Great triambic icosahedron
Introduction '''Great triambic icosahedron''''''Medial triambic icosahedron'''140px140pxTypesDual uniform polyhedraSymmetry groupIhNameGreat triambic icosahedronMedial triambic icosahedronIndex referencesDU47, W34, 30/59DU41, W34, 30/59Elements''F'' = 20, ''E'' = 60''V'' = 32 (χ = -8)''F'' = 20, ''E'' = 60''V'' = 24 (χ = -16)Isohedral faces120px120pxDuals120pxGreat ditrigonal icosidodecahedron120pxDitrigonal dodecadodecahedron'''Stellation'''Icosahedron: W34240pxStellation diagram3D model of a medial triambic icosahedronIn geometry, the '''great triambic icosahedron''' and '''medial triambic icosahedron''' (or '''midly triambic icosahedron''') are visually identical dual uniform polyhedra. The exterior surface also represents the '''De2f2''' stellation of the icosahedron. These figures can be differentiated by marking which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the even–odd rule, the internal structure of both shapes will differ.The 12 vertices of the convex hull matches the vertex arrangement of an icosahedron.
Great triambic icosahedron The '''great triambic icosahedron''' is the dual of the great ditrigonal icosidodecahedron, U47.It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed propeller. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges.The faces have alternating angles of  and . The sum of the six angles is , and not  as might be expected for a hexagon, because the polygon turns around its center twice. The dihedral angle equals .
Medial triambic icosahedron The '''medial triambic icosahedron''' is the dual of the ditrigonal dodecadodecahedron, U41.It has 20 faces, each being simple concave isotoxal hexagons or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.The faces have alternating angles of  and . The dihedral angle equals .Unlike the great triambic icosahedron, the medial triambic icosahedron is topologically a regular polyhedron of index two. By distorting the triambi into regular hexagons, one obtains a quotient space of the hyperbolic order-5 hexagonal tiling:250px
As a stellation 240px240pxIt is Wenninger's 34th model as his 9th stellation of the icosahedron
See also * Triakis icosahedron* Small triambic icosahedron* Medial rhombic triacontahedron
References * ** H.S.M. Coxeter, ''Regular Polytopes'', (3rd edition, 1973), Dover edition, , 3.6 6.2 ''Stellating the Platonic solids'', pp.96-104
External links *** gratrix.net  Uniform polyhedra and duals* bulatov.org   Medial triambic icosahedron  Great triambic icosahedron