In Heckballs: a laser-cuttable MDF set of building blocks I wrote a bit about Heckballs (Jecvals), although that dates to before I had done a bunch of the work I actually did on them.
Since then I've messed around quite a bit more with sheet cutting design (see for example Cardboard furniture) and come up with some other ideas.
One of the biggest improvements I haven't yet tested solves the problem of Heckballs falling apart once assembled due to the imprecision of the press fit. I saw this solution in a discarded liquor-box partition cardboard on the sidewalk one day. By extending the insertion bevel or chamfer or divot on one side of the mouth of a slit further down the slit, between one third and halfway down, it becomes possible to flex the inserted sheet significantly to one side after it's already inserted in the slit; the non-mouth end of the bevel becomes a fulcrum for this bending. Then, we add a roughly triangular projection to the opposite side of the slit, at the mouth; one side of it forms an insertion bevel at around 45°, while the other side is at right angles to the slit, forming a retaining clip. A corresponding hole is added to the sheet for it to insert into.
I've done this with cardboard and a scalpel, and in that medium it works beautifully, but MDF is quite a bit more rigid; will it work in Jecvals?
To be concrete about the dimensions, the octagon width in Heckballs is 100 mm, so the shallow slits are 25 mm. (There are also deep slits for joining two octagons into a ball.) I was cutting in 3 mm MDF with 2-mm chamfers (i.e., the chamfer forms a 2mm, 2mm, 21½mm right triangle) and 3.03 mm slit width. Extending the chamfer to 12.5 mm down the slit and 3 mm of extra slit width gives 12.5 mm of bendable span. We'd like to use a tab that presses on the full thickness of the inside surface of the hole, which would make it 3 mm tall, 3 mm deep, and a 3√2 diagonal if it uses 45°.
But how much can we bend without stress-relief cuts? We only have 12.5 mm - 3 mm = 9.5 mm of bend length to work with! According to Heckballs: a laser-cuttable MDF set of building blocks, MDF has elongation at break of 0.45%; if we use 0.3% to be safe, the inner part of a circular bend can be 0.3% shorter and the outer part 0.3% longer, so the bend radius must be 150 times the material thickness or more: 450 mm. 450 mm - √((450 mm)² - (9.5 mm)²) is, unfortunately, only 0.1 mm, which is so small that it might just slip out. If we extend the chamfer depth to 20 mm, leaving only 5 mm at the bottom of the slit to hold things in place, we might have 19 mm of bend; 450 mm - √((450 mm)² - (19 mm)²) = 0.4 mm, and an 0.4 mm tab is enough for some energy barrier to retention, but at a heavy cost to rigidity, and only about 10% of the crush strength of the material.
Also you'd probably want to chamfer the end of the retention tab triangle to keep it from breaking off, fillet the base one way or another to prevent a stress riser, and angle the retention tab contact surface so that it always makes contact over at least some of its surface despite manufacturing variations.
Three perhaps more viable approaches:
Have a full 3-mm-thickness tab that engages and disengages the hole without any elastic deformation directly, using a rigid-body relative motion, and some other much smaller normally-zero-load latching tab using elastic deformation to prevent that rigid motion. I'm not sure exactly how this would work yet.
Instead of relying on sheet flexion to provide the elastic engagement motion, cut an in-plane flexure to allow the latching tab to move, perhaps like an injection-molded plastic clip. This can easily achieve much thinner bending sections (200 microns is feasible) and thus much smaller bending radii, and tabs can grab from both sides of the hole, cutting the distance of flexion in half, and avoiding accidental disengagement from structure loading. This might require more complex cuts.
Use acrylic or cardboard or something instead of inflexible MDF.
Such clipping approaches would provide Jecvals structures with much-needed tensile strength and dimensional precision.
The Jecvals beams form a lattice of the balls 400 mm center to center; this lattice can contain 400-mm squares and 400-mm equilateral triangles. If we add some holes along the centerlines of the beams, we can make panels with tabs that slot into these holes, thus enabling the construction of things like shelves, screens (biombos), and tables. To keep adjacent panels from trying to fill up the same hole from opposite sides, there should be at least four different hole positions, none of which are at the center point of the beam, so that flipping a panel over will move it into a different hole in the beams.
In the case of the triangles, there will definitely be non-right angles between the panels and the beams, which means that the holes need to be wider than the panel material thicknes.
In some cases the panels can be made of other materials, such as 1.5-mm MDF, acrylic, or colored paper.
It would be desirable for the holes for the panels to permit the engagement of other parts as well, such as beam ends, but I'm not sure how to make that possible.