We’ve had a few folks ask about the “hidden hexagon” mentioned in the text, and we think it’s time to share the answer with everyone. This also means revealing a little bit more about what is going on (and not going on) in this geometric construction.
What is going on is this: Drawing lines from and through certain points seems to magically create a representation of one of the most important, not to mention useful, theorems for artisans in geometry: the Pythagorean Triplet. In the geometry of this particular interaction of a circle with a square, a triangle is formed in the upper half of the circle whose legs go on to generate a pair of squares that, when their areas are added together, equal the area of the lower square — and they do that in what looks to be a simple triplet ratio of leg lengths of three to four to five.
To arrive at the correct root lengths of the upper two squares to make this simple ratio happen, the trick from antiquity is to generate a hexagon inside the circle (by stepping the radius of the circle around its circumference) and to then draw a line from the lower left hand corner of the lower square through the vertex of the closest hexagon facet. Next, you continue the line to intersect the upper portion of the circle. This provides the point to which you then draw the legs of the triangle.
The results are leg length relationships of three segments to four segments to the five segments of the diameter line. We have just revealed the simplest of the countless Pythagorean triplets. But have we really? The answer is: Almost, but not really.
We had our friend Dr. Francis Natali take a look at it, and after a couple pages worth of quadratic equations, the truth was outed: The whole-number relationship just isn’t there – though it is, inexplicably, amazingly close. Another friend, Kit Africa, generated the drawing above via CAD, also revealing an oh-so-close 3-4-5 triplet. The bottom line: This drawing from antiquity is apparently symbolic: It celebrates the interaction of easily generated shapes that allowed artisans to intuitively design and build beautifully proportioned and aligned forms on the principles of simple plane geometry.
“This has been Uncle Sam’s Woodshop of the Air, transcribed in Washington, D.C., and I’m Calvin Cobb wishing that, as you slide down the banister of life — that all the splinters are going in your direction! So long!”
This past week I taught the first Moravian bench class of 2018 at Roy Underhill’s Woodwright’s School in Pittsboro, NC. This is the sixth year I have done the class at Roy’s; we have produced about 120 benches in that time.
Morning of the first day before the mayhem begins.
Over time, the class has changed and become more streamlined as my experience teaching it has increased. Of course, with each new group of students, there’s a different dynamic, even though we are building the same project. This keeps the class from ever becoming routine.
You never know what new layout method is going to turn up.
A first for this class was using a boring machine to hog out most of the waste from the long stretcher mortises. This was a huge improvement over a brace and bit.
Roy supervised the boring work.10 minuets later.
We had a young fella named J.D. Stevenson and his father stop in to observe the class for an afternoon. We promptly put him to work. He made us all look bad – he’s 13 years old.
“The Kid” J.D. Stevenson, paring a mortise.By the third day most everyone has good feel for the rip saw.George showing off.Some of Lake Erie Tool’s handiwork.
Ready for the vise.Katie using the 2 1/2″ auger.
As always, I never seem to get any pictures of the very end of the class. The last day is always a mad dash to get as far as possible!
Top: House of the Vettii, Pompeii, March 2017. Bottom: Saalburg, Germany, June 2017.
This time last year Chris Schwarz and Narayan Nayar were in Naples, Italy. In between consuming vast quantities of pizza they made a visit to Pompeii to study and photograph a fresco depicting a Roman workbench (Daedalus and Queen Pasiphae are also in the picture). Not long after Chris returned from Italy his limited edition letterpress book, “Roman Workbenches” went to press. And in June, Chris and his friends Görge Jonuschat and Bengt Nilsson traveled to the Roman fort at Saalburg to meet with archaeologist Rüdiger Schwarz to study and photograph two extant Roman workbenches.
The transformation of the planned expanded version of “Roman Workbenches” into “Ingenious Mechanicks” started in mid-July. Things, lots of things, started turning up in our research. While putting together a couple blog posts on Latin American workbenches during the Colonial era, this 18th-century workbench from Colombia turned up.
San Jose carpentero, artist unknown, Museo de Arte Religioso, Duitama, Boyaca, Colombia.
One of the mysteries of the Saalburg workbench is the two dovetail-shaped notches found on the long side of the bench. Half a world away, the Colombia bench had a similar notch and was equally perplexing. Was it for riving, securing a piece for tenoning, a place for a jig or other device? A few days later a different notch showed up, this time from Italy.
San Giuseppe nella Botega di Falegname, 1640-1692, Francisco Refini. Fondazione Zeri Photo Archive, Bologna.
A notch on the end of the bench was not so unusual and was normally used for riving or tenoning. This image went into an ‘X-file’ until we had other images or information to help decipher the possible uses of the notch.
Mid-July was blazing hot and humid and as I ran workbench searches (in air-conditioned comfort) a flurry of images were turning up. Anything without a date, artist, title or location went into a ‘Find It’ file. I sent Chris pdfs of benches from Italy, Spain, Germany and other European countries. While trying to verify if one particular painting was Italian or Spanish and its physical location, I just stopped to take a good look at the detail. I was drawn to the toolbox to left of Saint Joseph.
The Dream of Saint Joseph, ca. 1700, Luca Giordano (Neopolitan), completed in Madrid. Indianapolis Museum of Art. Photo by Christopher Schwarz.
Next, because Chris and I have discussed baskets for tools, I took a look at the tool basket…and there it was. Holy Cow! At the end of the bench a wedge was in a notch. I sent this off the Chris. He had it one of the pdfs I sent but now I was sure he had not yet seen this detail.
