Daisy Wheel Compass Geometry or Euclidean six point geometry

In the 10th century Persian mathematician Abul Wafa al-Buzjani at Baghdad wrote the treatise "On Those Parts of Geometry Needed by Craftsmen". Abul Wafa al-Buzjani writes about six point geometry and other types of geometry needed by craftsmen. You can see some of this geometry each time you step on your Persian rug and count the 3, 4, 5, 6, 7, 8, 10, 12, 16 or 24 point geometric shapes in the rug. Islamic geometry shows the influence that Euclid, Vitruvius, Archimedes had on Islamic mathematics after the late 8th century translation of Euclid's Elements into Arabic in Baghdad. Archimedes and Apollonius of Perga books on Conics, who named the conic section, ellipse, parabolas, hyperbolas , were also translated from Greek to Arabic and may be the reason for the medieval builders using conic and spherical geometry for their vaults instead of the Romanesque barrel vaults or groin vaults. Were medieval 
masons and carpenters, that used the daisy wheel geometry, influenced by the books from Baghdad?   Is the Elucidation of the 6 point geometry self evident ? 


Rotating the Euclidean six point  geometric drawing will result in a daisy wheel compass.
 It could have been used by carpenters to design hexagon or octagonal timber frames.

Trigon 3 sides Equilateral

Hexagon with 6 sides.

Square 4 sides.

Octagon 8 sides.

Dodecagon 12 sides.

Icosikaitetragon 24 sides

 

Polygon Crown Molding Miter Angles and Bevel Angles

We're in the process of remodeling our kitchen and the Rafter Tools app came in handy for calculating the miter angle and saw blade bevel angles for the polygon crown molding cuts.

My wife wanted a lite above the new gas range, so I used 5 sides of a hexdecagon polygon (16 sides) to bring the front of the new cabinet out to allow for the new Halo Four Inch LED Down light.

Here's a drawing of the plan view of the hexadecagon cabinet top. As you can see from the drawing there were 7 different corner angles I had to calculate the miter angle and bevel cut angles for the crown molding. 

The finished product was/is suppose to look like this 3D drawing of the 3 1/4 colonial crown molding. (Just got to love the follow me tool in Google SketchUp for the crown molding drawing.)

Here are the steps to use the Rafter Tools calculator to find the crown molding miter angles and saw blade bevel angles.

Step 1:

Scroll left in the Rafter Tools calculator to the Crown Molding Angles calculator. Then click on the drawing to open the Crown Molding Calculator.

Step 2: Scroll down to the bottom of the page. 

Step 3: Click on the Calculate Spring Angle button.

Step 4: Enter the rise and run of the crown molding. The 3 1/4 colonial crown molding I was using had a Crown Molding Run of 1 15/16" and a Crown Molding Rise of 2 5/8" . You will have to convert the inches to decimals. After you enter the crown molding run and rise in decimal inches press the Calculate Crown Spring Angle button. The calculator will then display the crown molding spring angle. Press the green line to have the crown molding spring angle transferred back to the Calculate Crown Molding Angles calculator.

Step 5: Enter the first corner angle. 135.00° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 135.00°.

 Step 6: Enter the first corner angle. 132.00° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 132.00°.

 Step 7: Enter the first corner angle. 154.00° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 154.00°.

 Step 8: Enter the first corner angle. 157.50° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 157.50°.

Step 9: Enter the first corner angle. 151.90° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 151.90°.

Step 10: Enter the first corner angle. 129.20° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 129.20°.

Step 11: Enter the first corner angle. 134.40° for the first corner.Then press the Calculate Crown Molding Angles button to display the crown molding miter and saw blade bevel angles for the spring angle of 36.43086° with a corner angle of 134.40°.

At any time you can click on the Email button to send yourself an email of the calculated results for the crown molding angles.

Hip Rake Walls Rotated Into The Roof Surface

+Tim Uhler framed a house with a hip rake wall that was rotated into the roof surface.  Wildcat This hip rake wall can be calculated using Rafter Tools, Plan A Calculator. To use this Rake Wall Calculator scroll left in Rafter Tools until the Rake Wall Angles calculator is displayed.

Next, click on the image to display the different  types of Rake Wall calculators.

Then scroll down to the Hip Rake Wall Plan A calculator.

Click on the Hip Rake Wall Plan A image to display the calculator and enter 120 for the Horizontal Wall Angle and enter 45.00 for the Roof Slope Angle.Then press the Calculate Plan A Rake Plate Angles button.

Because of the way the hip rake wall was framed, Tim only needed the Hip Rake Plate Stud Miter Angle -- 26.56505° and the Hip Rake Plate Stud Saw Blade Bevel Angle -- 37.76124° to cut the rake wall studs. He could have used the Hip Rake Wall Angle to cut the plumb cuts on the barge rafter. 

Note: the internal valley rafter framing for the sheet rock ceiling is not in my calculator. This is part of Tim's Funk City framing, or better known as a head scratcher to the rest of us.

Here's a of drawings showing the location of the hip rake wall.

Here's a drawing showing the hip rafter plumb backing angle that is used for the Hip Rake Plate Stud Miter Angle .

Saw Miter Angle = 26.5650

Saw Blade Bevel Angle = 37.76123

Bevel Angle (Compound Angle Developed) = 40.8934

Rake Stud Saw Blade Bevel Angle = arctan(cos(Hip Rafter Plumb Backing Angle)* tan(Hip Rafter Pitch Angle))

Rake Stud Saw Blade Bevel Angle = arctan(cos(26.56505°)* tan(40.89339°))= 37.76123°

Here's a drawing showing how the hip rafter plumb backing angle is developed geometrically.

You could scribe and cut the hip rake wall studs like all of us have done before, but the scribe mark you make for the bevel cut is the hip rafter angle, not the saw blade bevel angle that produces the compound angle of the hip rafter angle.

Segmental Arches - Gothic Arches - Lofting Ellipses - Groin Vaults

I'm adding Segmental Arches, Gothic Arches, Ellipse Lofting points for equal height ordinates and Groin Vaults to Rafter Tools. Here's a couple of the screen shot/drawings for these calculators.

Segmental Arch Calculator. Enter the width of the opening and the rise of the opening.

This arch calculator is Erik proof!

The Segmental Arch Calculator returns the radius of the arch, the arch apothem, arch segment height (arch rise), arch segment length (length of arch), arch segment angle and the arch mid point to layout the arch geometrically. 

Gothic Arch Calculator. Enter the width of the opening and the rise of the opening.

This Gothic arch calculator is Erik proof!

The Gothic Arch Calculator returns the radius of the arch, the arch apothem, arch segment height (arch rise), arch segment length (length of arch), arch segment angle and the arch mid point to layout the arch geometrically. It also returns the type of the Gothic arch. 

          Lancet Gothic Arch. A two-centered arch with the radius centers located outside the opening and on the spring line.

            Drop Gothic Arch. A two-centered arch with the radius centers located inside the opening and on the spring line.

            Equilateral Gothic Arch. An arch whose opening and radius of the two halves form an equilateral triangle.

I'm still working on the ellipse lofting and groin vault calculators. Their based on these drawings. Still need to makes these calculators Erik proof!