Roof Framing Geometry

Wednesday, September 3, 2025

Icosidodecahedron Fish Tank

Archimedean Solid Icosidodecahedron

Faces

12 Pentagons

20 Equilateral Triangles

Edge Length 2.75"

1/2" Polycarbonate material

Dihedral Angle

142.62263°

Polygon Saw Blade Bevel Angles

Equilateral Triangle: 10.8123169°

Pentagon: 26.56505°

Set the table saw blade tilt to 26.57° for the pentagon faces.

Rip the 1/2" polycarbonate to 5" in width with the edge bevel.

Cut the material into 5" pieces. Then use the Pentagon table saw sled to bevel one edge. The previous bevel is against the stop with the edge bevel facing up. Set a width, a stop, and a toggle clamp to cut all 12 pieces to the same width.

Then lay out one edge at 2.75" and place a stop with a toggle clamp to cut all of the pieces at the same width.

Rotate the pieces and cut all of the edges.

Cutting the 20 Equilateral Triangles.

Set the table saw blade tilt to 10.81° for the Equilateral Triangles.

Rip the material to about 3" with the edge bevels being parallel.

Lay out two triangles and cut the material to form a rhombus.

Lay out one edge of the material at 2.75" and set a width stop with the toggle clamp.

Cut the triangles.

Glue up

Tape up a top and bottom. They will be identical.

Tape the two pieces together.

Cut the tape at the top pentagon face and remove it. Then glue the pieces together on the inside of the Icosidodecahedron.

Tuesday, September 2, 2025

Cutting Solid Tetrahedrons out of wood

The slant angle of the Tetrahedron is 70.52°.

90° - 70.52° = 19.47° for the blade tilt.

First, create a bevel on one edge of the material. Here I have a 2" x 2" x 12" piece of Purpleheart wood. With one edge ripped to 19.47°.

Next, lay the material on the compound miter saw with the table saw edge bevel facing you. The miter angle is 30°, and the blade tilt is 19.47°.

After the first cut, turn the material upside down. The table saw edge bevel will be facing the compound miter saw fence. Lay out the desired Tetrahedron edge length at a 30-60° angle from the back edge of the material. In this example, the edge length is 2 1/8".

Then make the cut to cut the Solid Tetrahedron.

I was able to cut 4 solid Tetrahedrons out of each 12" piece of material.

Remember to flip the material back to the starting position each time you cut the Tetrahedron.

Video showing the complete process for cutting the Solid Tetrahedron.

Roman Dodecahedrons Made out of Wood

Platonic Solid Dodecahedron

12 Pentagonal Faces

Roman Dodecahedron made out of Ash.

The Roman Dodecahedron is painted with metallic Silver, and the rings are painted with carbon black.

This Roman Doecahedron made out of Walnut has a 2 1/2" hole in each face.

Cut 12 Pentagons with a 31.71° blade tilt for the edge bevel. (Note: This picture is from an Archimedean Solid. It does not have the 31.71° blade tilt.)

Drill a circle 1/8" deep with a 3" diameter drill bit.

Drill a circle 1/8" deep with a 2 1/4" diameter drill bit.

Drill a hole with a 1 1/2" diameter drill bit.

Glue up the pentagon pieces.

Add balls at each vertex of the Dodecahedron. I drilled a 3/16" hole at each vertex. Then, I use instant glue to attach the miller dowels, with a ball at each vertex.

I used Miller wood screw dowels with a 1/2" maple ball glue to each dowel.

Video showing the complete process for the Roman Dodecahedra made out of wood.

Truncated Octahedron Fish Tank

With Plato, Archimedes, and Euclid, goldfish.

Archimedean Solid Truncated Octahedron

6 Squares 8 Hexagons

Dihedral Angle

125.264389683°

Polygon Saw Blade Bevel Angles

Square: 19.47122°

Hexagon: 35.26439°

Here are some of the techniques I used to build the Truncated Octahedron fish tank. The Truncated Octahedron is built with 6 squares and 8 hexagons. You can use a hexagon table saw sled or a compound miter saw to cut the 8 hexagons with a 35.26 edge bevel angle. The 6 squares have a 19.47 edge bevel angle.

I used the

Amana Tool LB10801 Carbide Tipped Non-Melt Plastic 10 Inch D x 80T M-TCG, -2 Deg, 5/8 Bore, Circular Saw Blade

I felt it cut the 1/2” polycarbonate material like butter.

Cut a block of wood with a 32.26° angle, for the blade tilt for the Hexagons.

Before I used the table saw sled, I edge beveled one edge of the material with the table saw. Then I cut the material into 6" pieces. Then I set the miter saw to 30°, with a blade tilt of 35.26°, and cut one edge of the hexagon. So, I would have two edges mitered and beveled before I used the hexagon table saw sled.

Rotate the material until all its edges form a hexagon.

The squares were shaped in the same way as the hexagons were.

Use tape to glue the pieces of the Truncated Octahedron together.

After the pieces are cut, I used Acrylic Cement along the inside edges. It fuses the polycarbonate material together. Then I used fish-safe silicone along the inside edges.

This Truncated Octahedron fish tank is probably the only one in existence. You could also use any of these Archimedean Solids as a fish tank.

Cuboctahedron,

Small Rhombicuboctahedron,

Snub Dodecahedron,

Icosidodecahedron,

Truncated Cube,

Truncated Icosahedron

If I were to build the fish tank again, I would use 1/4" polycarbonate material. And try to get less glue on the faces of the pieces.

Wednesday, August 27, 2025

Building Wooden Archimedean Solids

da Vinic's Polyhedra Φ Project #5

All 13 of the Archimedean Solids are now built out of wood.

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]

Snub Dodecahedron: Edge Length 2.75": Circumradius 5.92856"

Truncated Tetrahedron: Edge Length 2.75": Circumradius 3.22465"