da Vinic's Polyhedra Φ Project

Golden Rectangle Rhombicosidodecahedron

Φ Rhombicosidodecahedron

Based on folio 735v (1495) of the Codex Atlanticus by Leonardo da Vinci

While I was studying Leonardo da Vinci's sketches in his Codex Atlanticus, one of his drawings of a polyhedron looked as if he had elongated the squares. All of the polyhedra must follow the mathematical property that all of the planes on the polyhedron must be 3-sided to 10-sided polygons. Rectangles are not used to build any polyhedra. However, da Vinci was bucking the norm, coloring outside the box, and was hinting at using rectangles based on the golden ratio, phi, 1.618, to draw and build his polyhedra. 

So, I drew out the Rhombicosidodecahedron using golden rectangles, instead of squares. And sure enough, the Rhombicosidodecahedron with golden rectangles followed one of the other rules of polyhedra. That all of the vertexs of the polyhedra must touch the circumradius of the inscribing sphere. This Φ polyhedra is a new set of polyhedra that all of the other mathematicians over the last 500+ years have ignored, because it doesn't follow the rule that all of the planes of the polyhedra must be 3-sided to 10-sided polygons.  This new Φ polyhedra is a non-uniform polyhedron. 

The diagonal of the pentagon is the dimension used for one side of the golden rectangle. Additionally, da Vinci enjoyed drawing cupolas on the polyhedra in his sketchbook. In my drawing, I added the red tetrahedron cupolas to the triangles. 

Building the wooden Golden Rectangle Rhombicosidodecahedron

Here I developed a glue-up template with all of the necessary information to cut and glue up the wooden Golden Rectangle Rhombicosidodecahedron. 

This vertex-deck plan view drawing allows us to develop the dihedral angles geometrically. It's one of the concept drawings that is missing from all of the books on polyhedra. Anyone studying Stereotomy should also study this drawing. 

Cutting the Pentagon Planes

Before I placed the material in the table saw sled, I ripped one edge of the material at a 22.02 ° angle. Then I ran the piece through the table saw with the blade tilted at 22.02°. Next, I placed a stop for the correct dimension of the pentagon on the table saw sled. Then I rotated the material and cut the edge until all of the edges of the pentagon plane were the exact same dimension. 

Cutting the Golden Rectangle Planes

I ran the material through the table saw at a 15.45° angle and cut the pieces to the correct width.

Next, I placed a dimension stop in the compound miter saw and cut the pieces with a 9.69° angle.

Cutting the Equilateral Triangle Planes

Here again, I ran the material through the table saw at 5.45° angle. Then cut the triangle in the compound miter saw with a 5.45° angle. 

Here, I'm using the drawing that is printed at a 1:1 scale to verify the correct dimensions of each piece.  

First glue up of the polyhedra. 

Golden Rectangle Rhombicosidodecahedron

Wooden Golden Rectangle Rhombicosidodecahedron and wooden nested Platonic Solids