Monday, March 21, 2022


HG IV Circular Work in Carpentry
Stereotomy Geometry

This book is a subset of the Holy Grail of Roof Framing Geometry & Trigonometry books.

More Info on HG IV


The book includes step-by-step instructions on developing stereotomic double curvature drawings for circular work in carpentry. Some of the techniques presented in this book are thousands of years in the making. As carpenters, we shouldn't have to re-invent the wheel each time we are faced with new challenges. There is a likely hood that someone, that came before us, has already refined a technique that we can use today. The tools might have changed, but the piece of wood that it is twisting and turning remains the same piece of wood.

This book also includes a zip file download of 930MB. There are some print, scribe, and cut drawings for double curvature in the zip file download. As, well as, PDFs, pictures of the process of cutting the wood, SketchUp files, and some AutoCAD files.

Book Contents

Cords, Midpoints, and Tangents
Lines and Midpoints
Unit Circle Arc Segment Functions
The Three Lost Points of Circular Work
Professor Louis Mazerolle's Conic Section Practice Page
Parabola Conic Section Practice Page
Hyperbola on Round Tower Roof
Ellipse Functions
Cylinder SkyLite
Helix Curve - Circular Stair Stringer
Helix Double Curvature Stair Stringer
Helix Double Curvature Stair Stringer Practice
Concial Helix
Double Curvature Arch
The intersection of Three Cylinders
Imperial - Dome Roof Orthographic Projection Exercise
The intersection of Cone and Inclined Plane
Layover Rafters on Elliptical Sleepers
The intersection of Vertical & Horizontal Cones
Hunting Miter Parabola
Circle Scribe Transfer Tool ™
Barrel Roof Framing - Elliptical Layover Roof Surface Geometry
Barrel Roof Framing - Trigonometry Purlin Rafters
Barrel Roof Framing - Circular Ribs
Barrel Roof Framing - Roof Surface Development
Barrel Roof Framing - Task Model
Barrel Roof Framing - Arch Top Dormer
Barrel Vault Elliptical Rib Layout
Eyebrow Roof Dormer with Ribs Task Model
Eyebrow Roof Dormer Inner Valley Plywood
Eyebrow Roof Dormer Math
Eyebrow Roof Dormer with Purlins
Curved Hip Rafters
Curved Bay Window Rafters
Ski Slope Rafters
Ogee-Imperial Hip Rafters Equal Slope
Ogee-Imperial Bay Window Hip Rafters
Ogee-Imperial Pentagon Pavilion
Ogee-Imperial Hexagon Pavilion
Ogee-Imperial Octagon Pavilion
Imperial - Straight Roof Task Model
Imperial - Dome Roof Task Model
Bell Roof
Dome Roof
Concave Roof
Belgian Oeil-de-Boeuf,
known as Bull's Eye windows
Medieval Gothic Cathedral Table
Round Tower Roof Intersecting A Straight Roof
Dome Roof Intersecting a Straight Roof
Round Tower Roof Intersecting A Dome Roof
Round Tower Roof Intersecting A Imperial Roof
French Capucine Dormer-Skylite
French Guitarde Dormer-Skylite
Guitarde and Capucine Solid Models Cut on the BandSaw
Single Curvature Knee Braces
Double Curvature Knee Braces
Fourneau's Guitarde Dormer built in 1765 in Rouen France
Round Tower on Square Plan
Pyramid Roof on Circular Plan
The Intersection of a Cone & Cutting Plane for the Ellipse cut on a bandsaw
Curved Purlins on Round Tower Roof
Saint-Andre's Cross on Round Tower
Round Tower Roof with Saint Andrew's Cross Task Models
Saint-Andre's Cross on Imperial Round Tower
Round Towers of California with Saint Andre's Cross
Professional Stair Builder Secrets
Enclosed Circular Stairs
Free Standing Circular Stairs
Enclosed Elliptical Stairs
Free Standing Elliptical Stairs
Tangent Handrailing
Round Tower with Ears
The Intersection of a Cone & Cutting Plane for the Ellipse
Groin Vaults
Barrel Vaults
Elliptical Vaults
Parabolic Arch Vaults
Gothic Arch Vaults

Wednesday, January 29, 2020

The Art of the Line for Carpentry

The Art of the Line for Carpentry
Stereotomy Geometry

This book is a subset of the Holy Grail of Roof Framing Geometry & Trigonometry books.
You can purchase the book HG-III at this link.

