular icosahedron (no surprise here!) which has the following key properties:
 20 faces (each an identical equilateral triangle, except for color)  an equilateral triangle is a regular polygon of 3 sides
 30 edges (each a line where two faces meet)
 12 vertices (each a point where five edges meet)
Like
all twisty puzzles discussed on my site, each face is divided into
multiple facets which are (usually for this class) colored triangles each
identical in shape and size. The
goal is (usually) to arrange the facets (by twisting layers) such that every face
shows a single
color. Most puzzles discussed on this page have twenty unique colors
(one for each face), but there are some which duplicate colors and thus have fewer unique colors.
I know of 2 variants of this puzzle, each described in some detail further below. But a quick summary is:
 Dogic [n=2] (2^{2}=4 triangles per face * 20 faces = 80 triangle facets; 60 tetrahedrons+ 20 octahedrons = 80 surface blocks)
 Icosaix [n=3] (3^{2}=9 triangles per face * 20 faces = 180 triangle facets; 12 hexahedrons + 60 octahedrons + 30 tetrahedrons = 102 surface blocks)
An important thing to note about this class of puzzles is, so far, no member puzzle is composed of icosahedron blocks (which is also true of the Dodecahedron family, but contrary to the Tetrahedron, Hexahedron and Octahedron families).
Another important thing about the icosahedron shape is that it is a
Platonic solid.
Two interesting things about any Platonic solid: all sides have the
same shape and each vertex (corner) has the same number of faces.
Trivia: the icosahedron is the mathematical "dual" of the dodecahedron (icosahedron has 12 vertices and 20 faces, while dodecahedron has 12 faces and 20 vertices).

Icosahedron 
Important Note: Although the corner blocks of these puzzles look like the "trivial" tips of the Pyramix family, the corner blocks of these puzzles should not rotate freely! The reason is because the corner blocks are not theoretically pyramids (like the corners of the Pyraminx family actually are). In other words, the corner blocks should be locked in place and unable to twist (this is analogous to the way that cornerblocks of the Rubik's Cube [family] are held inplace by 3 edgeblocks). However, both the Dogic and Icosaix puzzles allow the corners to rotate... although this breaks geometric theory (for a "regular" puzzle), it makes solving the puzzles easier (and because they are quite complex, this "trick" is greatly appreciated by some players [like me!]).
This cornerturning (n=2, doublelayer) puzzle seems moderately complex. Each face has only four triangular facets, and you are granted the freedom to twist a "corner block". Of course the fact that it has a daunting 20 sides makes the puzzle nontrivial. But there are two aspects of this puzzle which make it notably more difficult:
 Each "corner" is not a single block (contrary to almost every other twistypuzzle) but consists of 5 "tip" blocks
 Each "center" block is not a single color, but is "painted" as 3 isosceles triangles of different colors  this means the orientation of a "center" blocks is relevant to solving this puzzle (contrary to all other twistypuzzles discussed on my site)
It has 4 triangular facets per face: 3 with a single color (each a "tip" = 1/5 of a "corner"), and 1 (the center) with 3 colors. Thus, it is possible to have six (3*1+1*3) colors on a single face. Like all icosahedra, it has 20 faces. Thus it has 4*20=80 facets in total.
The Dogic is composed of 3? block types:
 60 tetrahedron "cornerslice" blocks (each with 1 visible facet showing 1 color)
 20 octahedron(?) "center" blocks (each with 1 visible facet showing 3 colors)
 unknown number of "core" blocks (each with no visible facet)
This puzzle has 80 surface blocks (60+20). It has an average of (exactly) 1 visible facet per surface block (80/80).
The Dogic has 21,991,107,793,244,335,592,538,616,581,443,187,569,604,232,889,165,919,156,829,382,848,981,603,083,878,400,000 combinations (about 21.99 sesvigintillion [10^{81}]). That value is calculated as: 60!/[5!^{12}] ("tips") * 20!/2*3^{19} ("centers") / 60 (no global orientation).
Important Note 1: Unlike all other twistypuzzles discussed on my site, the goal of this one is not
to get the same color on all blocks of each face! This would be
impossible due to the triangular facets at the center of each face
("center blocks") which have 3 different colors. Instead the object is
to get each "corner block" to have the same color (5 "cornerslices" of the same color) and also match the adjacent
color of each surrounding "center block" [see photograph] . Also, this is the only puzzle discussed
on my site where the orientation of the "centers" is relevant.
Important Note 2: I am describing the 12color version (a unique color per each corner). There also exists a 10color version (presumably the same color on opposite faces). There is probably a 20color version too (a unique color on every face).


This dualturning (n=3, singlelayer) puzzle is, well, big! Each face has 9 colored triangles (facets) per face and (like all icosahedron) has 20 faces. Thus it has an intimidating 180 "visible" facets (colored triangles). It is composed of 4? different block types:
 12 (irregular) hexahedron "corner" blocks (which do twist), each with 5 visible facets  a "pentagonbased pyramid"
 30 (irregular) tetrahedron "edge" blocks, each with 2 visible facets  the visible facets are equilateral triangles, but the hidden ones are isosceles
 60 octahedron "wing/center" blocks, each with 1 visible facet
 unknown number of "core" blocks (each with no visible facet)
So this puzzle has 102 visible "surface" blocks (12+30+60). Thus it has about 1.7647058823529411 visible facets per surface block (180/102 = 30/17).
I haven't been able to find a reliable source on the web for the number of combinations on this puzzle has. So based on my math and engineering experience, I theorize the that the Icosaix has approximately 15.79230654 googol sextillion [10^{121}] combinations. This value is calculated as 12!/2*5^{12} ("tips") * 30!/2*2^{29} ("edges") * 60!/[3!^{20}] ("center/wings") / 60 (no puzzle orientation).
Important Note 1: This puzzle is strictly a "singlelayer"
puzzle. This means that although a "corner block" may be twisted, no
layers below it may be twisted (contrary to what seems possible from a
visual inspection). The same also applies to face layers: only an
actual face may be twisted, layers below may not be twisted.

Icosaix

© 2017, H2Obsession
Diagram of transparent icosahedron © 2015 Tom van Zummermen ( http://stackoverflow.com/users/109774/tomvanzummeren)
Photograph of "Dogic" © 2015 Jaap Scherphuis ( http://www.jaapsch.net/puzzles/dogic.htm)
