After checking some designs I found online for boxes and other things I wanted to see if I can create a simple joining system myself.
Already did some hinges for another project and I quite like geometric forms that can be explained by math so I decided on a regular Icosahedron for my project.
If you not familiar with it:
The regular Icosahedron is made from 20 identical triangles, joining on their sides at an angle of around 138°.
There are other forms of it, like the great icosahedron and of course even more complicated structures.
But trust me, getting the 20 triangles together is a puzzle on it's own, might be a while till I create something more complex.
If you want to read more about the form itself, check this Wiki-link.

Step 1: The Design

The basics are really simple, just a triangle.
The Inkscape design is in such a way that you can add some flaps to glue a paper model together, which explains the wasteful placement of the triangles.
Like this all you need is fold them and glue them on the open sides of the triangles.
However the joints for the real thing were quite a different story.
Calculating the angle between two triangles in an Icosahedron is one thing, but getting the joints right so you can actually put the structure together took me a few tries.
Unlike a square box there is nothing that holds the triangles in the right spot until the entire structure is complete, which meant my first try of using a slightly different angle and some play for easy mounting ended in a disaster falling apart LOL
Next attempt was to design the joints really tight so everything will stay in place and allow a connection without glue.
You might have guessed already that this did not go too well either...
The last design, that was used for this Ible, was done from scratch:
The triangles are designed so they touch at their inner sides (inside faces only) and the joints are created so there is next to no play sideways once two (or more) triangles are joined correctly.
Like this it is still a challenge, especially the last triangle, but at least it is possible without breaking your fingers and using rubber bands, clamps and more.
Please ensure your cutting height is set so the focus of the laser is at the bottom of the material.
You want to cut so that the material is not totally seperated already, leave just enough material so it is easy to break the triangles out.
Of course if your setup uses a very long focal length this might not be an issue for you anyway.

Step 2: The Cutting / Engraving

I included some very basic vector lines that can be placed on the triangles.
The small circle can also be cut out to aid with the mounting as the holes allow you to use sticks or similar to push some loose joints back in place in the final assembly stages - but it is possible without holes.
A little raster art to cover the holes for the joints is included as well - it is the one you see on the pics.

The design is for 3mm material, if you want to use thicker material or thinner then you only need to create joints from 3mm material as only the inner faces meet.

All parts are included as a seperate Inkscape layer and all cutting / engraving lines are defined as paths.
How to sort the parts:
(Use some cardboard and low power for an anligment test if required)
1. Start with the holes for the joints.
2. Add some of the line art or the engravings - or create your own design.
3. Cut the triangles and joints out.

Tipps for better results:
The design is so that all triangles touch at the corners of the inner faces.
Covering the material with tape will make sure you don't get too much soot or smoke marks on the material.
If you can't adjust your focus point to be at the bottom of the material cut with multiple passes at a lower speed.
You want to keep a possible gap between the triangles as thin as possible.

Step 3: The Puzzling Part....

Ok, time to get the glue out, or the acetone if you use acrylic...

I tried quite a few ways to get this thing together so it will hold the shape and can tell you that good accuracy for the parts makes it much easier.
But the best way is to use a semi-quick drying glue so you can still adjust the parts as you go ahead.
Place one triangle with the inside up and join three more around it.
Add another triangle on one side of each these three triangles.
Now do the same on the other side of the first three triangles you added.
By now you can see how the shape forms and where to put the next triangles.
The tricky bit is the last triangle.
It gives the final shape and stability for the Icosahedron, so placement is quite critical.
(As it can be tricky you might sand the joint bits going into the last triangle a bit to the slide in with ease)
Glue the last three joint in place on the Icosahedron.
Check that all other joints are still in the right position, if in doubt use a chopstick or similar to push the flat against the triangles again - you can also check the holes to be evenly filled by the joints.
Place the last triangle on top of the joints and push the rest of of the Icosahedron to align the joints with the holes of the last triangle.
If you sanded the last joints it should slide right on, if not wait a bit for the glue on the rest of the model to set.

For acrylic:
Make sure with a few test pieces first that your joints form a tight fit.
Follow the procedures as above for the wood but don't use any glue.
Also sand the jast joint where the last triangle sits to allow for a very easy fit.
Once you are satisfied with the alignment push all joints that might be loose back in plase and secure with a drop of acetone into each joint hole after placing the last triangle - be careful and check the alignment every now and then.
Allow for the acetone to evaporate fully and try not to disturb the model.
If required repeat the acetone dropping into the holes and to finnish up use some clear glue (or acetone if tight enough) on the corners of the triangles.

Here is a little collection of images showing how to put the wooden model together: