In this sketch I did of a Masonic “Past-Master’s Jewels” medal, notice the representation of the pythagorean theorem. It is reported that its presence on the owner’s medal indicates that person was what we would likely now call a crew foreman. One of his many responsibilities was to ensure that all the layout tools were true – a clue as to why there’s the homage to Pythagoras. This theorem, codified later by Euclid into his “Proposition 47,” offers a logic proof that the area of the squares erected on the legs of a right triangle would equal the area of the square erected on its hypotenuse. That’s all well and good, but why would that particular equation be of vital interest to the foreman of a joiner’s or mason’s crew? To try to find out, I decided to construct an exact-as-possible, large-scale drawing of the graphic upon which I could explore with a pair of dividers.
The first thing I discovered was that the vertical line CL, which is fixed by the inherent baseline’s intersection points C and D, forms a right angle with the hypotenuse. Even though this result is likely nothing more than symbolic (there are a lot easier ways to generate a right angle with a compass and a straightedge), I believe this right angle – hidden in plain sight – is probably as important to the medal (and its wearer) as the theorem itself. The right angle (“recto” in Greek) is simply the right way to set a vertical post. (Wood’s superb resistance to compression happens when, and only when, the post is set at a right angle to level – an orientation that aligns the grain parallel to the force of gravity). It’s also the right angle to create symmetry to a baseline in common rectilinear structures (think cathedrals).
No reason to stop there, though. Exploring further revealed other attributes of this graphic that offer additional symbolic (and real) representations of the truths inherent in Geometry (note the traditional capital G). Print out the template (you’ll find it offered for free on the shopping page of www.byhandandeye.com) and take a look around on it for yourself. You’ll discover triangles with perfect 2:3 base-to-height proportions (one of the fundamental harmonics in music and architecture of the Medieval era); you’ll find sequences of the infamous triplet (the 3-4-5 triangle) revealed in the hypotenuse and even in the circumference of the circle that started it all; and you may find the module upon which the entire construction revolves. Have fun with this – I sure did!
To develop the curves in the various brackets – here the support for the back fence on the lid of a desk – I followed the ancient practice of melding arcs of a circle along a straight line.
I begin by making a few concept sketches to get an intuitive feel for the curve I would like to see transition the horizontal lid surface to the vertical back fence. I’m going to go with the shape in the first drawing.
From the sketch, it reveals that the overall form suits that of a 1:2 rectangle. (An octave, by the way – but that’s another story). Next, I divide the horizontal length into four equal segments. The first of these segments defines the flat at the top of the curve. I then draw a baseline for the sine curve from this segment point to the lower right hand corner, then divide that baseline into three equal segments.
To find the focal point of the arcs – which will each be one-sixth of a circle’s circumference – I set the dividers to the length of the segment (which is the chord of the arc) and swing out intersections to locate the focal point of the arc. Next, without changing the span of the dividers (because the chord equals the radius for sixth sector arcs as you may remember from Mr. Hammersmacker’s seventh grade geometry class), I swing the arc from the focal point to each segment point. The transition between the two arcs is seamless – proven to be so because a line connecting the two focal points will pass through the arc’s transition point.
While I mostly use the sector for doing design and layout work in my shop, I realized recently that it’s also a great tool for showing someone (especially your kids) an intuitive approach to understanding fractions. Here’s how I’d describe what’s going on in the drawing above:
Because I want to find out where a point four-sevenths of the width of a board would come to, I set the legs of the sector to touch each edge of the board to denominate (i.e. to name) the kind of divisions I’m looking for. Here, that would be seven – the denominator. Now I want to enumerate (i.e. give a number) to how many of those sevens I’m looking for – in this case the numerator is four. The job of the dividers is to grab this numerator above the denominator value on the legs of the sector in order to transfer the setting to the face of the board. For me (and my kid), this drawing offers a decent visualization of why the numerator goes over the denominator. You can learn more about the sector in excruciating detail in “By Hand and Eye;” and in a somewhat less excruciating matter in “By Hound and Eye.”