A Golden Tessellation of Quadrilaterals

In this tessellation, golden rectangles are shown in yellow. The orange darts are each made of two golden gnomons, joined at a leg — while the blue rhombi are each made of two golden triangles, sharing a base.

The Golden Rhombus, the Rhombic Triacontahedron, and the Rhombic Hexecontahedron

There’s a special rhombus which is called the “golden rhombus,” because its diagonals are in the golden ratio. To construct it with compass and straight edge, you first construct a golden rectangle (shown with blue edges and a yellow interior), and then connect the midpoints of its sides to form a rhombus (with edges shown in red).

Several polyhedra can be made which use golden rhombi as their faces. The most well-known of these polyhedra is the rhombic triacontahedron, which has 30 such faces. It is the dual of the icosidodecahedron.

If the rhombic triacontahedron is stellated 26 times, the result is the (non-convex) rhombic hexecontahedron. It has 60 golden rhombi as faces.

Both of these polyhedra can be constructed with Zometools (available at http://www.zometool.com). With white Zomeballs and red Zomestruts, these polyhedra look a lot like this:

The flat image at the top of this post was created using Geometer’s Sketchpad and MS-Paint. The four rotating polyhedral images were created using Stella 4d: Polyhedron Navigator, which you can purchase, or try for free, at http://www.software3d.com/Stella.php.

Cube-Based Polyhedron Featuring Regular Icosagons, Equilateral Triangles, Isosceles Trapezoids, and Rectangles

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Polyhedron Featuring Regular Icosagons, Equilateral Triangles, Trapezoids, and Rectangles

Created with software you can try and/or buy at http://www.software3d.com.Stella.php.

A Polyhedron with 122 Faces

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A Polyhedron with 122 Faces

The 122 faces are twenty blue, irregular hexagons; thirty red, irregular hexagons; 60 rectangles, and twelve regular pentagons.

Software credit: just visit http://www.software3d.com.Stella.php, and look for the free trial download of Stella 4d, the program I used to make this rotating image.