Was this a wedge in a dovetail- shaped notch? Could it be used as a planing stop? Could the wedge be taken out and the notch used for tenoning or something else?
Chris was on his way out of town but quickly replied. He was stunned. Very soon after he returned from his trip he drove the two hours to Indianapolis to see the painting for himself. In the blog post he wrote about it he said he almost wet his pants. Huh. I am pretty sure, although he was over 500 miles from me, I heard him shriek like a little girl, a 6 foot-3 inch-ish little girl.
One detail, well to one side of a painting, opened the door to further workbench explorations and discoveries.
On the end: notch, wedge in notch.
One of the paintings in my ‘Find It’ file was found on a Spanish site. It turned out to be German, part of a ten-panel work by 16th-century artist Bernhard Strigel of Memmingen and in the collection of the German National Museum in Nürnberg. It has a squared notch on the long side of the bench the painting is dated in the same year as the Löffelholz Codex.
Top: Löffelholz Codex, 1505, Nürnberg. Bottom: Hans Kipferle guild table, 1561, Bolzano. Right: Holy Family, 1505-1506, Bernhard Strigel, Memmingen.
Two other images of workbenches with the straight-side and square notch have long been known to woodworkers: the Löffelholz Codex from Nürnberg and the guild table from Bolzano/Bozen.
Strigel’s painting helps to confirm the presence of the notch on workbenches, at least in this southwestern part of Germany and/or Löffelholz wasn‘t crazy. Additionally, the Hans Kipferle table tells us that a half-century later the side notch was still in use.
Map by Brendan Gaffney.
If we step back to the time of the Roman Limes Germanicus and the Roman road network you can see another dimension to the European workbenches with side notches: where the workbenches are located.
So what happened next? Chris went to the shop to try out the theory of the ‘notches and wedges’ on the Saalburg and Holy Roman (Löffelholz) workbenches.
Saalburg workbench. Dovetail-shaped notch ready to cut.
The side notches with the wedge in place serve as side stops for traverse planing.
Wedges in side notches and Roman iron planning stop in place.
On the Holy Roman workbench notches were cut on the end and one on the side.
The notch cuts in progress and the finished bench.
Several weeks before he finished “Ingenious Mechanicks” Chris invited some woodworker friends to a Benchfest. He challenged them to use and critique the three workbenches, French belly and shaving horse attachment that he built for the book. He took notes and Narayan Nayar took photos. The notches with wedges and the notch as vise (with a small wedge) worked beautifully. It was another example of a seemingly simple workbench feature having multiple uses in the shop.
Since the publication of “Roman Workbenches” and the Saalburg visit a cornucopia of workbench and workholding ideas have surfaced and are packed into the forthcoming “Ingenious Mechanicks.”
If you are still on the fence, undecided or torn about adding “Ingenious Mechanicks” to your library Chris will post a short video later in the week to illustrate some of the features of the workbenches.
DRAWING THE SYMMETRICAL ‘S’ CURVE This example works with multiple curves and shows how to draw a symmetrical design that identifies the midrib of the leaf for Chapter 5. By using a compass and a horizontal line divided into eight segments, a slow, gentle “S” curve (also referred to as a cyma curve) can easily be drawn.
• Draw a horizontal line and divide it into eight equal segments.
• Set your compass to Length A (four segments).
• Position the compass point on Compass Point 1 and strike a mark above the line as shown.
• Position the compass point on Compass Point 2 and strike another mark above the line, creating Compass Point 4 where the lines intersect.
• Place the compass point at Compass Point 4 and draw an arc intersecting the horizontal line at Compass Point 1 and also Compass Point 2.
• Notice that the distance between Compass Point 1 and Compass Point 2 is only three segments, thereby causing the arc to become slower, or more open.
• Position the compass point on Compass Point 2 and strike a mark below the line as shown.
• Position the compass point on Compass Point 3 and strike a mark creating Compass Point 5 where the lines intersect.
• Place the compass point at Compass Point 5 and draw an arc intersecting the horizontal line at Compass Point 2 and also Compass Point 3.
DRAWING A SPIRAL Now let’s draw a more challenging curve – the spiral, scroll or volute. Many acanthus leaves terminate with a spiral shape at the base of the leaf, while other designs have entire leaves curling in the shape of a spiral.
This process shows the technique of drawing a basic spiral using straight lines and a compass.
• Draw a vertical and horizontal line with a small square at the intersection. The larger the square, the more open the spiral; the smaller the square, the tighter the spiral.
• Begin by placing the compass point at the upper right corner of the small square and strike a curve in only the upper left quadrant. The size is your choice – try experimenting with various sizes.
• Moving counterclockwise around the small center square, reposition the compass point to the upper left of the square and strike a curve in the lower left corner, beginning and connecting with the end of previous curve.
• Continue repositioning the compass point counter-clockwise around the small, center square and repeat the steps above, forming a continuous spiraling line.
The illustration above shows an incorrect flow of a spiral, a correct flow using the compass method shown and the completed drawing of the Baroque style leaf based on the curves drawn (Chapter 16). Notice how your eye is invited to follow along the gentle curves of the correctly flowing line, where the incorrect lines are jarring and uncomfortable to view. The lines represent the approximate center of the leaf. Notice the line that joins the two scrolls is drawn freehand and should appear to be flowing directly from the top scroll as shown.
“This has been Uncle Sam’s Woodshop of the Air, transcribed in Washington, D.C., and I’m Calvin Cobb wishing that, as you slide down the banister of life — that all the splinters are going in your direction! So long!”
This is an excerpt from “
