Building Wooden Archimedean Solids

da Vinic's Polyhedra Φ Project #2

Cuboctahedron: Edge Length 2.75": Circumradius 2.75"

Small Rhombicuboctahedron: Edge Length 2.75": Circumradius 3.84714"

Snube Cube: Edge Length 2.75": Circumradius 3.6952"

Rhombicosidodecahedron: Edge Length 2.75": Circumradius 6.1406"

The study of polyhedra develops spatial reasoning and imagination while providing the foundation for points in 3D  space to a vertex-plan view drawing of 2D points. Precise geometric forms are powerful examples to show off one's mastery, as they require a clear understanding of the relationship between 3D objects and 2D projections.[George Hart: 2022]

Left to right: Golen Rectangle Rhombicosidodecahedron, Standard Rhombicosidodecahedron, Small Rhombicuboctahedron, Snube Cube, and Cuboctahedron.

 Not museum quality, but they look great. I gave it my best shot with my old eyes. I found it easy to cut the polygon pieces, but a real challenge to glue the polygon shapes with a perfect fit. Then I found myself over-sanding on some of the polygon faces, which creates ridges on the adjoining polygon faces. 

Rhombicosidodecahedron

Material: Squares-Yellowheart, Equilateral Triangles-Bubinga, Pentagons-Bolivian Rosewood

Archimedean Solid Rhombicosidodecahedron table saw blade tilt angle settings:

12 Pentagons

20 Equilateral Triangles

30 Squares

Dihedral Angle 

159.09484°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 7.62263°

Square: 13.28252°

Dihedral Angle 

148.28252°

Polygon Saw Blade Bevel Angles

Pentagon: 18.43495°

Square: 13.28251°

Polygon Table Saw Sleds

Glue up Templates for Rhombicosidodecahedron

Cut and glue time: 3 Days

The next three Archimedean Solids were built using this printout for the blade tilt angles. You lay a piece of wood on the paper and scribe the blade tilt angles. Cut the piece of wood with a compound miter saw. Basically, sneak up on the scribe mark on the wood for the blade tilt angles.  

Small Rhombicuboctahedron

Material: Equilateral Triangles-Yellowheart, Squares-Bubinga

Archimedean Solid Small Rhombicuboctahedron table saw blade tilt angle settings:

18 Squares

8 Equilateral Triangles

Dihedral Angle 

144.73561°

Polygon Saw Blade Bevel Angles

Square: 22.5°

Equilateral Triangle: 12.76439°

Dihedral Angle 

135.00°

Polygon Saw Blade Bevel Angles

Square: 22.5°

Square: 22.5°

Small Rhombicuboctahedron Glue Up

Cut and glue time: 1 Day

Snub Cube

Material: Equilateral Triangles-Yellowheart, Squares-Bubinga

Archimedean Solid Snub Cube table saw blade tilt angle settings:

6 Squares

32 Equilateral Triangles

Dihedral Angle 

153.2345877°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 13.382706°

Equilateral Triangle: 13.382706°

Dihedral Angle 

142.98343007°

Polygon Saw Blade Bevel Angles

Square: 23.63386°

Equilateral Triangle: 13.382706°

Snub Cube Glue Up

Cut and glue time: 1 Day

Cuboctahedron

Material: Equilateral Triangles-Yellowheart, Squares-Bolivian Rosewood

Archimedean Solid Cuboctahedron table saw blade tilt angle settings:

6 Squares

8 Equilateral Triangles

Dihedral Angle 

109.471220635°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 19.47122°

Square: 35.26439°

Cuboctahedron Glue Up

Cut and glue time: 4 Hours

Only 9 more Archimedean Solids to build. 

 da Vinic's Polyhedra Φ Project #1

Golden Rectangle Rhombicosidodecahedron

Φ Rhombicosidodecahedron

The study of polyhedra develops spatial reasoning and imagination while providing the foundation for points in 3D  space to a vertex-plan view drawing of 2D points. Precise geometric forms are powerful examples to show off one's mastery, as they require a clear understanding of the relationship between 3D objects and 2D projections.[George Hart: 2022]

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 appeared to have had the squares elongated. 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

Wooden Golden Rectangle Rhombicosidodecahedron with Tetrahedron cupolas and the phi symbol burned into the golden rectangles with a branding iron. 

Cutting the Tetrahedron cupolas.

The Tetrahedron is cut using the slant angle of a 3-sided pyramid.  Which is 70.52°.  Set the table saw blade tilt to 19.47° and bevel one edge of the material. 

Set the Compound Miter saw miter angle to 30°, and the bevel angle to 19.47°. Make the cut with the table saw, bevel facing you. 

Turn the material upside down and lay out the bottom of the Tetrahedron using the desired edge length. 

Then cut the material with the bottom of the tetrahedron facing up and the table saw bevel facing the fence.