This excerpt from our latest book, “From Truths to Tools,” speaks to a rather esoteric, but highly useful, rule for use with scaled drawings:
Here’s a typical, traditionally drawn small boat plan:
To find the dimension of any particular part of the boat, we simply set the divider to the part – here the cap on the centerboard trunk:
Then we transfer the dimension (what the Greeks called, more precisely, a “magnitude”) to the graphic rule…
…and read the numerical distance of 3′, 6″. This technique eliminates the need for an awkward and hard-to-read scaled ruler and, furthermore, works no matter what scale the drawing is made to.
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 is an excerpt from “From Truths to Tools” by George Walker and Jim Tolpin; Illustrated by Andrea Love.
Just out of curiosity, let’s see what happens when we draw a circle, then switch the dividers’ legs around. Being sure to keep the same setting (i.e. the radius of the first circle), we set the point anywhere on the rim and swing the other leg around to construct a second circle.
We now have before us two circles of the same size, which yields the birth of “symmetria” (symmetry) – one of the most useful and foundational principles in geometry (not to mention keeping the universe itself intact).
The intersection of the symmetrical circles at each other’s focal points is the geometric truth underlying a powerful layout tool called a spiling batten. To see how this wool works, follow the steps in the drawing.
1.) Swing an arc (about one-third of a circle) from a focal point.
2.) Keeping the same radius. swing back a little arc from any place on the first arc.
3.) Swing back another arc from a second point on the first arc. The intersection of these two small arcs is the location of the original focal point.
Be aware that you need to be careful to maintain the same setting for all these arcs.
A common application of spiling in boatbuilding is in the fitting of a boat plank perfectly between two other previously installed planks. We begin by tacking in place a thin piece of wood (the spiling batten) in the opening between the planks. Next, from station points we’ve made on the upper and lower planks (usually at the centerline of frame locations) we swing an arc onto the batten.
To avoid errors due to a change in the divider setting, we will record the divider span somewhere on the the batten to provide a double-check.
When we are done making arcs from all the station points, we remove the batten and lay it on the stock to be cut to shape. Then we swing two arcs from each arc drawn on the batten.
The intersection of these arcs will be the location of the original station point. Finally, we’ll use a bendable length of wood to connect the transferred station points onto the stock. Cut to the line and we are rewarded with a ready-to-plane-to-perfect fit.
In the spirit of the holidays, let’s perform some simple, ancient geometry to create the iconic symbols of the two religions celebrating major holidays this month. You’ll need only a compass, a straightedge, a piece of paper and a couple of candles to illuminate your work. In chronological order (in more ways than one) let’s start with Judaism’s Star of David:
Begin with a circle and mark the focal point. We have actually started with the symbol for Ra, the ancient Egyptian sun god for whom winter solstice was celebrated for thousands of years prior to Judaism – but that may or may not be another story.
Now draw a line vertically through the focal point (i.e. a diameter) and mark its intersection points at the rim.
Next set the compass to span from one of the rim intersection points to the focal point and swing an arc through the rim as shown. Mark the arc’s intersection points.
Repeat from the other rim intersection and mark two more rim points.
Connect all the rim points across the circle.
Erase the circle rim, diameter line and interior arcs and you are left with the Star of David.
Now let’s create the Christian cross – also from the intersection of line and circle:
Again we’ll start with a circle (which came to represent the heavens), but this time we’ll draw the diameter line at about a 45° angle.
Construct another diameter line at a right angle to the first. Use the intersecting arcs method (or just fudge it, I won’t tell).
Connect the rim intersection points to create a square (which traditionally represents the four directions, the four seasons and the earth itself).
Now bisect the lower horizontal line and extend the bisection line from the focal point down past the lower rim of the circle.
We’ll set our compass to the span between the rim intersection point and focal point, and swing a second circle. (A second of a pair of circles traditionally represented the Dyad … the reflection, the knowing of the first circle called the Monad (all one/alone).)
When we erase most of the lines we are left with a cross … a symbol of the melding of heaven with earth. Or for the math geeks: a pairing of a diameter line (2) with the non-terminating (i.e. irrational) square root of two.
Note: This geometric construction of the cross is not historical but rather the product of my imagination.
I am pleased to announce that Mary May and George Walker will be at the Lost Art Press storefront on Dec. 9 to celebrate the release of their new books.
Mary, the author of “Carving the Acanthus Leaf,” and George, one of the authors of “From Truths to Tools,” will each give a short presentation on their work that evening, answer your questions and sign books. Lost Art Press will provide drinks and snacks for this free event.
Only a limited number of people can attend (fire marshal’s orders), so we will offer free tickets to this event starting at Friday at noon Eastern time.
Note that Saturday, Dec. 9, is also the last open day for 2017. So if you need books signed by me (note: I am happy to fake any signature, including: Tommy Mac, Roy Underhill and André Roubo) that’s the day to do it.
