We have a new kitten. Her name is Hexagon.

# Month: July 2018

## Four Polyhedra Featuring Heptagons

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

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.

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.

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.

## 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.

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

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:

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.