Make the Platonic Solids with LightsKarl Sims
A Platonic solid is a regular convex polyhedron with a single type of regular polygon for its faces. Each vertex is also similar and joins an equal number of edges. These are named after the Greek philosopher Plato. Only five such shapes exist. 
Tetrahedron  Cube  Octahedron  Dodecahedron  Icosahedron  
4 triangles 4 vertices 6 edges 
6 squares 8 vertices 12 edges 
8 triangles 6 vertices 12 edges 
12 pentagons 20 vertices 30 edges 
20 triangles 12 vertices 30 edges 
These polyhedra are constructed using wooden poles for spokes that
connect each vertex to a small cube at the center, and lights are
strung between the spokes along each edge. "Fairy lights" have LEDs
that align nicely along their wire and are visible from all sides.
They are available in strings of various lengths, and seem to
consistently have a 4" distance between LEDs, so we'll use edge
lengths of even multiples of 4" to get an equal number of lights
per edge. The specific spoke lengths and number of lights required
will depend on the sizes you choose to make each shape below.
Materials:
Tools:
Drill holes in the cubes
The spokes of each Platonic solid can conveniently be attached to the corners and/or faces of a cube at its center.
Tetrahedron 4 spokes 
Cube 8 spokes 
Octahedron 6 spokes 
Dodecahedron 20 spokes 
Icosahedron 12 spokes 

Drill a 1/4" hole for each spoke into the 5 cubes using the patterns shown. If your dowel is too tight in a 1/4" hole try a 17/64" drill bit instead. Drill to a depth of about 1/4" from the center of the cube.
It will help to create a simple jig to support a cube while drilling a hole into its corner straight along its 3D diagonal. Clamp a board to your drill press and drill a ½" hole in it. Rest a corner of the cube in that hole and align the opposite top corner with a smaller drill bit so the cube's diagonal is precisely vertical. Then glue 3 small blocks to the board pushed against the lower faces of the cube to hold it steady at that angle.
The center cube for the dodecahedron also requires two holes drilled into
each face at a 20.9° angle, and for the icosahedron at a 31.7°
angle. Tilt your drill press platform or create two small ramp jigs
to hold the cube at these angles when drilled. Position each hole so
it is aimed towards the center of the cube, and turn the cube 180°
to drill the second hole on the same face. Align the pairs of holes
on each face in alternating directions as shown above.
For the dodecahedron  For the icosahedron  
Make the spokes
Saw a ¼" notch into one end of each wooden dowel to
support the strings of lights.
Select a size for each polyhedron in the tables below, and cut the dowels to the corresponding spoke length for each. Note that the spoke lengths may need to be adjusted slightly based on the actual depth of the holes in the cube, and the actual depth of the notches in the spokes. Insert a spoke and check the distance between the notch and the cube center, or to the opposite notch. Adjust the spoke lengths carefully so your light strings will not be too tight or too loose. Insert the spokes into the cubes, and paint if desired. The examples shown below were painted with flat black spray paint. 
Attach the lights
Wrap the lights between the ends of the spokes. Secure the wire to the first spoke with some hitches, and align the lights equally on each edge. If you put one light right next to the spoke, each vertex should end up with multiple lights as the examples show.
To reach all the edges of a polyhedron with a single string of lights, you'll need to wrap the lights twice along some of the edges. The LEDs on one of the double wires can be individually disabled by pinching them with pliers, so all the edges still have an equal number of active lights. It should be easy to find a path that uses the minimum number of doubled edges for the tetrahedron, cube, and octahedron, but here are suggested paths for wrapping the lights on the dodecahedron and icosahedron:
After snipping off the extra lights at the end of the string, you may need to reconnect two of the wire strands, if your wire is 3stranded. Use a piece of metal to test which two ends need connecting and solder them together.
Secure the end of the wire to the last spoke.
Notes on the values in these tables:  
Edge length:  assumes 4" between lights. 
Total lights:  Lights per edge × (Edges + Edges needing double lights) + 3 extra lights for sufficient wire at the ends. 
Total width:  predicts the smallest doorway the structure could fit through once assembled. 
Spoke length:  the centertovertex radius of the polyhedron. 
Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

The cubeoctahedron requires holes drilled diagonally into each of the 12 edges of the center cube. Its Total width / Edge length ratio is 1.41, and its Spoke length is exactly equal to its Edge length.
The rhombic dodecahedron requires holes in each face and corner of the center cube, like for the octahedron and cube combined. Its Total width / Edge length ratio is 1.73, and it has two different spoke lengths: 6 with a Spoke length / Edge length ratio of 1.15, and 8 spokes with a length exactly equal to the Edge length.
The stellated octahedron also requires holes in each face and corner of the center cube. Its Total width / Edge length ratio is 1.41, and it also has two different spoke lengths: 6 with a Spoke length / Edge length ratio of .707, and 8 spokes with a Spoke length / Edge length ratio of 1.22. This shape can be simplified by omitting the interior octahedral edges and using just 8 spokes to make two intersecting tetrahedra.
The truncated octahedron requires 4 holes in each face of the center cube at a 26.6° angle, like for the icosahedron but with holes near all 4 edges of each face in a diamond pattern. Its Total width / Edge length ratio is 2.45, and its Spoke length / Edge length ratio is 1.58.
Further reading:
Wooden Book by Daud Sutton: Platonic & Archimedean Solids
Wikipedia pages for:
Dual polyhedra,
Archimedean solids,
Catalan solids,
stellated polyhedra,
compound polyhedra,
and 4 dimensional polytopes.
Back to other work by Karl Sims