California Bay Window Hip Rafter Head Cuts

A Rafter Tools app user asked me to make an app that calculated a bay window roof that has two of the hip rafters landing on top of the California roof sheathing. The question I'm now facing is which type of hip rafter head cuts should I use in the Rafter Tools app?

1. Head Cut Option # 1
2. Head Cut Option # 2
3. Head Cut Option # 3

I'm leaning towards option number 3.

Making A Wood polyhedron or Wooden Polyhedra

This model is the Tetrahedron (3 equilateral triangles at each vertex) . See my last article on the geometry of the Platonic Solid Tetrahedron for the angles of the Platonic Solid.

I was trying to see if it was possible for Leonardo da Vinci (1452-1519) to have made wooden polyhedra models  having solid edges for his illustrations in Luca Pacioli's 1509 book The Divine Proportion. Conclusion, no. He might have made wooden polyhedra models out of twigs, but to have cut all the edge pieces exactly would have been beyond his tool-man-ship. This Platonic Solid Tetrahedron has 6 edge pieces. That's 6 x 4 = 24 compound cuts on the wooden model. In his Dodecahedron model / drawing it would been 30 x 4 = 120 compound cuts on the wooden model and all of them would have to be exact. If the  compound cuts on the wooden model are not exact, then you'll never be able to put all of the pieces together. This isn't even counting the 30 x 2 = 60 edge bevels, hip rafter backing.

Leonardo da Vinci's Polyhedrahttp://www.georgehart.com/virtual-polyhedra/leonardo.html

1: Rip 6 pieces of wood to the same dimension. In this model I had 4x4 redwood post left over from a job and I ripped them to 2 1/2 x 2 1/2.

2: Cut a scrap of wood to 35.2°. This is half of the Dihedral angle.

3: Use the scrap of wood to set the table saw blade tilt to 35.2°.

4: The hip rafter backing angle is 54.73572°.  Since the table saw blade tilt is limited to 45° we're ripping  the material with the blade tilt at 35.2° , half of the Dihedral angle, on both sides of the material to form the Dihedral angle on each edge piece of the Tetrahedron. This is the same as backing out the hip rafter.

5: Check the Dihedral angle. It should be 70.5°.

6: Set the blade tilt of your miter saw to 45° and the miter to 45°. These two angles are for cutting the edge piece on the top edge of the material. If we were cutting the edge piece on the side of the material, like we do with hip rafters, then we would use 54.73561° as the miter angle and 30° as the saw blade bevel tilt. It would result in a compound  bevel angle of 45.00°.

7: Cut one end of each of the 6 edge pieces of the Tetrahedron.

8: Setup a stop on the miter saw. All the edge pieces of the Tetrahedron must be exactly the same dimension.

9: Reverse the edge piece and cut the other end using the stop.

10: Assemble the 6 edge pieces. Assemble the 3 bottom pieces first. This will tell you if your miter angle and blade tilt were exact. 

In this photo you can see one edge of the tetrahedron model does not fit perfectly. This is due to the blade tilt or miter angle not being exactly 45°.

These pictures make it look easy to build the platonic solid, but it's not. Assembling the 6 edge pieces perfectly is harder than cutting the 6 edge pieces.

Platonic Solid Icosahedron

With the material laying flat on the compound miter saw
Your miter angle is = 31.71747°
Your Saw Blade Bevel Angle = 54.00°

or you can cut the material on edge in your compound miter saw using
Saw Miter Angle = 58.2825°
Saw Blade Bevel Angle = 18.00°

Left and right miter angle jigs for Icosahedron.

The saw miter and saw blade bevel angles for the Icosahedron are beyond the capabilities of most compound miter saws.  You will have to build a jig to cut the edge pieces of the Icosahedron.

It's been a couple of years since I built the Icosahedron, but I think the jig triangles are 30° - 60° triangles and the saw miter angle would be 58.2825° - 30° = 28.2825° with a saw blade bevel angle of 18.00° for cutting the edge pieces on edge in the compound miter saw.

Roof Framing Angles

 for the

 Great Pyramid

Hip Rafter Backing Angle

33.78620°

also used for the plywood

saw blade bevel angle

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.

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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

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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

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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

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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

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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

Stereotomic & Descriptive Geometry for Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Tetrahedron - Tetrahedral Ground Plan

Roof Slope Angle 33.69007°

Eave Angle 60°

Stereotomic Geometry for Tetrahedral Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Square Rectangular -  Pyramidal Ground Plan

Roof Slope Angle 50°

Eave Angle 90°

Stereotomic Geometry for Pyramidal Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Pentagonal Ground Plan

Roof Slope Angle 40°

Eave Angle 108°

Stereotomic Geometry for Pentagonal Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Hexagonal Ground Plan

Roof Slope Angle 40°

Eave Angle 120°

Stereotomic Geometry for Hexagonal Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Octagonal Ground Plan

Roof Slope Angle 40°

Eave Angle 135°

Stereotomic Geometry for Octagonal Hip Rafter Bevel Angles

Descriptive Geometry for Hip Rafter Bevel Angles at peak & seat of hip rafter

Trapezoidal Ground Plan With Three Different Roof Pitches

Roof Slope Angles 40° , 44.07884°, 59.72722°

Eave Angles 90°, 90°, 115° , 65°

Stereotomic Geometry for Trapezoidal Ground Plan With Three Different Roof Pitches

Tetrahedron Slice of Hip Rafter Head Cut
Miter Angle = Hip Rafter Slope Angle = 37.00039°
Saw Blade Bevel Angle = 63.90458°
Hip Rafter Side Cut Bevel Angle = 21.36377°

Tetrahedron Slice of Hip Rafter Head Cut

Miter Angle = Hip Rafter Slope Angle = 37.00039°

Saw Blade Bevel Angle = 26.09542°

Hip Rafter Side Cut Bevel Angle = 58.47947°

Treatise on Roof Framing Geometry for Hip Rafter Side Cut Bevel Angles