Four Polyhedra Featuring Heptagons

Heptagons only appear infrequently in interesting polyhedra. I recently found a few that I like.

Icosidodeca 2nd tetstell with heptagons and triangles is pyritohedral.gif

To form the first of these solids, shown above, I started with the icosidodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it twice using Stella 4d (available here). To obtain the model shown below, which also features heptagons and triangles, I stellated it once more. Both of these polyhedra have pyritohedral symmetry.

Icosidodeca 3rd tetstell with heptagons and triangles is pyritohedral.gif

To form the next model shown, I began with an rhombicosidodecahedron, set it to tetrahedral symmetry, and stellated it eight times. This produces a chiral solid with tetrahedral symmetry.

Stellated Rhombicosidodeca 8th tetstell features 8 heptagons tet symmetry.gif

For the last of these four polyhedra featuring heptagons, I began with the snub dodecahedron, dropped the symmetry of the model down from icosahedral to tetrahedral, and then stellated it sixty-one times. The resulting solid is chiral, with tetrahedral symmetry.

Snub Dodeca 61st tetstell.gif

A Pyritohedral, Stellated Polyhedron, and Its Convex Hull

To make this polyhedron using Stella 4d (available here), I began with the dodecahedron, dropped the symmetry of the model from icosahedral to tetrahedral, and then stellated it thirteen times. 

Dodeca 13th tetstell.gif

This stellated polyhedron has pyritohedral symmetry, but this is easier to see in its convex hull:

Convex hull of the dodecahedron's 13th tetstell.gif

The eight blue triangles in this convex hull are equilateral, while the twelve yellow ones are golden isosceles triangles.

A Chiral Polyhedron Made of Kites and Triangles, Along with Its Dual, Made of Triangles and Isosceles Trapezoids

To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:

chiral polyhedron made of triangles and kites -- found while exploring tetstells of the icosahedron.gif

The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.

Faceted Dodeca.gif