  • Art du Trait de Charpente
  • Art of the Line for Carpentry
  • German Schiften
  • Basic Task Models
  • TC - DP Shadow Line
  • Canted Hip and Valley Rafters
  • Net Roof Surface
  • Folding Roof Surface Planes
  • Footprint Planes
  • Net Planes
  • Sloping Ridges
  • Rafter Claws
  • Purlins
  • Dormers
  • Tréteau à devers
  • Canted Trestles
  • Joiners Table
  • Sawhorses
  • Saint Andrews' Cross
  • WorldSkills Carpentry Competition Practice Task Model
  • Twisted Crosses
  • Canted Cross
  • Hexagon Canted Cross
  • Pentagon Canted Cross
  • Witches Cut
  • Warlock Cut
  • MoonWalk Lines
  • Roof Framing Geometry
  • Roof Framing Stereotomy
  • The Aurora Roof Framing Challenge
  • American Journeyman Carpenters Examination
  • German Schiften Roof Averaging Exam

Sunday, September 2, 2018

Holy Grail of Roof Framing Geometry & Trigonometry Book

Book I & Book II

With all the 3D drawings we use today, a craftsman who spends his whole life doing a few simple operations has little chance of exercising his intelligence and thus corrupts the spirit of his mind. These books are for anyone interested in learning the geometric process to roof framing that can be used on the job site. As well as the trigonometric formulas that have evolved from the geometry. A lot of the geometry in the books will be the first time there’s an English explanation to the geometric drawings. French Art du Trait and German Shiften drawings are intellectual exercises in three-dimensional lines. However, precise rafter layout can only be accomplished with knowledge of the geometry that develops the parallelogram in the compound joint.

Most roofs can be calculated with a CMC calculator. For complex roof framing, trigonometry or geometry must be used. There is more trigonometry in this book than all of the roof framing books written in English combined. However, it has over 1,000 geometric roof framing drawings in the book as well, for carpenters that are not comfortable using trigonometry.

This book does not fit in your sisters' lunch pail. It was written to make complex roof framing simple. 
5 different techniques to cut the Witches cut. 
Roof framing base knowledge starts with geometry, not the CMC calculator. 
Precise rafter layout is accomplished with plumb lines and level lines.
Bastard Hip Roof Framing Simplified to compete with the King of Bastards -- Ryan Smith
Grasshopper method for  canted hip rafters.

Link to Purchase the book:


Precise rafter layout can only be accomplished with knowledge of the geometry that develops the parallelogram in the compound joint. Take away the carpenters Framing Square, CMC calculator, and circular saw to show you what they really know about cutting rafters. Can they still layout and cut a hip rafter without the use of these tools? These books will help you acquire the knowledge that has been passed down by the builders of the past.

The information shown in some roof framing books can be overwhelming. My primary objective in writing these books is to share the knowledge I've gained from reading and studying books on roof framing. It's easy to think you know everything about roof framing, big fish in a little pond syndrome until you see or examine the roof framing from around the world and from the builders of the past. I'm as guilty as anyone on the big fish in a little pond syndrome, roof framing in the San Francisco Bay area during the height of conventional roof stick frame era, before roof trusses. Now that I've studied a lot of geometry and trigonometry, I know that using the Pythagorean theorem in the 1980's was like taking bunny steps compared to what I know and can do now.
The king, queen, and knight of roof framing are
1: Hip Rafter Backing Angle
2: Valley Sleeper Layover Bevel Angle
3: Roof Surface - Sheathing Angles.

All of the roof framing angles we use are based on these three angles. For plumb hip rafters, the hip rafter backing angle and roof sheathing angles define the cut on the timber. For canted rafters, the valley sleeper layover bevel angle and roof sheathing angles define the rafter cuts for canted rafters.

