For those who might have seen the mesmerizing Kawai Tsugite joint (which fits perfectly in three different ways) pop up somewhere on the internet every now and again, you were likely one of many who were looking for a deeper explanation.
Thankfully, engineer and woodworking master Matthias Wandel (yes, the same Matthias who built the Never-Ending Escalator of Slinky Steps) has taken to his personal blog to breakdown this complex joint – starting by rebuilding it from scratch.
“The three-way aspect of this joint is based on the rotational symmetry of a cube about an axis going through opposite corners,” explains Wandel. “If I spin this cube between my fingers by 120 degrees (1/3 of a turn), it will occupy exactly the same space as before. Now, if I cut the cube in half, through a plane perpendicular to that axis, I can rotate one side of the cube by 120 degrees and the two halves will still form a cube. This is the basis of this joint.”
Although Wandel is quick to note that this isn’t necessarily the most functional joint for most woodworking purposes (because it doesn’t take advantage of the wood grain and compromises strength), it is however an interesting exercise in geometry.
For anybody who may want to create their own Kawai Tsugite joint, Wandel has generously created and shared a SketchUp file – however he recommends that the best way to make this joint is to “mount the wood at a 35.26 degree angle on a CNC machine or pantorouter”.
You can see Wandel’s full process of creating the joint from scracth over on his blog.