## A Chiral Tessellation of the Plane Using Regular Hexagons and Parallelograms

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In addition to the sixty heptagons, there are twelve regular pentagons and sixty quadrilaterals in this polyhedron, along with 380 triangles of various shapes and sizes. That’s 512 faces in all.

I used *Stella 4d: Polyhedron Navigator *to make this. You may try this program, for free, at http://www.software3d.com/Stella.php.

The blue kites have sides which are in the golden ratio (~1.618:1), while the yellow kites’ sides are in a ratio equaling the square of that number, or approximately ~2.618:1.

In addition to the eighteen regular octagons, this polyhedron also has eight equiangular hexagons and twenty-four isosceles trapezoids among its fifty faces. I made it using *Stella 4d*, which you can try, for free, right here.

Here’s a single dodecahedron.

A new “cluster polyhedron” can be made by augmenting each pentagonal face with another dodecahedron.

If you can do it once, you can do it again, augmenting each pentagon with a new dodecahedron.

Once more.

I made these polyhedral clusters using *Stella 4d: Polyhedron Navigator*, which you can try for free at http://www.software3d.com/Stella.php.

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