I always want to make something with the layer look of baltic birch plywood, so I made a series of platonic solids. The icosahedron is my favorite one and fairly hard to make, so here is how I made it and with this method you can make dodecahedron and octahedron as well. Hexahedron, aka cube is easy to make anyways, while tetrahedron is a different story because its dihedral angle is smaller than 90 deg, I made it in different way and will cover it in another instructable.
All tools you need are table saw, miter saw, and an angle gauge.
Step 1: Deconstruction and Calculation
Platonic solids are all made up by regular polygons, so all you need is to make the right amount of them and figure out the dihedral angle, which is 2 times of the bevel angle of the edge.
An icosahedron has 20 equilateral triangles, with dihedral angle of 138.189685°, means each triangle should have 3 edges with bevels of (138.189685°/2) ≈ 69.1°
It's also important to know the size of the final piece you want in order to calculate the diameters of the parts. I made multiple versions of different sizes and thicknesses of the edges. The smaller one in the picture is approx. 12" diameter with edges length of 7" and width of 3/4" (the problem of this one is the triangle hole is too small that I can put my hand inside the geometry to do any kine of finish or decoration.) The bigger one as you can see the size in the comparison, is made of 2 sheets of 3/4" plywood glued together. For more reference, see Regular icosahedron Dimensions .
Step 2: Wood Preparation and Bevel Cut
I'll use the bigger icosahedron as example for the next steps.
First glue 2 sheets of 3/4" plywood in to one 1.5" board and let cure - simple and regular step.
Bevel the table saw blade to 69.1° (or 20.9°, i don't know how exactly to describe it, please see the diagram.) It's better to make the angle rather smaller if you not confident about the accuracy of your angle gauge. i'll explain this later.
Cut the 1.5" sheet into bars - the cross section of the bars should be parallelogram. Theoretically the width of each cut doesn't matter, but i'd say it feels right to make it 1/2 - 1/4 to the thickness for a good balance between the weight and tensile strength.
In this icosahedron we need 20 triangles with are 60 edges. My board is about 36", so I decided to make the edge 12"ish so I only need to cut 20 bars, or 21, 22...
Step 3: Miter
Cut one end of the bars to 60º (which on most of the miter saws this is the max. and only one side can turn to (some smaller saws may only reach 45º so they don't work.).) Since parallelogram is centrosymmetric, you can start the cut from either end.
Turn the bar upside-down and the cut on the 1/3, you'll get a trapezium, turn and cut again for another trapezium, turn again cut the end piece off.
Make sure you have 60+ pieces.
Arrange the piece like a corrugated board and use miter saw to trim the end, make sure they are same length. I used mask tape and grouped them in 4s for every cut.
Step 4: Make Triangles
Glue 3 bars to one triangle. Find a big and flat work bench, use tape instead of clamps to hold the shape while the glue is curing. Forgot to take pictures of the triangles, but the pentagons are same idea:)
Step 5: Test Assembly
Just simply attach one triangle to another, if everything was going right, you would get a perfectly fit icosahedron without effort. If not, the highest possibility is inaccurate bevel as shown in the last picture. This is why I said it's better to cut bevel smaller if uncertain. Luckily wood is flexible enough to tolerate a little error.
Remove the mask tape and sand the surface.
Step 6: Assembly
Glue the triangles one to another and hold with tape like the last step.
Step 7: Finishing
Clean the excessive glue, use nature color wood filler fill gaps and sand again, seal with wood conditioner.
I also made some ones painted in solid color and/or attached LED inside.
Runner Up in the