Hip rafter shift - offset, plumb line shift, DP-shadow line, and roof averaging design are the other four techniques that need to be studied-learned to graduate from the big fish in a little pond to compete in the world skills of roof framing. There are other areas in the roof that need to be studied, but these seven angles-techniques should be the base for all roof framers.

Don't re-invent the wheel, like I've been guilty of, learn these seven angles - techniques. That has been passed down through the centuries. The builders of the past have made it pretty easy for us. All we have to do is study the techniques that they used from the middle ages to the present day. And we get to use circular saws, instead of broad axes, handsaws, draw knives, chisels, Bisaiguë, etc... to cut our rafters.

Pages from the book:
Book Table of Contents

    Book I
  • Unit Circle Basics for developing Trigonometric Functions
  • The "Unit Circle" is a circle with a radius of 1
  • Tetrahedron Trigonometric Identity Formulas by Joe Bartok
  • Fundamental Hip-Valley Roof Ratios by Joe Bartok
  • Trigonometric Identities
  • Law of Sines Formulas
  • Law of Cosines Formulas
  • Roof Framing Angles in an Unfolded Tetrahedron
  • Canadian & American Geometric Roof Framing
  • Development using a Steel Framing Square
  • Definitions of Roof Framing Angles by Joe Bartok
  • Timber Framing Trigonometric Identity Formulas by Joe Bartok
  • Timber Framing Hawkindale Angle Formulas
  • Frank L. Martindale - H. L. McKibben - L. E. Gray
  • Rees Hawkins Acheson
  • Ed Levin
  • Curtis Milton
  • Timber Framing Angles: S, D , R1 , R2 , R3 , R4 , R5 , R6 , R7
  • P1 , P2 , P3 , P4 , P5 , P6 , P7
  • C1 , C2 , C3 , C4 , C5 , C6, C7 , C8 ,
  • Timber Framing Angles: A5 , A7 , A8 , A9
  • R5B , A5B , R5P , A5P , R4B , R4P ,
  • P4BV , P5BV , R6P , R6PBV , R7BV , VSB ,
  • Roof Averaging Formula: RA
  • Saw Blade Bevel Angle Trigonometric Formulas
  • Tréteaux Angles, tréteau à devers
  • Tetrahedron Slice for Trigonometric Formulas
  • Platonic Solids Trigonometric Formulas
  • Platonic Solid Tetrahedron Trigonometric Formulas
  • Platonic Solid Hexahedron Trigonometric Formulas
  • Platonic Solid Octahedron Trigonometric Formulas
  • Platonic Solid Dodecahedron Trigonometric Formulas
  • Platonic Solid Icosahedron Trigonometric Formulas
  • Auxiliary Drawings for Platonic and Archimedean Solids
  • Archimedean Solids with Jack Rafter Side Cut Angles
  • Archimedean Solids Compound Joinery
  • Archimedean Solid Truncated Tetrahedron
  • Archimedean Solid Rhombicuboctahedron
  • Archimedean Solid Cuboctahedron
  • Archimedean Solid Truncated Hexahedron
  • Archimedean Solid Truncated Octahedron
  • Archimedean Solid Truncated Cuboctahedron
  • Trigonometric Formulas Quick Reference
    Book II
  • Treatise On Those Parts of Geometry Needed by Craftsmen
  • A lesson in Applied Geometry and Euclidean Geometry
  • Pentagon Developed from 5 Circles
  • daisy - stumpEnNailen Test
  • Glossary - Vocabulary
  • Folding Roof Planes
  • Démaigrissement Method
  • Irregular Hip Roof Development - Billy Dillon
  • When Roofs Collide - Will Beemer
  • Hip and Valley Roof Compound Angle Formulas and Geometry
  • Fundamental Hip-Valley Roof Ratios by Joe Bartok
  • Bunny - Grasshopper Test
  • Snake - Two Headed Serpent Test
  • Warlock Cut and the Secret Line
  • Prism Planes in Roof Framing
  • Pythagorean Theorem
  • Hip Rafter Backing Angles
  • Basic Hip Rafter Shift
  • Basic Hip Rafter Shift On The Job Site
  • Hip Rafter Plumb Line Shift
  • Jack Rafter Board
  • Hip Rafter Plumb Line Shift Example
  • Equal Overhang for Unequal Sloped Roofs
  • Steel Framing Square Usage for Top Bevel Cuts
  • Jack Rafter Plumb Line Shift With Framing Square
  • Jack Rafter Plumb Line Shift
  • Trigonometric Formulas Quick Reference
  • Hip Rafter Plumb Line Shift On The Job Site
  • Basic Rafter Lengths
