## Wednesday, March 27, 2013

### Platonic Solid Stereotomic & Descriptive Geometry for Edge Bevel Angles

Wood polyhedron or Wooden Polyhedra
The 5 Platonic solids:
The Tetrahedron (3 equilateral triangles at each vertex)
The Hexahedron (3 squares at each vertex, cube)
The Octahedron (4 equilateral triangles at each vertex)
The Dodecahedron (3 pentagons at each vertex)
The Icosahedron (5 equilateral triangles at each vertex)

The so-called Platonic Solids are regular polyhedra. “Polyhedra” is a Greek word meaning “many faces.” There are five of these, and they are characterized by the fact that each face is a regular polygon, that is, a straight-sided figure with equal sides and equal angles: All of the edges of the Platonic Solids are Hip Rafters.

Platonic Solid TETRAHEDRON

Platonic Solid TETRAHEDRON
Deck: Equilateral Triangle
Roof Surface Faces: 3 Equilateral Triangles
Deck Angle = 60°
Plan Angle DD = 30.00°
Common Rafter Slope Angle SS = 70.52878°
Hip Rafter Slope Angle R1 = 54.73561°

Dihedral Angle Between Faces = 70.52855°
Hip Rafter Backing Angle = 54.73572°

Platonic Solid Edge Cuts
Hip Rafter Miter Angle  = 54.73561°
Hip Rafter Bevel Angle = 45.00°
Hip Rafter Saw Blade Bevel Angle  = 30.00°

Stereotomic & Descriptive Geometry for the Tetrahedron (3 equilateral triangles at each vertex)
Hip Rafter Edge Bevel Angles

Platonic Solid Hexahedron (Cube)

Platonic Solid CUBE
Deck: Equilateral Triangle
Roof Surface Faces: 3 x Isosceles Triangle
Deck Angle = 60°
Plan Angle DD = 30.00°
Common Rafter Slope Angle SS = 54.73561°
Hip Rafter Slope Angle R1 = 35.26439°

Dihedral Angle Between Faces = 90°
Hip Rafter Backing Angle = 45°

Platonic Solid Edges
Hip Rafter Miter Angle  = 35.26439°
Hip Rafter Bevel Angle = 35.26439°
Hip Rafter Saw Blade Bevel Angle  = 30.00°

Stereotomic & Descriptive Geometry for the Hexahedron (3 squares at each vertex, cube)
Hip Rafter Edge Bevel Angles

Platonic Solid Octahedron

Platonic Solid OCTAHEDRON
Deck: Square
Roof Surface Faces: 4 x Equilateral Triangles
Deck Angle = 90°
Plan Angle DD = 45.00°
Common Rafter Slope Angle SS = 54.73561°
Hip Rafter Slope Angle R1 = 45.00°

Dihedral Angle = 109.47174°
Hip Rafter Backing Angle = 35.26413°

Platonic Solid Edges
Hip Rafter Miter Angle  = 45.00°
Hip Rafter Bevel Angle = 54.73561°
Hip Rafter Saw Blade Bevel Angle  = 45.00°

Stereotomic & Descriptive Geometry for the Octahedron (4 equilateral triangles at each vertex)
Hip Rafter Edge Bevel Angles

Platonic Solid  Icosahedron

Platonic Solid ICOSAHEDRON
Deck: Pentagon
Roof Surface Faces: 5 x Equilateral Triangles
Deck Angle = 108°
Plan DD = 54.00°
Common Rafter Slope Angle SS = 37.37737°
Hip Rafter Slope Angle R1 = 31.71747°

Dihedral Angle = 138.16038°
Hip Rafter Backing Angle = 20.91981°

Platonic Solid Edges
Hip Rafter Miter Angle  = 31.71747°
Hip Rafter Bevel Angle = 58.2825°
Hip Rafter Saw Blade Bevel Angle  = 54.00°

Stereotomic & Descriptive Geometry for the Icosahedron (5 equilateral triangles at each vertex)
Hip Rafter Edge Bevel Angles

Platonic Solid Dodecahedron
Platonic Solid DODECAHEDRON
Deck: Equilateral Triangle
Roof Surface Faces: 3 x Isosceles Triangle
Deck Angle = 60°
Plan Angle DD = 30°
Common Rafter Slope Angle SS = 37.37737°
Hip Rafter Slope Angle R1 = 20.90516°

Dihedral Angle = 116.56506°
Hip Rafter Backing Angle = 31.71747°

Platonic Solid Edges
Hip Rafter Miter Angle  = 20.90516°
Hip Rafter Bevel Angle = 31.71747°
Hip Rafter Saw Blade Bevel Angle  = 30.00°

Stereotomic & Descriptive Geometry for the Dodecahedron (3 pentagons at each vertex)
Hip Rafter Edge Bevel Angles

1. Hi,

please would you mind giving me some pointers how to calculate mitre and bevel cuts for an octohedron with butt joints?

I'm also using square cross section timber so I'd like to include the 45 degree rotation of the timber into the compound angle calc so that I don't have to worry about the timber slipping down the 45 degree angled fence on its edge (or I could make a jig for that, but I don't trust arris rail to be perfectly 45 degrees).

The way I saw octahedron butt joints done was that each piece in the 4-way joints rotates 45 degrees around its central longitudinal axis. I hope that makes sense.

Thanks! Kester

1. Kester,

Miter angle = 60
Saw Blade Bevel Angle = 54.73561
or
Miter angle = 31.48216
Saw Blade Bevel Angle = 16.77865

I don't think it will work if you rotate the material 45. The joints would end up with what I refer to as claws.

Sim