The 3/4/5 Rule is an easy way to determine a 90 degree angle. In a “right triangle”, if the Leg 1 equals 3 units, and Leg 2 equals 4 units, then the hypotenuse will necessarily equal 5 units. In practice you put Leg 1 and Leg 2 at what you think is a right angle and vary the angle between the two ends until the hypotenuse equals 5 units. The “units” can be any uniform measuring distance from inches to varas.
The following values will produce a close approximation of the Tello Barbed Quatrefoil Arch at Mission Espada: W = 72” Rq = 18” Ra = 52.5” Cs = 13.5”
This is the theoretical design of the Tello Arch. In real life it is very difficult to make all the voussoir identical and would require shims, thicker mortar, and finally varying the angle on the keystone to make it all fit.
6) Adding the Barb:
From the center point on the horizontal line draw a diagonal reference line that passes through the intersection of the upper lobe and the right lobe, extending to the outer arch. Do the same for the left lobe. These will be half way (45 degrees) between the horizontal and vertical lines. Measure the section of the circumference between the two lines (Cq) and divide by six (6) (Cs). This number will be the length of the top of each of the upper seven voussoirs. Working along the outer circumference, make two marks ½ Cs on either side of both diagonals. Draw a radius from each of those marks to the center point of the horizontal line. Draw lines parallel to the horizontal and then to the vertical from where those lines intersect the lobes to define the barb. This will define the two “barbed voussoirs”
7) Finishing the Upper Voussoirs
Again working along the outer circumference mark the distance Cs from the edge of the barbed voussoir (not the diagonal line) to the other four times and draw radii from those marks to the center of the horizontal line. This will define the design of the top seven voussoirs.
8) Measure the circumference distance from the edge of the barbed voussoir down on both sides. Put a mark ½ way of that distance and draw radii from the mark to the center point of the horizontal. This will define the shape of the bottom two voussoirs, or springers on either side.
Paraguay Example: Nine voussoirs to a side instead of five as in the Tello Example. Notice that the barb is divided among three voussoirs instead of one.
Design Your Own Barbed Quatrefoil Roman Arch
Thanks to Ed Day for sharing his drafting skills to develop this exercise.
The following exercise could be done using formal geometry, but the challenge is to do the whole thing with a straight edge, divider, a compass (any device that lets you draw a measured circle) and something flexible to measure with (like a piece of rope). The only “math” they would have needed would be the 3/4/5 rule which is a practical application of the Pythagorean Theorem to determine 90 degree angles (see explanation below). Algebra like symbols are used just for simplicity of explanation and are not essential. Fractions of measurements can be determined by "folding the rope" or using a divider. See accompanying diagrams.
Attn: Voussoir is an architectural term for an arch stone. The top voussoir is referred to as the keystone, the most bottom ones as springers.
The only info you need to design this particular arch would be the opening width and the side column width.
1) Determine inside width (W) of the opening as well as width of each column (Wc).
2) Quatrefoil Lobe Radius = Rq = 1/4 W
Outer Arch Radius = Ra = 1/2 W + Wc
3) Draw a horizontal line = 2Ra
At the midpoint draw a vertical line = Ra
4) Drawing the quatrefoil lobes:
Using a compass set to Rq, measuring from the midpoint of the horizontal line make marks on either side on the horizontal line and on the vertical line.
Using those marks as centers. Draw as much of a circle as you can around all three marks using the same radius.
5) Drawing the outer arch:
Using a compass set to Ra. Draw semicircle from the center point on the horizontal line.
Defining the position of the barb