The Cartesian node

"A lot of people are afraid of heights. Not me, I'm afraid of widths."  Stephen Wright

A Cartesian node . . . sounds like some sort of obscure part in the human anatomy, doesn't it? Or maybe some unsavory thing hanging off of said obscure body part that would require the assistance of a surgeon for safe removal? Actually, it's none of that.

Named for the 17th century French mathematician, Rene Descartes  - or, in actuality, for his Latinized name, Cartesius - a Cartesian node can be explained in proper mathematical terminology. But, dear reader, you're going to have to find someone other than me to do that for you. Much like Stephen Wright's aversion to widths, I get more than a bit flustered when letters like X and Y are part of equations that also contain numbers (this was all explained in my one of my junior high geometry classes, but I think I was home sick that day and it never did quite sink in.) So, in the interest of not to making an even bigger fool of myself by trying to communicate in a language I can't fluently speak, I'll simply give you some pretty pictures to look at and elaborate in plainer English from this point forward . . .

OK, that's much better. The 3 dimensional version of a Cartesian node (drawing, left) can be plainly described as a vertical post with couple of overlapping horizontal posts butted up against it (the word "adjacent" would be the more refined one to use here, but there's just something a bit more visceral and earthy about the phrase "butted up against.") One could also take that vertical post, by the way, and just as easily place it behind the two overlapping horizontal ones to get the same effect. Likewise, you can also reverse the order of the two overlapping horizontal posts and still have a Cartesian node.

Examine just about any piece of Gerrit Rietveld furniture from the De Stijl period and you'll see lots of Cartesian nodes. The vertical post ends up becoming one of the chair legs, for instance, and the overlapping horizontal posts  - variously named laths or cross braces - bring the four legs together. The beauty of this thing is that you can come up with a fairly strong wood joint by simply gluing your piece together this way. To get real and usable strength out of it, however, you'll need to also join together the adjacent pieces with dowels while gluing.

Take a look at the second drawing (left) for additional clarity. Well, actually, that drawing is still more grounded in mathematical theory than it is in carpentry reality. Sorry. The carpentry parlance for the technique its attempting to convey is blind doweling: dowels that are present on the inside bridging the two pieces, but which can't be seen after assembly. Unfortunately, the drawing gives the impression that it's possible to blind dowel all three. Not so. In reality, you can blind dowel the two horizontal laths protruding into the vertical leg. But, once that's done, the third dowel vertically tying the two horizontal laths together has to be drilled in such a way as to leave at least one end of that last dowel exposed. No big deal, really, if the resulting work is done cleanly. Rietveld, in fact, wasn't afraid to use rather prosaic looking carriage bolts or machine screws in place of a third dowel in these "gotta drill it all the way through" scenarios.

Back to reality . . . the photo (left) shows a closeup perspective of the rear leg joinery on a 1924 Rietveld "Military" chair, which is very similar to the chair I'm building. In this variation of the basic Cartesian node, there a two smaller laths present to tie the two rear legs together. Two more laths do the same for the front legs. These smaller laths (four in total) also capture the heavier horizontal side braces, which are there to bridge the front and rear leg assemblies. Look into the background of the photo and observe, on this particular Rietveld design, that the front leg is mounted on the outside of the horizontal brace instead of on the inside. Finally, the top laths - both front and rear - also provide a resting platform for the seat plank. This whole business of overlapping wood post Cartesian nodes really defined Rietveld's efficient furniture design philosophy and played into his larger goal of providing labor-saving carpentry techniques that could be accomplished largely with the use of (then) new and inexpensive machine tools.

I'll leave you this week with another unusual word to kick around: dado (pronounced DAY-doh,) which is yet another carpentry term. Imagine, for a moment, that the chair legs in the above photo have slots cut into them - maybe about a quarter or a third depth - that is just wide enough to allow those perpendicularly placed horizontal side braces to snugly rest inside. That's a dado. It considerably strengthens the already very strong doweled/bolted Cartesian node joint . . . and we'll be using this dado technique for our chair building project.

This sort of dado cut can be obtained with the aid of either a table saw or with a router. For reasons that I'll explain later, I'll be using a router on this particular project. Obviously, the width of the slot is critical. No need to be fearful of widths, mind you, but you'll surely want to pay attention as to what you're doing to make them as clean and accurate as possible. Accurate widths - and depths - will involve carefully setting up the router with something called a template collar, or guide collar, which has to be precisely centered around the cutting bit. A template - typically a flat piece of wood, metal or plastic, with a precisely dimensioned rectangular hole cut through it - then needs to be fashioned and securely clamped onto the piece of wood needing to be cut. If all goes well, you get a perfect dado. As to the exact details of how it's all done, with some nice pictures to go with it? Well, that's the subject of another post- or another couple of posts - for another time.