  • Rafter Lengths
  • Purlin Rafters - Frieze Blocks - Panne - Pfetten
  • Purlin - Roof Sheathing Cuts in the Roof Surface
  • Purlin Lip-Cut - Seat Cut- Lower Claw Angle
  • Shadow Line - Purlin Rafters - Geometrically
  • Saw Blade Bevel Angles
  • Purlin Rafters Geometrically
  • Jack Rafter Lower Claw Angle
  • 4 Hits of the Square for Lower Claw Angles
  • Witches Cut - Hexenschnitt
  • Square Tail Fascia
  • Roof Averaging - Roof Design - Dachausmittlung
  • Sloping Ridge and Roof Averaging - Roof Design
  • Roof Design - Roof Surface - Dachausmittlung
  • Polygonal Roof Framing
  • Polygonal Roof Framing Octagonal
  • Octagon Roof Framing Geometric Development
  • Octagonal 4 Butt Roof Framing
  • Octagonal Roof Framing Head Cuts
  • Octagonal Roof Framing Example
  • Polygonal Roof Framing Hexagon
  • Hexagon Roof Framing Geometric Development
  • Polygonal Roof Framing Pentagon
  • Pentagon Roof Framing Geometric Development
  • Round Tower Roof Framing
  • Round Tower 4 Butt Head Cuts Roof Framing
  • Round Tower Mitered Head Cuts Roof Framing
  • Round Tower Rafter Layout Roof Framing
  • Pyramid Roof Framing
  • Pyramid Square Equal Sloped Roof Framing
  • Pyramid Rectangular Unequal Sloped Roof Framing
  • Bay Window Roof Framing
  • Bay Window Roof Framing Example
  • Bay Window Match Gable Roof Framing
  • Bowed Bay Window Roof Framing
  • Unequal Side Length Bay Window Roof Framing
  • Equal Overhang Bay Window Roof Framing
  • Unequal Sided Octagon Geometric & Trigonometric
  • Layover Valley Sleeper Roof Framing
  • Valley Rafter Roof Framing - de noue - Kehlsparren
  • Irregular Valley Rafter Roof Framing
  • Bisecting Hip & Valley Rafters Roof Framing
  • Cripple Hip Rafter Roof Framing
  • Valley Rafter to Hip Rafter Jack Sliders Roof Framing
  • Valley Sleeper Roof Framing
  • Valley Layover Rafter Geometrically
  • Off Angle Lay-Over California Valley Roof Framing
  • Snub Nose Hip Roof Framing
  • Clipped Hip Roof Framing
  • Alhambra Granada Hip Roof Framing
  • Rake Wall Roof Framing
  • Exterior Hip Roof Rake Wall Framing
  • Rake Walls Rotated into Roof Surface
  • Hopper Angles
  • Crown Molding Angles
  • Shed Dormer - Pitch Break Roof Framing
  • CutIn Dormer Shed Roof Framing
  • Vaulted Ceiling Dormer Roof Framing
  • Unequal Slope Vaulted Ceiling Dormer Roof Framing
  • Trapezoid With Unequal Overhang
  • Trapezoid With Equal Overhang
  • Prow Roof Framing Angles
  • Prow Rafter Perpendicular Framing Angles
  • Unequal Pitched Roofs using The Law of Sines
  • Unequal Pitch Roof With Unequal Height Plate Line Trigonometry
  • Unequal Pitched Gable Rafters with Unequal Plate Heights Example
  • Hip Rafter Diamond Post Angles
  • Chanlattée Hip Rafter Roof Framing
  • Saint Andrews Cross Roof on Plumb Hip Rafter Framing
  • Devers Hip Rafters
  • Devers Hip Rafter Tables for Equal Sloped Roofs
  • English, French, German, Russian Translations
  • Equal Pitched Roof Steel Framing Square Usage
  • Rafter Tables for Equal Sloped Roofs
  • TFG - Roof Framing Kernel
  • Complex Roof Layout Geometry
  • Purlin Miter Angle Framing Square Usage
  • Purlin Run Multiplier for Purlin Miter Angle Framing Square Usage for Pentagon
  • Purlin Run Multiplier for Purlin Miter Angle Framing Square Usage for Hexagon
  • Purlin Run Multiplier for Purlin Miter Angle Framing Square Usage for Octagon
  • Level 1 Roof Cutter's Exam, Equal Pitched 90° Deck Angle
  • Level 2 Roof Cutter's Exam, Unequal Pitched 90° Deck Angle
  • Level 3 Roof Cutter's Exam, Pentagon with 75° Rafter Tails
  • Level 4 Roof Cutter's Exam, Fan Tail Rafters with Square Rafter Tails
  • Level 5 Roof Cutter's Exam, Unequal Pitched Gable
  • Level 6 Roof Cutter's Exam, Multi-Gon Equal Pitched
  • Level 7 Roof Cutter's Exam, Unequal Pitched with Deck Angles Not at 90°
  • Book List - Reference
  • Book III - Book IV Reference
  • Trigonometric Formulas Quick Reference

