Precision Table Saw Settings and Techniques for the Archimedean Solids built in wood

Constructing complex polyhedra in wood presents a unique challenge, requiring exceptional accuracy in cutting compound angles. This article details the necessary miter and saw blade bevel angle (blade tilt) settings, along with essential techniques and tools, for precisely building these captivating wooden solids using a table saw or compound miter saw.

Archimedean Solids Vertex Drawings drawn out using geometry

The Archimedean Solids are made up of different types of polygons at each vertex, unlike the Platonic Solids, which only have one type of polygon meeting at each vertex. The vertices of the Archimedean Solids have Equilateral Triangles, Squares, Pentagons, Octagons, and Decagons meeting at the vertices. The intersection of each of these polygon planes at the vertex requires us to define the dihedral angle for each type of polygon meeting at the vertices. Then we need to determine the triangle perpendicular to the dihedral angle edge. The triangle perpendicular to the dihedral angle edge is then drawn out geometrically to determine the Saw Blade Bevel Angle (blade tilt), which forms cuts on the polygon planes to achieve precision mating pairs. These blade tilt settings can be used on a table saw or a compound miter saw.

The first step is to draw a vertex diagram with the polygon planes unfolded. These unfolded polygon planes are referred to as planner nets. Once the planner nets are unfolded, you draw a line perpendicular to the edge of the planar net, which we will call line one. Then draw a line perpendicular to the edge line in plan view from line one, until it intersects the vertex plan view line of the adjacent planner net. We will call this line two. Draw a line perpendicular to the edge of the adjoining planner net and connect it to line two. We will call this line three. 

Example drawing for geometrically drawing out the dihedral angle triangles.

Example drawing for geometrically drawing out the table saw blade tilt angles for the Archimedean Solids using the dihedral angle triangle.

The Archimedean Solids have Equilateral Triangles, Squares, Pentagons, Octagons, and Decagons meeting at the vertices. The miter angles on the table saw will be :

Equilateral Triangles: 30°

Squares : 0°

Pentagons: 36°

Hexagon: 30°

Octagons: 22.5°

Decagons: 180°

The miter angles using a compound miter saw :

Equilateral Triangles: 30°

Squares : 0°

Pentagons: 18°

Hexagon: 30°

Octagons: 45°

Decagons: 54°

Example drawings of how to draw out the vertex-deck plan view drawings of the Archimedean Solids

Archimedean Solid Truncated Tetrahedron table saw blade tilt angle settings:

4 Hexagons

4 Equilateral Triangles

Dihedral Angle 

109.471220635°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 15.7931690484°

Hexagon: 54.7356103172°

Dihedral Angle

70.528779366°

Polygon Saw Blade Bevel Angle

Hexagon - 54.7356103172°

Hexagon - 54.7356103172°

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°

Archimedean Solid Truncated Cube table saw blade tilt angle settings:

6 Octagons

8 Equilateral Triangles

Dihedral Angle 

125.264389683°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 9.73561°

Octagon: 45°

Archimedean Solid Truncated Octahedron table saw blade tilt angle settings:

6 Squares

8 Hexagons

Dihedral Angle 

125.264389683°

Polygon Saw Blade Bevel Angles

Square: 19.47122°

Hexagon: 35.26439°

Dihedral Angle 

109.471220635°

Polygon Saw Blade Bevel Angles

Hexagon: 35.26439°

Archimedean Solid Small Rhombicuboctahedron table saw blade tilt angle settings:

8 Squares

18 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°

Archimedean Solid Truncated Cuboctahedron table saw blade tilt angle settings:

12 Squares

8 Hexagons

6 Octagons

Dihedral Angle 

144.73561°

Polygon Saw Blade Bevel Angles

Hexagon: 22.5°

Square: 12.76439°

Dihedral Angle 

125.26438°

Polygon Saw 

Blade Bevel Angles

Octagon: 32.23561°

Hexagon: 22.5°

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°

Archimedean Solid Icosidodecahedron table saw blade tilt angle settings:

12 Pentagons

20 Equilateral Triangles

Dihedral Angle 

142.62263°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 10.8123169°

Pentagon: 26.56505°

Archimedean Solid Truncated Dodecahedron table saw blade tilt angle settings:

12 Decagons

20 Equilateral Triangles

Dihedral Angle 

142.62263°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 5.6599°

Decagon: 31.71747°

Archimedean Solid Truncated Icosahedron table saw blade tilt angle settings:

12 Pentagons

20 Hexagons

Dihedral Angle 

142.62263°

Polygon Saw Blade Bevel Angles

Pentagon: 16.47221°

Hexagon: 20.90515°

soccer ball

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°

Archimedean Solid Truncated Icosidodecahedron table saw blade tilt angle settings:

12 Decagons

20 Hexagons 

30 Squares

Dihedral Angle 

159.09484°

Polygon Saw Blade Bevel Angles

Square: 7.62263°

Hexagon: 13.28252°

Dihedral Angle 

142.62263°

Polygon Saw Blade Bevel Angles

Decagon: 24.09°

Hexagon: 13.28251°

Archimedean Solid Snub Dodecahedron table saw blade tilt angle settings:

12 Pentagons

80 Equilateral Triangles

Dihedral Angle 

164.17552°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 7.91223°

Equilateral Triangle: 7.91223°

Dihedral Angle 

159.93022°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 7.91223°

Pentagon: 19.15756°

This article serves as an excellent foundation for those wishing to explore further complex polyhedral constructions using geometry.