Some of you might recall how back in January, I’d published a 3D model for a workbench I’d been considering building. Its benchtop featured a highly flexible clamping system comprised of Kreg Klamp Traks. The basic idea was to have two orthogonal tracks, one running length-wise, and the other across the width. Each track would divide its opposing dimension into golden sections, as a means of introducing a degree of skew that might also be useful from the standpoint of construction geometry:

A key point of my design, however, was that width and length would not follow golden proportions, because I wanted a bench somewhere along the dimensions of 30″ x 6′ or 7′, so as to easily accommodate large panels and frames. I ended up settling on 33^{3/4}” for its width, the length of a single Klamp Trak extrusion (to eliminate one cut, basically), and a length of 7′.

But recently, some one asked me: “What would your workbench look like if its benchtop were completely conformed to the golden ratio?” Well, I wasn’t about to leave that one alone. So I modified my original design, leaving the benchtop width 33^{3/4}“, but shortening its length to 54^{39/64}” ( 33^{3/4}” x 1.618 ≈ 54^{39/64}), and re-dimensioning everything else, accordingly. And here’s how it looked:

[ Note: You can freely download this revised golden ratio workbench SketchUp model from the Trimble 3D Warehouse, by clicking here. ]

The revised bench is about two and a half feet shorter in length than the original design, and a bit more stout looking, as a result. But still comparable to what I’d had before, certainly stronger, and completely adequate for all but the longest panels or frames.

Then, I compared this new design against Kreg’s own Klamp Table product, whose length is 33^{3/4}” (likewise based on the length of a single, extruded Klamp Trak), with a width of 21^{3/4}“, and whose tracks run along adjacent sides of the benchtop, rather than intersecting somewhere inside its perimeter:

Interestingly enough, the length-to-width ratio of the Klamp Table is 1.552, which is very close to the golden ratio (1.618), so Kreg Tool clearly put some good thought into the design of this product (as they do all their products, quite frankly). But I’m biased, of course, and still prefer my own design: I have about two a half times the square area, and, in my opinion, my intersecting tracks support a wide variety of clamping configurations, including the simultaneous clamping of multiple work pieces.

I must admit, however, that the Kreg Media Center has a rather impressive video of the Kreg Table being used to build a very large face frame, using pocket screw joinery. So perhaps my concerns about benchtop size are somewhat unfounded (at least for frame construction, anyway).

Now, here’s another interesting ramification of my redesigned benchtop (you might conclude I’ve finally lost it upon first read, but please bear with me here…). Many of you are probably familiar with diagrams of *golden spirals*. Here’s someone who clearly is:

The golden spiral is an infinitely recursive, logarithmic curve (or, logarithmic spiral), based on the golden ratio. Even though the golden ratio is a transcendental number (like *pi*, or *e*), there are a number of practical ways of computing (actually, approximating) golden spirals (and hence, golden proportions), using rational numbers. One method is to use Fibonacci numbers. Yet another is to fit together a collection of quarter circles that are tangential to a series of golden ratio proportioned squares. This is the green curve shown below (the red curve is the actual golden spiral):

Now, what happens if I take my benchtop layout in SketchUp, similarly enhance it with a series of squares based on successive applications of the ratio 1/φ^{n }(but only up to a point, of course, and beginning with the first three squares defined by my Kreg Klamp Trak positions), and then draw connected, tangential circles within those squares? Yup. You guessed it — a very precise approximation of a golden spiral, on the surface of my benchtop:

[ Note: You can freely download this golden spiral approximation SketchUp model from the Trimble 3D Warehouse, by clicking here. ]

Now, you might ask, rightfully so: Why do this? And, who cares? Well, beyond the profound aesthetics of it all, there’s also a practical consideration at hand here. Let’s assume that I actually were to build this bench, etch the golden spiral into its surface (say, about 1/32″ wide and 1/16″ deep), and provide measurements along the edges of the squares (for example, using a ruled adhesive tape). I now have the basis for mechanically calculating golden ratio proportions, right there on my benchtop.

I can easily envision creating a simple jig for transferring golden proportioned lengths to a work piece, leveraging an articulated arm, with a trammel point at one end, that simply follows the etched spiral. No calculating necessary — just have the fixed dimension of your work piece cut to the length you want, and the mechanical calculator gives you the length of the opposing dimension (well, maybe…).

Anyway, that’s the idea. Whether I’ll actually attempt this, or not, is a completely different story. Funny how a simple question can inspire such a lengthy progression of ideas, isn’t it?

## Some related points

The mechanical calculation of φ-proportioned dimensions is certainly not a new idea. Check out these Phi (Golden Ratio) Rules, sold by Lee Valley (they’re rather quite nice):

At this time, I’m still pretty much committed to my original design; that is, the lengthier 33^{3/4}” x 7′-0″ benchtop. However, it occurred to me that I could easily build two of the smaller workbenches (what I’m now calling “conformed” golden ratio benches) and place them end-to-end if I need a longer surface:

[ Note: You can freely download this double workbench SketchUp model from the Trimble 3D Warehouse, by clicking here. ]

Since I wrote about my initial design back in January, Kreg Tool introduced a new side clamping option, called the Klamp Vise, which seems pretty cool. The only thing I don’t care for is that it’s mounted on a small Klamp Vise Plate, which would need to be re-positioned if you wanted the Klamp Vise somewhere else on your workbench. However, I believe that were I to fasten Klamp Trak sections along the perimeter of my workbench, I could then have a completely movable side clamping option. Something certainly worth at least trying out.

I’m also still committed to the use of inset vises, as well. Unlike the side clamps, inset vises can also help with spreading (that is, using a vise to assist in taking an assembly apart).

And finally, it occurred to me today that, were I to actually install Klamp Trak sections along the perimeters of my workbenches in support of side clamping, I could also use those sections to temporarily fasten two workbenches end-to-end, by inserting, on each side, the small hex machine bolt and flat nut that comes with each Klamp Trak section. And the resulting separation between the adjacent bench ends needs to be there, anyway, to facilitate the insertion of clamps into the middle tracks. (See the diagram above.) Whew!

That’s it for now…Happy Fourth of July, and happy spiraling to everyone!

I love how your mind works on so many levels. Even when I don’t understand the technical bits, I still get the overall point. And the golden spirals definitely have an aesthetic appeal, although I’d never want one as a tattoo. But then, I don’t want any tattoos.

Thanks Alexandra!

I’m glad you love how my mind works. Although it does have a way of getting me in trouble, too (well, sometimes!). And yes, I’m no fan of tats, but I thought it would be refreshing to include a photo of a human being in one of my more technical articles, and that photo seemed spot-on to the topic! Now, if only I could figure out what a Higgs Boson tattoo might look like…

~John

I’ll just say it — reading the math bits kinda makes my head hurt . . . But … it does seem noble (as in the golden ratio) and I do really like the idea of clamp tracks . . . plus . . . digging Kreg more and more as I see their stuff … and . . . I wish I could teach myself enough Sketchup to do table designs like this… one day.

interesting stuff my friend.

Hey jb!

Yeah, I would be nowhere without either Kreg’s pocket screw and clamping tools, or SketchUp. I’ve come to regard both as truly indispensable tools. Learning to use either requires changing your thinking a bit, but once you get the feel for it, and your skill builds up, you wonder how you got by without it!

Thanks for the comment!

~ John

Golden Geometry

It’s the best kind there is! (Unless, of course, you’re really into the non-Euclidian space-time sorta stuff )

Thanks, Ruby!

~John