Monday, March 12, 2018

Holy Grail for Roof Framing Geometry

There are no decimal points in Euclidean Geometry using a straightedge and Compass.

I deciphered what Euclid was drawing on the plaster tablet in 200 BC.  
The Euclidean Geometry for the Devers-canted hip rafter head cuts. 
Using the roof surface angles, the profile rafter slope angle and the valley rafter sleeper angle you can draw out the Devers hip rafter head cut and foot cut angles. As well as the jack rafter and purlin rafter head cuts.

Saturday, October 28, 2017

Freestanding Circular Stair Stringer Framing

On our current rough frame project in Orinda California, this custom residential house has a round tower with circular stairs.  The inside and outside stair stringers are freestanding so you can walk under the stairs.

The first time I saw a set of freestanding stairs being built was in 1982, by Glen Watkins. This was on a tract of homes in the San Jose California area. Glen wouldn't let anyone in the house he was working on until he was done building the stairs. He didn't want anyone stealing his trade secrets. 

The second time I saw a set of freestanding stairs being built was in 1992, by Tom, can't remember his last name, and no it's not Tom White, Glen's Watkins nephew. This was on high-end tract homes in the Black Hawk area in Danville California. The set he was working on was elliptical. I was more interested in how he laid out the ellipse than his method of building the freestanding stairs.  

The third time I saw a set of set of freestanding stairs being built was about 5 years ago by Tim Uhler in an article he wrote for JCL magazine based on Will Holladay's method. Good article to get you going on building a set of freestanding stairs. Tim's stairs had a freestanding stair stringer on both sides of the stairs and he used   1/4" CDX or 1/4" ACX plywood for the laminated stair stringers.  He said he had about 1/4" deflection with someone standing on the stairs. 1/4" deflection is acceptable, but I was concerned about the deflection I would have on the set of stairs I was building with 18 stair treads.  

Billy Dillon told me to used George R. diCristina's dropped riser method on page 81 in his book  "A Simplified Guide to Custom Stairbuilding and Tangent Handrailing". The dropped riser method acts as a cantilevered joist system and supports the inside stringer to minimize deflection in the stairs.  Our dropped riser is 15 1/2" tall. George diCristina gives an example in his book using a 1" thick outside stringer attached to the wall and a 1 1/2" inside freestanding stringer supported by the dropped risers. I thought the 1 1/2" inside stair stringer was too narrow. However, when we were building our set of stairs my son Erik said we didn't need to add anymore 1/4" layers of oak plywood to the stair stringer after installing 6 layers, but he said the stairs was already extremely stiff with the 6 layers of oak plywood at 1 1/2"  thick. So, George diCristina is probably correct about only needing a 1 1/2" thick inside stair string using the dropped riser method. 

I used my StairTangent app to layout the circular stairs on the subfloor. The app makes it pretty easy and quick to layout your stair tread with the correct 10" width at the 12" line of travel. 

