Most (all?) hobby sized Laser cutters cutters only really have 2 axis of operation X and Y (the Z doesn't move during operation). This makes them perfect for cutting out 2D profiles, it's very much what they're designed for. It's not great however for creating more complex shapes.

Polyhedra are 3D shapes with flat faces, straight edges and sharp corners. The edges where the faces meet align at their own angles. If we cut these faces as standard lasercut 2D profiles there would be big gaps between the faces. We want the edges to be flush and that means laser cutting through the material at an angle.

Over at JustAddSharks.co.uk.co.uk we've been experimenting with various mitre jigs designed to cut materials at exactly the right angles to create these basic polyhedral shapes.

(read the original post here)

Step 1: The Theory, the Solution

Changing the angle of the laser beam is always going to be tricky, it would need to be done after the third mirror and be attached to the flying optics part of the laser. If we can't change the angle of the laser beam then we can always change the angle of the material.

The focal point of the laser is rather important, as the beam comes through the collimating lens it gets concentrated into a single point which makes the beam powerful enough to vaporise the material. The material needs to be placed an exact distance from the lens. With the material at an angle this distance becomes harder to measure.
The bottom of the cut needs to line up exactly with the corner of the material being cut. This ensures that the panel remains the same size once it has been cut and the item can still be assembled.
The laser needs to cut in a straight line along the material, keeping the laser beam aligned to the bottom corner for the entire length of the panel

A jig was designed to hold the material at the desired angle under the laser beam. Flat spots on the either side of the provides a guide for all the alignment requirements. The top of these flats are the same height as the surface of the cut making it easy to set the focal distance. Lines across flats are directly above the bottom corner of the material and the two flat surfaces ensure that the beam is in line as it travels from left to right. The jig is held in place using magnets to lock it down to the honeycomb.

Step 2: Making the Polyhedra

The faces of each shape should be laser cut in advance of setting up the mitre jig. This will often be regular shapes cut to the desired size but in the case of the cuboid the sides can be changed to make any sized box. The panels should form a basic net for the cuboid.

The actual program downloaded to the laser cutter should be a single line with a length slightly longer that the panels you intend to cut.

Place each panel into the jig, run the laser and create a mitred cut on one side. Manually rotate the panel until all the sides of the panel have been mitred, then it is simply a case of gluing the panels together with a little bit of wood glue.

We constructed 3 different angled jigs for testing purpose. The 45 degree cuboid jig, the 32.5 Octahedron jig and the 20.9 degree Icosahedron jig.

Currently each jig has the angle fixed and it only suitable for one thickness of material, there is a planned upgrade to make a jig that adjusts to any angle and any thickness.