For the outside stringer against the wall, I used a laser to mark off plumb lines on the stair stringer, from the lines you snapped on the subfloor for the stair tread location. Then from the top of the subfloor on the second-floor mark off the stair riser height at each of these plumb lines.  These marks are the top of the stair string attached to the wall.  Use 0.131 by 2 1/2' gun nails to attach the CDX 5 ply to the wall. About 4 nails per stud. On the last layer, 3 total, use 0.131 x 3 1/4" gun nails to attach the last layer of plywood to the wall  Make sure you get lots of glue, TiteBond III,  on each layer your installing, too much glue is the correct amount. 

The StairTangent app also gives you the stair housing angle that you will use to cut the head and foot cuts of the laminated stair stringer. 

I didn't take any pictures of this next step of laying out the stair stringer riser and treads on the stringer, but here's how  I laid it out. First, lay out the riser height at the top of the stairs. My riser height was 7 1/2" and I'm using 1 1/8" treads, so I measured down 8 5/8" from the top of the second floor subfloor to the top of the first riser. I cut a stud using the dimension from the bottom of first floor subfloor to this line I just made. Then I cut the rest of the riser studs incrementing the dimension by the riser height of 7 1/2". Cut the studs all the way down to the last riser stud of 6 3/8". Use your riser height stud to mark off the riser height on the stair stringer attached to the wall. Place the riser stud next to the stringer and use a 16" level to mark off the bottom of the treads at the stringer. Cut the stringer with a SkilSaw and finish off the cut with a multi-Tool. 

For the first tread and riser. Attach a dropped riser against the stair joist head out beam.  Then attach the next dropped riser, 15 1/2" tall, on our stairs. We glued behind the location of the dropped riser using LockTite PL subfloor glue, then Erik attached it to the wall with about 3 gun nails. Then he added about 4  -- 3 1/8" screws to the side of the dropped riser. Then use your riser height stud for the other end of the dropped riser. Nail- screw it together after you check it for level and nail the bottom of the riser stud to the subfloor on your arc line. Which should be 3 1/2", 14 layers of 1/4", back from your original stair radius line. Repeat until your down to the last couple of stair treads and determine if you need to cut the dropped riser down.  

For the stair tread, we laid a piece of plywood on the snapped arc lines on the subfloor and scribed the pattern for the tread. Both ends of the tread have a radius cut. We also used a laser on the arc line at each tread to measure the length of each riser, theoretically, they should all be the same length. We also used the laser to check the location of the end of the tread-riser location and adjusted the two as necessary. 

Next, mark off level lines from the bottom of the outside stringer to the riser height studs for the location of the bottom of your inside laminated stringer. 

Glue the back side of the first piece of 1/4" oak and use staples, 1 1/2", to attach it to the stairs. 

We used a flush cut trim router bit on the first piece 1/4" oak plywood we installed.  Then decided we could wait and flush cut the plywood when we had about six layers on. That didn't work that well. Four layers were also too much. 3 layers of the 1/4" oak plywood seemed to be the best way to flush cut the plywood.

Use a roller and put the glue on as much as you can get on the plywood, but only about 30 seconds before you staple the next piece of plywood on.

Because this stringer was going through the window we stapled a piece of 1/4" oak plywood to the outside stringer at the top edge only and clamped the bottom edge until the glue dried. There's only a 1 1/2" gap, between the stair stringer and glass in the window.

We installed 3x4 blocks at the top of the riser on the inside stringer to screw the stair tread too and to screw the 1/4" laminated stringers too. On the outside stringer, we install 3x4's with 3 -- 6" SDS screws attached to the outside wall. Two 3 1/2" SDS screws through the dropped riser into the 3x4 block and three screws above the tread through the riser into the 3x4 block. 

Even though we stapled the 1/4" oak plywood as we installed each layer, 14 in total, we installed 4 3/4" screws in the last layers into the riser height studs that we installed. We also install screws about 3" O.C. into the bottom of the stringer and some into the 3x4 block at the top of each riser.

I had Erik stand on the middle of the stairs and there was zero deflection. This set of stairs feels stiffer than a lot of the straight set of stairs I've built.